Volume
LIPIcs, Volume 392
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)
Events
- APPROX 2026 August 19-21, 2026, Boston University, Boston, Massachusetts, USA
- RANDOM 2026 August 19-21, 2026, Boston University, Boston, Massachusetts, USA
Editors
Publication Details
- published at: 2026-09-09
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-449-9
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Complete Volume
LIPIcs, Volume 392, APPROX/RANDOM 2026, Complete Volume
Abstract
LIPIcs, Volume 392, APPROX/RANDOM 2026, Complete Volume
Cite as
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 1-1630, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@Proceedings{singh_et_al:LIPIcs.APPROX/RANDOM.2026,
title = {{LIPIcs, Volume 392, APPROX/RANDOM 2026, Complete Volume}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {1--1630},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026},
URN = {urn:nbn:de:0030-drops-278458},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026},
annote = {Keywords: LIPIcs, Volume 392, APPROX/RANDOM 2026, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 0:i-0:xxvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{singh_et_al:LIPIcs.APPROX/RANDOM.2026.0,
author = {Singh, Mohit and Gur, Tom},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {0:i--0:xxvi},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.0},
URN = {urn:nbn:de:0030-drops-278443},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
APPROX
Improved Approximation Algorithm for Maximum Balanced Biclique
Abstract
We study the Maximum Balanced Biclique (MBB) problem: Given a bipartite graph G with n vertices on each side, find a balanced biclique in G with maximum size. We give a polynomial-time (n/Ω̃((log n)³))-approximation algorithm for the problem, which improves upon an (n/Ω((log n)²))-approximation by [Chalermsook et al., 2020] and answers their open question. Furthermore, our approximation ratio matches that of the maximum clique problem by [Feige, 2004] up to an O(log log n) factor.
Cite as
Pasin Manurangsi. Improved Approximation Algorithm for Maximum Balanced Biclique. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 1:1-1:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{manurangsi:LIPIcs.APPROX/RANDOM.2026.1,
author = {Manurangsi, Pasin},
title = {{Improved Approximation Algorithm for Maximum Balanced Biclique}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {1:1--1:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.1},
URN = {urn:nbn:de:0030-drops-277184},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.1},
annote = {Keywords: Maximum Balanced Biclique, Approximation Algorithm, Semidefinite Programming}
}
Document
APPROX
Stochastic Caching via Subset Entropy
Abstract
A classic approach to beyond worst-case algorithm design is to impose stochastic assumptions on the input. However, a limiting feature of such stochastic analyses is that, by the min-max principle, performance on worst-case distributions mirrors that of randomized algorithms on worst-case inputs. In other words, the same shortcoming of worst-case analysis - its inability to distinguish between "easy" and "hard" instances - reappears as an inability to distinguish between "easy" and "hard" distributions. This raises a natural question: Can we characterize "easy" input distributions with useful beyond worst-case bounds?
A canonical example is the stochastic caching problem (Aho, Denning, and Ullman, 1971). When the page requests are drawn i.i.d. from the uniform distribution, the best achievable competitive ratio is O(log k), matching the performance of the best randomized algorithm on worst-case instances (Fiat, Karp, Luby, McGeoch, Sleator, and Young, 1991). However, when the input distribution has less entropy, intuition suggests that we should be able to do better by exploiting the information provided by the distribution. In this paper, we formalize this intuition by defining a new information-theoretic parameter of probability distributions called subset entropy. We then use this parameter to give a fine-grained characterization of the competitive ratio of stochastic caching, including a new analysis for the well-known LRU algorithm on stochastic inputs.
While our technical results are for the caching problem, we believe the broader principle - parameterizing algorithmic performance by an entropy measure of the input - is of independent interest and might apply to other online and stochastic optimization problems. Indeed, for many other fundamental problems such as (comparison-based) sorting, online matching, online load balancing, etc., the hardest stochastic instances involve high-entropy distributions. We hope our work is a step toward a broader theory of fine-grained algorithmic performance for this class of problems.
Cite as
Ravi Kumar, Roie Levin, Joseph (Seffi) Naor, and Debmalya Panigrahi. Stochastic Caching via Subset Entropy. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 2:1-2:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kumar_et_al:LIPIcs.APPROX/RANDOM.2026.2,
author = {Kumar, Ravi and Levin, Roie and Naor, Joseph (Seffi) and Panigrahi, Debmalya},
title = {{Stochastic Caching via Subset Entropy}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {2:1--2:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.2},
URN = {urn:nbn:de:0030-drops-277194},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.2},
annote = {Keywords: Online Algorithms, Beyond Worst-Case Analysis, Caching, Paging, Entropy}
}
Document
APPROX
Socially Fair Clustering: Parameterized Approximation and Local Search
Abstract
We study the Socially Fair Clustering problem introduced by Abbasi, Bhaskara, and Venkatasubramanian [Abbasi et al., 2021] and by Ghadiri, Samadi, and Vempala [Ghadiri et al., 2021], along with its extension, the (p,q)-Socially Fair Clustering problem. This problem generalizes k-median and k-means to settings where data points are partitioned into 𝓁 groups, and the goal is to find a fair clustering that is simultaneously good for all groups. We present several algorithms for this problem.
1) For 𝓁_p-Socially Fair Clustering, we give the first constant-factor FPT-approximation parameterized by the number of groups 𝓁, resolving the open question raised by Ghadiri, Singh, and Vempala [Ghadiri et al., 2022]. Our main ingredient is a new algorithm for closing additional centers in parameterized time inspired by local search.
2) We then turn to the more general (p,q)-Socially Fair Clustering problem. The known algorithm for this problem, proposed by Chlamtáč, Makarychev, and Vakilian [Chlamtáč et al., 2022], achieves a very good approximation but is complex, slow and difficult to implement. We analyze the performance of a simple local search algorithm and show that it provides an O(q)-approximation in the worst case.
3) Finally, we design approximation algorithms for the facility location variant of the problem, where the number of facilities (centers) is not fixed in advance, and opening each facility incurs an opening cost. Unlike in previous work, we do not assume these opening costs are the same for all groups.
Cite as
Aditya Anand, Yury Makarychev, and Liren Shan. Socially Fair Clustering: Parameterized Approximation and Local Search. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{anand_et_al:LIPIcs.APPROX/RANDOM.2026.3,
author = {Anand, Aditya and Makarychev, Yury and Shan, Liren},
title = {{Socially Fair Clustering: Parameterized Approximation and Local Search}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {3:1--3:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.3},
URN = {urn:nbn:de:0030-drops-277207},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.3},
annote = {Keywords: Socially fair clustering, Approximation algorithms, Fixed-parameter tractability, Local search, Facility location, k-median and k-means}
}
Document
APPROX
Near-Optimal Streaming Approximation for Max-DICUT in Sublinear Space Using Two Passes
Abstract
The Max-DICUT problem has emerged as a canonical problem for understanding the approximability of constraint satisfaction problems in the streaming model. A seminal result of Kapralov and Krachun [STOC 2019] shows that it is impossible to beat 1/2-approximation for Max-DICUT in sublinear space in the single-pass streaming setting, even on bounded-degree graphs. In a recent work, Saxena, Singer, Sudan, and Velusamy [SODA 2025] prove that the above lower bound is tight by giving a single-pass algorithm for bounded-degree graphs that achieves (1/2-ε)-approximation in sublinear space, for every constant ε > 0. For arbitrary graphs of unbounded degree, they give an O(1/ε)-pass O(log n) space algorithm. Their work left open the question of obtaining 1/2-approximation for arbitrary graphs in the single-pass setting in sublinear space. We make progress towards this question and give a two-pass algorithm that achieves (1/2-ε)-approximation in sublinear space, for every constant ε > 0.
Cite as
Santhoshini Velusamy. Near-Optimal Streaming Approximation for Max-DICUT in Sublinear Space Using Two Passes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 4:1-4:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{velusamy:LIPIcs.APPROX/RANDOM.2026.4,
author = {Velusamy, Santhoshini},
title = {{Near-Optimal Streaming Approximation for Max-DICUT in Sublinear Space Using Two Passes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {4:1--4:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.4},
URN = {urn:nbn:de:0030-drops-277211},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.4},
annote = {Keywords: Streaming algorithms, Approximation algorithms, Graph algorithms}
}
Document
APPROX
Incremental Dominating Set
Abstract
Dominating Set is a fundamental problem in graph theory: given a graph, find a minimum-weight subset of vertices such that every vertex is either selected or adjacent to a selected vertex. In online settings where vertices arrive sequentially, comparing algorithms against an offline optimum with full knowledge of the input leads to extremely strong lower bounds, where even a simple star graph shows that any online algorithm must have competitive ratio Ω(Δ), with Δ the largest degree of any vertex in the graph, matching the trivial strategy of selecting all vertices.
We study the incremental dominating set problem, where the optimal algorithm is constrained to the same choices available to online algorithms. This introduces a benchmark that enables a meaningful comparison between algorithms.
We present the first results for vertex-weighted graphs and randomized algorithms in this model. For incremental dominating set, we give an O(Δ)-competitive deterministic algorithm and an O(log²Δ)-competitive randomized algorithm. We extend these results to the Connected Dominating Set problem using a linear-programming formulation that captures connectivity through local constraints. When the neighborhood of each arriving vertex is known in advance, deterministic algorithms achieve similar polylogarithmic competitive ratios as their randomized counterparts. Finally, we establish matching lower bounds, showing that our results are optimal up to constant factors.
Cite as
Ilan Doron Arad, Jonathan Gal, and Joseph (Seffi) Naor. Incremental Dominating Set. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{arad_et_al:LIPIcs.APPROX/RANDOM.2026.5,
author = {Arad, Ilan Doron and Gal, Jonathan and Naor, Joseph (Seffi)},
title = {{Incremental Dominating Set}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {5:1--5:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.5},
URN = {urn:nbn:de:0030-drops-277229},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.5},
annote = {Keywords: Algorithms, Online Algorithms, Dominating Set}
}
Document
APPROX
Bichromatic Geometric Spanners
Abstract
For an edge-weighted graph G = (V,E) and a stretch parameter t ≥ 1, a t-spanner is a subgraph H ⊆ G such that the shortest path distances in G and H satisfy δ_H(u,v) ≤ t δ_G(u,v) for all u,v ∈ V. In metric spanners, V is a finite metric space, and G is the complete graph with edge weights corresponding to the distances between the endpoints. When G is the complete graph on n points in the plane, O(n)-size t-spanners are possible for any t > 1: For every ε > 0, there is an (1+ε)-spanner with O(n/ε) edges (the stretch can be arbitrarily close to 1).
When G = K(R,B) is the complete bipartite graph on n bichromatic points in the plane, in general, no spanner construction can achieve stretch t < 3 with o(n²) edges. Bose et al. (SICOMP 2009) constructed a (3+ε)-spanner with O(nlog n) edges for any constant ε > 0. Our main result is a new construction for a (3+ε)-spanner with O(√{1/ε} ⋅ n) edges. Eliminating the O(log n) factor resolves a problem left open for more than 17 years, and raises a new research problem about optimizing the dependence on ε. We also study spanners for G = K(R,B) on n bichromatic points on the real line: In this case, we show that the MST of K(R,B) is a 7-spanner, and we construct a 3-spanner with at most 2n-3 edges.
Cite as
Theodore Fung and Csaba D. Tóth. Bichromatic Geometric Spanners. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fung_et_al:LIPIcs.APPROX/RANDOM.2026.6,
author = {Fung, Theodore and T\'{o}th, Csaba D.},
title = {{Bichromatic Geometric Spanners}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {6:1--6:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.6},
URN = {urn:nbn:de:0030-drops-277233},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.6},
annote = {Keywords: Euclidean spanner, bichromatic points, computational geometry}
}
Document
APPROX
Improved Approximation Algorithms for Bounded-Length Path Augmentation
Abstract
The Forest Augmentation Problem (FAP) asks for a minimum set of additional edges (links) that makes a given forest 2-edge-connected while spanning all vertices. A key special case is the Path Augmentation Problem (PAP), where the input forest consists of vertex-disjoint paths. Grandoni, Jabal Ameli, and Traub [STOC'22] recently broke the long-standing 2-approximation barrier for FAP, achieving a 1.9973-approximation. A crucial component of this result was their 1.9913-approximation for PAP; the first better-than-2 approximation for PAP.
In this work, we present a 1.9412-approximation for bounded-length PAP, in which each path of the input forest has constant length p_max. The running-time of our algorithm is O(n^p_max). One of our key innovations is a (11/6 + ε)-approximation preserving reduction to so-called structured instances, which simplifies the problem and enables our improved approximation. Additionally, we introduce a new relaxation inspired by 2-edge covers and analyze it via a corresponding packing problem, where the relationship between the two problems is similar to the relationship between 2-edge covers and 2-matchings. Using a factor-revealing LP, we bound the cost of our solution to the packing problem w.r.t. the relaxation and derive a strong initial solution. We then transform this solution into a feasible PAP solution, combining techniques from FAP and related connectivity augmentation problems, along with new insights.
Cite as
Felix Hommelsheim. Improved Approximation Algorithms for Bounded-Length Path Augmentation. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 7:1-7:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hommelsheim:LIPIcs.APPROX/RANDOM.2026.7,
author = {Hommelsheim, Felix},
title = {{Improved Approximation Algorithms for Bounded-Length Path Augmentation}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {7:1--7:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.7},
URN = {urn:nbn:de:0030-drops-277247},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.7},
annote = {Keywords: Approximation algorithms, path augmentation, connectivity augmentation, survivable network design}
}
Document
APPROX
Capacitated Partition Vertex Cover and Partition Edge Cover
Abstract
We study generalizations of the classical Vertex Cover and Edge Cover problems that incorporate group-wise coverage.
Our first focus is the Capacitated Partition Vertex Cover (C-PVC) problem in hypergraphs. In C-PVC, we are given a hypergraph with capacities on its vertices and a partition of the hyperedge set into ω distinct groups. The objective is to select a minimum size subset of vertices that satisfies two main conditions: (1) in each group, the total number of covered hyperedges meets a specified threshold, and (2) the number of hyperedges assigned to any vertex respects its capacity constraint. A covered hyperedge is required to be assigned to a selected vertex that belongs to the hyperedge. This formulation generalizes classical Vertex Cover, Partial Vertex Cover, and Partition Vertex Cover.
We investigate two primary variants: soft capacitated (multiple copies of a vertex are allowed) and hard capacitated (each vertex can be chosen at most once). Let f denote the rank of the hypergraph (i.e., the maximum number of vertices contained in any single hyperedge). Our main contributions are: (i) an (f+1)-approximation algorithm for the weighted soft-capacitated C-PVC problem, which runs in n^O(ω) time, and (ii) an (f+ε)-approximation algorithm for the unweighted hard-capacitated C-PVC problem, which runs in n^O(ω/ε) time.
We also study a natural generalization of the edge cover problem, the Weighted Partition Edge Cover (W-PEC) problem, where each edge has an associated weight, and the vertex set is partitioned into groups. For each group, the goal is to cover at least a specified number of vertices using incident edges, while minimizing the total weight of the selected edges. We present the first exact polynomial-time algorithm for the weighted case, improving runtime from O(ω n³) to O(mn + n²log n) and simplifying the algorithmic structure over prior unweighted approaches (that rely on the tropical matching problem).
Cite as
Rajni Dabas, Samir Khuller, and Emilie Rivkin. Capacitated Partition Vertex Cover and Partition Edge Cover. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 8:1-8:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dabas_et_al:LIPIcs.APPROX/RANDOM.2026.8,
author = {Dabas, Rajni and Khuller, Samir and Rivkin, Emilie},
title = {{Capacitated Partition Vertex Cover and Partition Edge Cover}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {8:1--8:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.8},
URN = {urn:nbn:de:0030-drops-277257},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.8},
annote = {Keywords: Approximation algorithms, capacitated vertex cover, iterative rounding}
}
Document
APPROX
A Bad Example for Jain’s Iterative Rounding Theorem for the Cover Small Cuts Problem
Abstract
The Cover Small Cuts problem is a key problem in Network Design. In the Cover Small Cuts problem, we are given a capacitated (undirected) graph G = (V,E,u) and a threshold value λ, as well as a set of links L with end-nodes in V and a non-negative cost for each link 𝓁 ∈ L; the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than λ is covered by a link.
Jain’s iterative rounding theorem is a well-known result in the area of approximation algorithms and, more broadly, in combinatorial optimization. The theorem asserts that LP relaxations of several problems in network design and combinatorial optimization have the following key property: for every basic feasible solution x there exists a variable x_e that has value at least a constant (e.g., x_e ≥ 1/2).
We construct an example showing that this property fails to hold for the standard LP relaxation of the Cover Small Cuts problem. This indicates that the polyhedron of feasible solutions to the LP (for Cover Small Cuts) differs in an essential way from the polyhedrons associated with several problems in combinatorial optimization. Moreover, our example shows that a direct application of Jain’s iterative rounding algorithm does not give an O(1) approximation algorithm for Cover Small Cuts.
We mention that Bansal et al. [Ishan Bansal et al., 2024] showed that the WGMV primal-dual algorithm, due to Williamson et al. [David P. Williamson et al., 1995], applied to the same standard LP relaxation achieves approximation ratio 16 for the Cover Small Cuts problem. That is, the WGMV primal-dual algorithm, applied to the same LP relaxation, finds an integer solution of cost ≤ 16 times the optimal value of the LP.
Cite as
Miles Simmons, Ishan Bansal, and Joe Cheriyan. A Bad Example for Jain’s Iterative Rounding Theorem for the Cover Small Cuts Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{simmons_et_al:LIPIcs.APPROX/RANDOM.2026.9,
author = {Simmons, Miles and Bansal, Ishan and Cheriyan, Joe},
title = {{A Bad Example for Jain’s Iterative Rounding Theorem for the Cover Small Cuts Problem}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {9:1--9:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.9},
URN = {urn:nbn:de:0030-drops-277263},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.9},
annote = {Keywords: approximation algorithms, capacitated network design, covering small cuts, edge-connectivity of graphs, f-connectivity problem, iterative rounding method, primal-dual method}
}
Document
APPROX
Hardness of the Binary Covering Radius Problem in Large 𝓁_p Norms
Abstract
We study the hardness of the γ-approximate decisional Covering Radius Problem on lattices in the 𝓁_p norm (γ-GapCRP_p). Specifically, we prove that there is an explicit function γ(p), with γ(p) > 1 for p > p₀ ≈ 35.31 and lim_{p → ∞} γ(p) = 9/8, such that for any constant ε > 0, (γ(p)-ε)-GapCRP_p is NP-hard. This shows the first hardness of GapCRP_p for explicit p < ∞. Work of Haviv and Regev (CCC, 2006 and CJTCS, 2012) previously showed Π₂-hardness of approximation for GapCRP_p for all sufficiently large (but non-explicit) finite p and for p = ∞.
In fact, our hardness results hold for a variant of GapCRP called the Binary Covering Radius Problem (BinGapCRP). The Binary Covering Radius Problem trivially reduces to both GapCRP and the decisional Linear Discrepancy Problem (LinDisc) in any norm in an approximation-preserving way. We also show Π₂-hardness of (9/8 - ε)-BinGapCRP in the 𝓁_∞ norm for any constant ε > 0.
Our work extends and heavily uses the work of Manurangsi (IPL, 2021), which showed Π₂-hardness of (9/8 - ε)-LinDisc in the 𝓁_∞ norm.
Cite as
Huck Bennett and Peter Ly. Hardness of the Binary Covering Radius Problem in Large 𝓁_p Norms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bennett_et_al:LIPIcs.APPROX/RANDOM.2026.10,
author = {Bennett, Huck and Ly, Peter},
title = {{Hardness of the Binary Covering Radius Problem in Large 𝓁\underlinep Norms}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {10:1--10:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.10},
URN = {urn:nbn:de:0030-drops-277274},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.10},
annote = {Keywords: Covering radius problem, linear discrepancy, hardness of approximation}
}
Document
APPROX
Improved Algorithms and Lower Bounds for Parametrized Metrical Service Systems
Abstract
We consider the parametrized setting of the classical metrical service system (MSS) problem first studied by Bubeck and Rabani (APPROX/RANDOM 2020). In this setting, the adversary is restricted to a set of m distinct request types, known to the algorithm in advance. The goal is to obtain competitive ratio bounds in terms of m. In this work, we make significant progress in understanding the landscape of parametrized MSS and resolve several open problems from Bubeck and Rabani.
Our first main result is a tight bound for parametrized MSS on weighted stars. Previously, Bubeck and Rabani gave a randomized lower bound of Ω(m) and deterministic upper bound of O(2^m). We show that, surprisingly, a deterministic O(m)-competitive algorithm exists, matching the randomized lower bound. Our key insight is an interval covering formulation of MSS on weighted stars which enables an application of the primal-dual method.
Our second main contribution is an improved lower bound construction for parametrized MSS on hierarchically separated trees (HSTs). Bubeck and Rabani’s construction gave a ω(1) lower bound when m ≥ 6. Our improved lower bounds are tight for 2-level HSTs and also rule out O(1)-competitive algorithms on HSTs when the parameter m ≥ 4. We also complement these results by giving a deterministic O(1)-competitive algorithm on general metrics when m = 2 while showing that it is impossible when m ≥ 3.
Cite as
Junhao Gan, Xiao Sun, and Seeun William Umboh. Improved Algorithms and Lower Bounds for Parametrized Metrical Service Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gan_et_al:LIPIcs.APPROX/RANDOM.2026.11,
author = {Gan, Junhao and Sun, Xiao and Umboh, Seeun William},
title = {{Improved Algorithms and Lower Bounds for Parametrized Metrical Service Systems}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {11:1--11:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.11},
URN = {urn:nbn:de:0030-drops-277289},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.11},
annote = {Keywords: online algorithms, competitive analysis, metrical service systems, metrical task systems}
}
Document
APPROX
Uncrossed Multiflows and Applications to Disjoint Paths
Abstract
A multiflow in a planar graph is uncrossed if the curves identified by its support paths do not cross in the plane. Recently, uncrossed flows have played a role in approximation algorithms for maximum disjoint paths in "fully-planar" instances, where the combined supply-plus-demand graph is planar. They are also used in algorithms to find low-congestion unsplittable flows for both fully-planar and single-source instances. For these two instance classes, any fractional multiflow can be converted into one that is uncrossed, which these algorithms then exploit to obtain their results.
We investigate the utility of uncrossed flow more generally and ask three key questions. First, are there other interesting planar multiflow instances that admit uncrossed flows (beyond fully-planar and single-source)? We answer affirmatively, demonstrating a new family of "pairwise-planar" instances whose fractional flows can be uncrossed. This family subsumes fully-planar but includes substantially more, such as (2-connected) fully-compliant series-parallel instances and some instances that have large clique demand graphs. Second, can we always round a fractional uncrossed flow to a "good" integral flow? We again answer positively. For maximization problems, we show any fractional uncrossed flow can be rounded to an integral flow with a constant fraction of its value. For congestion problems (where we must fully route all given demands), we give a rounding procedure that yields an integral multiflow with edge congestion 2. Consequently, we obtain constant-factor approximation algorithms for maximum disjoint paths and minimum congestion integer multiflow for pairwise-planar instances, and show such instances have a constant integral flow-multicut gap. Finally we ask, given an arbitrary planar instance, can we determine if there exists a congestion-1 uncrossed fractional flow (congestion setting) or find the maximum value uncrossed fractional flow (maximization setting)? For congestion, we show this problem is NP-hard, but finding uncrossed edge-disjoint paths is polytime solvable if the demands span a bounded number of faces. For maximization, we present a strong (almost-polynomial) inapproximability result.
Cite as
Chandra Chekuri, Guyslain Naves, Joseph Poremba, and F. Bruce Shepherd. Uncrossed Multiflows and Applications to Disjoint Paths. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 12:1-12:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chekuri_et_al:LIPIcs.APPROX/RANDOM.2026.12,
author = {Chekuri, Chandra and Naves, Guyslain and Poremba, Joseph and Shepherd, F. Bruce},
title = {{Uncrossed Multiflows and Applications to Disjoint Paths}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {12:1--12:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.12},
URN = {urn:nbn:de:0030-drops-277298},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.12},
annote = {Keywords: Network Flows, Disjoint Paths, Planar Graphs, Crossing, Flow-Multicut Gap, Approximation Algorithms, Combinatorial Optimization}
}
Document
APPROX
Approximating (Weighted) Chromatic Correlation Clustering via Cluster LP
Abstract
Correlation Clustering is a fundamental clustering problem that is generalized to Chromatic Correlation Clustering to incorporate categorical data. Both problems have been intensively studied, and recently, substantial improvements were obtained in the approximation algorithms for Correlation Clustering. At the heart of this success lies a new linear program (LP) formulation called the cluster LP; a natural question was whether this LP can be extended to Chromatic Correlation Clustering to enable similar success.
We answer this question in the affirmative by presenting a (2+ε)-approximation algorithm for the problem using a chromatic cluster LP. We then consider Weighted Chromatic Correlation Clustering, in which edges have fractional weights satisfying the probability constraints, to show that our algorithm extends to this weighted version to yield the same approximation guarantee.
Cite as
Fateme Abbasi, Hyung-Chan An, Jarosław Byrka, Changyeol Lee, and Yongho Shin. Approximating (Weighted) Chromatic Correlation Clustering via Cluster LP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 13:1-13:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{abbasi_et_al:LIPIcs.APPROX/RANDOM.2026.13,
author = {Abbasi, Fateme and An, Hyung-Chan and Byrka, Jaros{\l}aw and Lee, Changyeol and Shin, Yongho},
title = {{Approximating (Weighted) Chromatic Correlation Clustering via Cluster LP}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {13:1--13:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.13},
URN = {urn:nbn:de:0030-drops-277301},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.13},
annote = {Keywords: Chromatic correlation clustering, Chromatic cluster LP, Preclustering}
}
Document
APPROX
Hitting Axis-Parallel Segments with Weighted Points
Abstract
We study a geometric hitting-set problem in which the input consists of a set P of weighted points and a family 𝒮 = ℋ∪𝒱 of axis-parallel segments in the plane. The goal is to select a minimum-weight subset of P that hits every segment in 𝒮. Even restricted geometric hitting-set problems are known to be computationally hard, and for axis-parallel segments the standard decomposition into horizontal and vertical sub-instances yields only a simple factor-2 approximation.
We present an LP-rounding algorithm that breaks the factor-2 barrier. For the weighted problem, we obtain a randomized (1+2/e)-approximation by combining systematic rounding on horizontal lines with an exact repair step on residual vertical sub-instances. In the unweighted case, a sharper analysis gives a (1+1/(e-1))-approximation. Finally, we consider the case where one of the sub-instances consists of lines instead of line segments, a problem considered by Fekete et al. (Geometric Hitting Set for Segments of Few Orientations, Theor. Comp. Sys., 62 (2) 2018),. In this case, we improve their result to obtain an approximation factor of 1+1/e and show that the problem is APX-hard. We also present algorithms for the generalization to d orientations, as well as PTASes for bounded-complexity subclasses of the unweighted Hitting Set problem.
Cite as
Rajiv Raman, Siddhartha Sarkar, and Jatin Yadav. Hitting Axis-Parallel Segments with Weighted Points. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 14:1-14:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{raman_et_al:LIPIcs.APPROX/RANDOM.2026.14,
author = {Raman, Rajiv and Sarkar, Siddhartha and Yadav, Jatin},
title = {{Hitting Axis-Parallel Segments with Weighted Points}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {14:1--14:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.14},
URN = {urn:nbn:de:0030-drops-277313},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.14},
annote = {Keywords: Geometric Hitting Set, Approximation Algorithms, Computational Geometry, LP Rounding, Axis-Parallel Segments, PTAS, APX-hardness}
}
Document
APPROX
Length-Constrained Network Design in Planar Digraphs
Abstract
We study length-constrained generalizations of Directed Steiner Tree (DST) and Directed Steiner Forest (DSF) in planar digraphs. In both problems, the input is a directed graph with edge costs. DST asks for a min-cost subgraph connecting a root to a given set of terminals, and DSF asks for a min-cost subgraph connecting each of a given set of source-sink terminal pairs. In the length-constrained setting, each edge has both a cost and a length, and the input includes a length bound h; the goal is to find a min-cost subgraph connecting each terminal pair via a path of length at most h. Our work is motivated by a recent line of results showing that several network design problems that are traditionally hard in directed graphs admit polylogarithmic approximation ratios in planar digraphs. We give polylogarithmic bicriteria approximation algorithms for length-constrained analogues of DST and DSF in planar digraphs. Our approximation ratios match the best known for DST and DSF in planar digraphs, with an O(log k) violation of the length constraint, where k denotes the number of terminals (or terminal pairs). As corollaries, we obtain polylogarithmic approximations for buy-at-bulk DST and DSF in planar digraphs.
Cite as
Chandra Chekuri and Rhea Jain. Length-Constrained Network Design in Planar Digraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 15:1-15:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chekuri_et_al:LIPIcs.APPROX/RANDOM.2026.15,
author = {Chekuri, Chandra and Jain, Rhea},
title = {{Length-Constrained Network Design in Planar Digraphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {15:1--15:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.15},
URN = {urn:nbn:de:0030-drops-277323},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.15},
annote = {Keywords: Approximation algorithms, network design, directed graphs, length constraints}
}
Document
APPROX
Directed Low Diameter Decomposition for Structured Digraphs
Abstract
Low diameter decompositions, or LDDs for short, are a fundamental primitive in the design of efficient graph algorithms. Roughly speaking, an LDD is a distribution over partitions of the vertices into bounded-diameter clusters such that nearby vertices are likely to be clustered together. Recently, there has been growing interest in lifting the notion of LDDs into directed graphs. In particular, there are two natural directed analogues. The first is a directed LDD, where after removing a random subset of edges, every strongly connected component has a small diameter. The second is a quasipartition, which imposes the stronger requirement that whenever one vertex can still reach another after the edge removal, the two vertices must be close in the original directed metric. Every quasipartition yields an LDD, but the converse does not necessarily hold.
In this work, we initiate the systematic study of LDDs in structured directed graphs. As our first main result, we show that any directed graph with pathwidth pw admits an (O(pw), Δ)-LDD. This improves upon the previous best-known (2^O(pw²), Δ)-LDD construction, which was implicitly derived from the quasipartition result of Salmasi, Sidiropoulos, and Sridhar [SODA'19].
As our second result, we show that the integrality gap of the Directed Non-Bipartite Sparsest-Cut LP relaxation on an n-vertex graph with treewidth tw is O(tw log n). This improves upon the O(tw log²n) bound of Mémoli, Sidiropoulos, and Sridhar [ICALP'16, Algorithmica'18]. We obtain this result through the refined analysis of the quasipartition construction of Mémoli et al. for bounded treewidth graphs.
Cite as
Shinwoo An and Arnold Filtser. Directed Low Diameter Decomposition for Structured Digraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 16:1-16:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{an_et_al:LIPIcs.APPROX/RANDOM.2026.16,
author = {An, Shinwoo and Filtser, Arnold},
title = {{Directed Low Diameter Decomposition for Structured Digraphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {16:1--16:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.16},
URN = {urn:nbn:de:0030-drops-277334},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.16},
annote = {Keywords: Low diameter decomposition, quasipartition, metric embedding, pathwidth}
}
Document
APPROX
Bi-Lipschitz Extensions and Outlier Embeddings into Trees
Abstract
We develop low distortion embeddings with outliers from arbitrary metrics into hierarchically separated trees (HSTs). In particular, we develop an efficient algorithm that for any ε > 0, given an input metric (X,d), and a probabilistic embedding of all but k points from X into HSTs with distortion c, samples from a probabilistic embedding of all but O((k/ε)log k) points into HSTs that achieves distortion at most (32+ε)c.
Our results are based on two key technical components. First, we extend an algorithm of Munagala et al. [Munagala et al., 2023] for minimizing the distortion of embeddings without outliers into HSTs to the setting with outliers. We combine this with new results on bi-Lipschitz extensions into trees and 𝓁₁ space. In particular, we show that any probabilistic embedding into HSTs can be extended to k additional points with only a factor O(log k) of additional distortion. This bi-Lipschitz extension result utilizes a new probabilistic partitioning scheme that we call onion partitioning.
Cite as
Shuchi Chawla, Arnold Filtser, Kristin Sheridan, and Yonatan Trachtenberg. Bi-Lipschitz Extensions and Outlier Embeddings into Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 17:1-17:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chawla_et_al:LIPIcs.APPROX/RANDOM.2026.17,
author = {Chawla, Shuchi and Filtser, Arnold and Sheridan, Kristin and Trachtenberg, Yonatan},
title = {{Bi-Lipschitz Extensions and Outlier Embeddings into Trees}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {17:1--17:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.17},
URN = {urn:nbn:de:0030-drops-277346},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.17},
annote = {Keywords: metric embeddings, hierarchically separated trees, outliers}
}
Document
APPROX
Tight Approximation Results for Matroid Optimization with a Linear Constraint
Abstract
We study the following class of matroid optimization problems with a linear constraint (𝒫-MOL). Given a matroid ℳ = (E,ℐ), two nonnegative weight functions v,w:E → ℝ_{≥ 0}, and a threshold L ∈ ℝ_{≥ 0}, find opt v(S) where S is either an independent set or a base of ℳ satisfying a budget-type constraint: w(S) ≤ L or w(S) ≥ L, and opt ∈ {min,max}. 𝒫-MOL provides a unified representation for a broad family of NP-hard optimization problems, including budgeted matroid independent set, constrained minimum-basis, and knapsack-cover variants with a matroid constraint. Also, it naturally extends to multiple matroid constraints. In particular, we consider the matroid intersection cover (MIC) problem, where feasibility is defined by the common independent sets of two matroids and one seeks minimum v(S) subject to w(S) ≥ L.
Our main result is a unified efficient polynomial-time approximation scheme (EPTAS) for all nontrivial 𝒫-MOL variants, obtained by generalizing a technique of Hassin and Levin (SIAM J. Comput., 2004) for solving the constrained minimum spanning tree problem. Specifically, for any fixed ε > 0, we present an algorithm running in time |E|^O(1) ⋅ (1/ε²)^O(1/ε) that outputs a feasible solution S whose value is at most (1+ε)OPT for minimization variants and at least (1-ε)OPT for maximization variants. This resolves the complexity status of all members of 𝒫-MOL, as none of these problems admits a fully polynomial-time approximation scheme (Doron-Arad, Kulik and Shachnai, ICALP'24). Finally, we separate the 𝒫-MOL family from its extension to matroid intersection. We show that an EPTAS is unlikely to exist for the matroid intersection variant of 𝒫-MOL under a covering constraint, whereas an EPTAS is known to exist under a budget constraint. This highlights a qualitative difference between these two types of linear constraints that does not arise in the single-matroid setting.
Cite as
Ilan Doron-Arad, Hadas Shachnai, and Gilad Shmerler. Tight Approximation Results for Matroid Optimization with a Linear Constraint. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 18:1-18:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{doronarad_et_al:LIPIcs.APPROX/RANDOM.2026.18,
author = {Doron-Arad, Ilan and Shachnai, Hadas and Shmerler, Gilad},
title = {{Tight Approximation Results for Matroid Optimization with a Linear Constraint}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {18:1--18:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.18},
URN = {urn:nbn:de:0030-drops-277354},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.18},
annote = {Keywords: Matroids, Knapsack, Linear Constraints, EPTAS, Budgeted Optimization, MOL Problems}
}
Document
APPROX
Strong Inapproximability for a Promise Rank Problem
Abstract
Given a linear subspace of n × n matrices over 𝔽_{2^r} that is promised to contain a matrix of rank 1, we prove that it is hard to find a matrix of rank n^o(1/log log n), assuming NP doesn't have sub-exponential algorithms. In addition to being a basic problem, the hardness of this problem, even for the exact version, drove recent PCP-free inapproximability results for minimum distance and shortest vector problems concerning codes and lattices.
The proof combines the concept of superposition soundness introduced by Khot and Saket with moment matrices. To produce a rank-gap of 1 vs. k, the reduction runs in time n^O(log k). We also give another moment-matrix-based construction which runs in time n^O(k) but works for any finite field F_q.
Cite as
Venkatesan Guruswami, Xuandi Ren, and Shaoxuan Tang. Strong Inapproximability for a Promise Rank Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 19:1-19:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{guruswami_et_al:LIPIcs.APPROX/RANDOM.2026.19,
author = {Guruswami, Venkatesan and Ren, Xuandi and Tang, Shaoxuan},
title = {{Strong Inapproximability for a Promise Rank Problem}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {19:1--19:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.19},
URN = {urn:nbn:de:0030-drops-277360},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.19},
annote = {Keywords: rank minimization, inapproximability, promise problems, PCP, moment matrices}
}
Document
APPROX
On the Approximability of Parameterized Minimum Monotone Satisfying Assignment
Abstract
The parameterized Minimum Monotone Satisfying Assignment (k-MMSA) problem asks whether a monotone Boolean circuit admits a satisfying assignment of Hamming weight at most k. The MMSA hierarchy is defined by allowing a bounded number of alternations between AND and OR gates in the circuit. While the polynomial-time approximability of the MMSA hierarchy has been studied extensively, much less is known in the parameterized setting. In particular, k-MMSA₂ is the well-known k-SetCover problem, whose parameterized inapproximability lies in the polylog(n) regime. In contrast, k-MMSA₄ captures k-MinLabel, for which known lower bounds give poly(n) inapproximability. Sandwiched by k-MMSA₂ and k-MMSA₄, the inapproximability of k-MMSA₃ remained comparatively unexplored.
In this paper, we give an FPT-time O(2^k log n)-approximation algorithm for k-MMSA₃, suggesting that in the fixed-parameter regime, the third level of MMSA remains surprisingly close to the second level. Complementing this algorithm, we also give an FPT-time gap-preserving reduction from k-MMSA₃ to k-MMSA₂. Thus, stronger inapproximability for k-MMSA₃ would imply new hardness for k-MMSA₂, potentially offering a route around the current barriers for the latter problem.
Revisiting Marx’s reduction from k-MMSA_t to gap k-MMSA_{t+2}, we also show that k-MMSA₄ admits no n^o(1)-factor FPT approximation unless W[2]=FPT, and no n^O(1/k)-factor approximation running in n^o(k) time under ETH. These results separate the parameterized approximability behavior of the third and fourth levels and clarify where stronger inapproximability enters the k-MMSA hierarchy.
Cite as
Venkatesan Guruswami, Bingkai Lin, Xuandi Ren, and Xin Zheng. On the Approximability of Parameterized Minimum Monotone Satisfying Assignment. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{guruswami_et_al:LIPIcs.APPROX/RANDOM.2026.20,
author = {Guruswami, Venkatesan and Lin, Bingkai and Ren, Xuandi and Zheng, Xin},
title = {{On the Approximability of Parameterized Minimum Monotone Satisfying Assignment}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {20:1--20:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.20},
URN = {urn:nbn:de:0030-drops-277379},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.20},
annote = {Keywords: Parameterized approximation, Minimum Monotone Satisfying Assignment, Set Cover, inapproximability}
}
Document
APPROX
Cycle Cancellation for Submodular Fractional Allocations and Applications
Abstract
We consider discrete allocation problems where m indivisible goods need to be allocated among n agents. When agents' valuation functions are additive, the well-known cycle canceling lemma [Lenstra et al., 1990; Shmoys and Tardos, 1993; Plotkin et al., 1995] plays a key role in the design and analysis of rounding algorithms. In this paper, we prove an analogous lemma for the case of submodular valuations. Our algorithm removes cycles in the support graph of a fractional allocation while guaranteeing that each agent’s value, measured using the multilinear extension, does not decrease.
We demonstrate applications of the cycle-canceling algorithm, along with other ideas, to obtain new algorithms and results for three well-studied allocation objectives: max-min (Santa Claus problem), Nash social welfare (NSW), and maximin-share (MMS). For the submodular NSW problem, we obtain a 1/5-approximation; for the MMS problem, we obtain a 1/2(1-1/e)-approximation through new simple algorithms. For various special cases where the goods are "small" valued or the number of agents is constant, we obtain tight/best-known approximation algorithms. All our results are in the value-oracle model.
Cite as
Chandra Chekuri, Pooja Kulkarni, Ruta Mehta, and Jan Vondràk. Cycle Cancellation for Submodular Fractional Allocations and Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chekuri_et_al:LIPIcs.APPROX/RANDOM.2026.21,
author = {Chekuri, Chandra and Kulkarni, Pooja and Mehta, Ruta and Vondr\`{a}k, Jan},
title = {{Cycle Cancellation for Submodular Fractional Allocations and Applications}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {21:1--21:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.21},
URN = {urn:nbn:de:0030-drops-277384},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.21},
annote = {Keywords: Submodular Optimization, Allocation Problems, Fair Division}
}
Document
APPROX
A Configuration-LP Framework for Connected k-Median Clustering
Abstract
We study the connected k-median clustering problem, a clustering problem that augments the classical k-median objective with connectivity constraints. We focus on the overlapping variant of the problem, where clusters are allowed to share vertices. In addition to a metric space (V,d), the input contains a connected graph G on the same vertex set V of size n. The goal is to select at most k centers C and assign vertices to them so as to minimize the k-median cost (i.e., ∑_{v ∈ V} d(v,C)), subject to the constraint that each cluster induces a connected subgraph of G. Since the metric space and the connectivity graph are independent, the problem is significantly more challenging than standard clustering. Eube et al. [Eube et al., 2025] showed that even the assignment version is Ω(log n)-hard to approximate and gave approximation algorithms with guarantees depending polynomially on k.
We develop a configuration-LP-based framework that combines covering LP techniques with a rooted minimum-density oracle. For the assignment version, we obtain an O(log² n)-approximation. For the general version, we develop a bicriteria framework that opens O(klog n) centers while achieving an O(log² n)-approximation in cost.
Cite as
Kushagra Chatterjee, Rojin Rezvan, and Ali Vakilian. A Configuration-LP Framework for Connected k-Median Clustering. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chatterjee_et_al:LIPIcs.APPROX/RANDOM.2026.22,
author = {Chatterjee, Kushagra and Rezvan, Rojin and Vakilian, Ali},
title = {{A Configuration-LP Framework for Connected k-Median Clustering}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {22:1--22:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.22},
URN = {urn:nbn:de:0030-drops-277392},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.22},
annote = {Keywords: Connected Clustering, k-Median Clustering, Configuration LP}
}
Document
APPROX
Approximation Algorithms for Matroidal Prerequisite Systems
Abstract
Optimal selections in a decision process are often constrained by prerequisites. However, such prerequisites can encode functional rather than literal dependencies, so a required dependency may be supplied by one or several interacting alternatives. We introduce matroidal prerequisite systems (MPS), a combinatorial constraint structure where a poset specifies prerequisites while a matroid determines when those prerequisites have been satisfied by its span. This creates an order-sensitive notion of feasibility over words, where feasible words are associated with independent sets, while dependencies may be fulfilled through substitutable functionality.
Our main contribution is approximation algorithms for nonnegative additive maximization and monotone submodular maximization over the feasible words of an MPS. The guarantees are determined by two structural parameters: the maximum matroid rank Δ of a principal ideal in the poset and the maximum matroid connectivity λ_max. These measure the distance an MPS is from encoding a matroid or a poset antimatroid, respectively, both of which are generalized by an MPS. For additive maximization, we obtain deterministic Δ- and (1+λ_max)-approximation algorithms. By extending these techniques, we obtain efficient deterministic (2+λ_max)-approximation and randomized (Δ²⋅(1-1/e-δ)^{-1})-approximation algorithms for all δ > 0 for submodular maximization. The algorithm design and analysis use the theory of polymatroid greedoids, via a cryptomorphism we prove between an MPS and a strong polymatroid greedoid. Finally, a reduction from densest k-subgraph shows it is not possible to efficiently compute a min{Δ,λ_max}^o(1)-approximation to nonnegative additive maximization over the feasible words of an MPS under the Gap Exponential Time Hypothesis. Thus, an MPS provides a tractable, but provably nontrivial, framework for combinatorial optimization with interacting prerequisites, independence, and substitution.
Cite as
Robert P. Streit and Vijay K. Garg. Approximation Algorithms for Matroidal Prerequisite Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 23:1-23:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{streit_et_al:LIPIcs.APPROX/RANDOM.2026.23,
author = {Streit, Robert P. and Garg, Vijay K.},
title = {{Approximation Algorithms for Matroidal Prerequisite Systems}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {23:1--23:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.23},
URN = {urn:nbn:de:0030-drops-277405},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.23},
annote = {Keywords: matroids, posets, polymatroid greedoids, submodular maximization}
}
Document
APPROX
The Code Distortion Problem
Abstract
Two linear error-correcting codes 𝒞₁, 𝒞₂ ⊆ F_qⁿ are called linearly equivalent if there is a linear isometry mapping 𝒞₁ to 𝒞₂. In this work, we generalize the notion of linear equivalence and study the minimum distortion 𝒟(𝒞₁, 𝒞₂) of a linear mapping between codes 𝒞₁, 𝒞₂ ⊆ F_qⁿ, which quantifies how similar 𝒞₁ and 𝒞₂ are. We introduce and study the Code Distortion Problem (CDP), which asks to find a minimum distortion mapping between two input codes 𝒞₁ and 𝒞₂. CDP generalizes the Linear Code Equivalence Problem (LCE), which is essentially the special case of CDP where 𝒟(𝒞₁, C₂) = 1 and which is well-studied because of its role in cryptography.
We prove that (decisional) CDP is NP-hard to approximate to within any constant factor, and that it is in Σ₂^P. We also give a single-exponential-time k²-approximation algorithm for CDP, where k is the dimension of the input codes. Furthermore, we give a single-exponential-time ((2k + 1)/3)²-approximation algorithm for a natural special case of CDP, and we show that our analysis is tight in this case.
We use techniques from analogous work on the Lattice Distortion Problem (LDP) by Bennett, Dadush, and Stephens-Davidowitz (ESA, 2016). We also introduce or study a number of additional concepts that might be of independent interest. These include an adaptation of the celebrated reduction of Goldreich, Micciancio, Safra, and Seifert (IPL, 1999) from the Shortest Vector Problem (SVP) to the Closest Vector Problem (CVP) on lattices to the analogous problems on codes; successive minima bases for codes; and the matrix 0 → 0 "norm" on subspaces.
Cite as
Huck Bennett, Matthew Fox, and Bryant Morrell. The Code Distortion Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 24:1-24:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bennett_et_al:LIPIcs.APPROX/RANDOM.2026.24,
author = {Bennett, Huck and Fox, Matthew and Morrell, Bryant},
title = {{The Code Distortion Problem}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {24:1--24:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.24},
URN = {urn:nbn:de:0030-drops-277415},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.24},
annote = {Keywords: Code Distortion, Code Equivalence, Error-Correcting Codes, Metric Embeddings, Approximation Algorithms, Hardness of Approximation}
}
Document
APPROX
Online Matching with Size-Based and Convex Delays
Abstract
We study the online min-cost perfect matching with delay (MPMD) problem where m requests arrive in a metric space of n points. In MPMD, an algorithm can choose to match a request or to delay, and the objective is to minimise the sum of connection and delay costs. The connection cost of a match is the distance between the locations of two matched requests in the metric, and the increase of the delay cost is a function of the set of unmatched requests at every moment. In this paper, we study two different types of delay functions, size-based (MPMD-Size) and convex delays (MPMD-Convex).
The study of MPMD-Size was initiated by Deryckere and Umboh (APPROX/RANDOM 2023) where the instantaneous delay increment is a non-negative monotone function of the number of unmatched requests. We give an exponential improvement on the lower bounds for the deterministic and randomized competitive ratios. We also give improved upper bounds in terms of n, as opposed to Deryckere and Umboh’s upper bounds that were functions of m, which can be much larger than n. Our results settle the deterministic competitive ratio (up to constants). At the heart of these results is a succinct encoding scheme of MPMD-Size on a given n-point metric as a metrical task system problem on a 2^{n-1}-point metric.
We also consider MPMD-Convex proposed by Liu et al. (ISAAC 2018) where the delay cost incurred by each request is a uniform convex delay function of the time difference between its arrival time and the moment that it is matched by the algorithm. They focused on delay functions f that are unbounded, non-decreasing, continuous, and satisfy f(0) = f'(0) = 0, and showed that the deterministic competitive ratio is Ω(n) for n-point uniform metrics. We show that, surprisingly, when f is a non-negative, monotone polynomial with f'(0) > 0, there is an O(1)-competitive deterministic algorithm for uniform metrics. Our result completes our understanding of MPMD-Convex on uniform metrics for a broad class of functions.
Cite as
Junhao Gan, Xiao Sun, and Seeun William Umboh. Online Matching with Size-Based and Convex Delays. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 25:1-25:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{gan_et_al:LIPIcs.APPROX/RANDOM.2026.25,
author = {Gan, Junhao and Sun, Xiao and Umboh, Seeun William},
title = {{Online Matching with Size-Based and Convex Delays}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {25:1--25:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.25},
URN = {urn:nbn:de:0030-drops-277428},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.25},
annote = {Keywords: online algorithms with delays, competitive analysis, matching}
}
Document
APPROX
Online TCP Acknowledgment Under General Delays
Abstract
In a seminal work, Dooly, Goldman, and Scott (STOC 1998; JACM 2001) introduced the classic Online TCP Acknowledgment problem. In this problem, a sequence of n packets arrives over time, and the objective is to minimize both the number of acknowledgments sent and the total delay experienced by the packets. They showed that a natural greedy algorithm, which acknowledges when the delay of pending packets equals the acknowledgment cost, is 2-competitive.
Online TCP Acknowledgment is the canonical online problem with delay, capturing the fundamental tradeoff between reducing service cost through batching and the delay incurred by pending requests. Prior work has largely focused on richer service-cost models, e.g., Joint Replenishment and Multi-Level Aggregation. However, other than the work of Albers and Bals (SODA 2003), which studies maximum delay and closely related objectives, not much is known about general delay costs beyond the sum of delay costs of requests.
In this work, we study Online TCP Acknowledgment under two generalized delay-cost models that we call batch-aware and batch-oblivious. In the batch-aware model, each batch incurs a delay cost that depends on the packet delays within that batch. For the max-over-batches objective, which generalizes Albers and Bals, we show that greedy remains 2-competitive for every monotone batch-delay function. For the sum-over-batches objective, the picture changes sharply: greedy is Ω(n)-competitive, and the optimal deterministic competitive ratio is Θ(log n). Our matching upper bound requires only the minimal assumption that the batch delay function is monotone.
In the batch-oblivious model, the delay cost is a function of the global packet-delay vector. We show that greedy is 2-competitive for continuous submodular delay costs, and more generally under a weaker zero-coordinate diminishing-marginals condition. This yields 2-competitive algorithms for 𝓁_p norms, Top-k norms, and ordered norms. Using the submodular-norm approximation of Patton, Russo, and Singla, we also obtain an O(log n)-competitive algorithm for arbitrary symmetric norms.
Cite as
Sujoy Bhore, Michał Pawłowski, and Seeun William Umboh. Online TCP Acknowledgment Under General Delays. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 26:1-26:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bhore_et_al:LIPIcs.APPROX/RANDOM.2026.26,
author = {Bhore, Sujoy and Paw{\l}owski, Micha{\l} and Umboh, Seeun William},
title = {{Online TCP Acknowledgment Under General Delays}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {26:1--26:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.26},
URN = {urn:nbn:de:0030-drops-277435},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.26},
annote = {Keywords: Online Algorithms, TCP Acknowledgment, General Delay Functions}
}
Document
APPROX
Approximation Algorithms for Discounted Graph Search with Norm Objectives
Abstract
We introduce a unified framework for classical search and routing problems, including pathwise search, expanding search, the minimum spanning tree problem, and the traveling salesperson problem. The framework is based on two parameters. The first is a discount factor α ∈ [0,1]: the first traversal of an edge incurs its full cost, whereas each subsequent traversal incurs only an α-fraction of this cost. For a path starting at a designated root vertex, the α-latency of a vertex is the discounted cost accumulated until the vertex is first visited. The second parameter is a norm parameter p ≥ 1. The objective is to find a root-starting path that visits all vertices and minimizes the p-norm of the resulting vector of α-latencies.
The model interpolates between several well-studied objectives. For p = 1 and α = 1, it recovers pathwise search; for p = 1 and α = 0, it recovers expanding search. As p tends to infinity, the objective converges to a makespan-type criterion. At the endpoints α = 1 and α = 0, this limiting objective corresponds to TSP-type and MST-type behavior, respectively. For p = 1, we give polynomial-time constant-factor approximation algorithms for all α ∈ [0,1], matching the best known guarantees for expanding search at α = 0 and pathwise search at α = 1. For general p ≥ 1, we obtain a randomized constant-factor approximation algorithm and a derandomized pseudo-polynomial-time algorithm with the same guarantee.
Cite as
Svenja M. Griesbach, Felix Hommelsheim, and Max Klimm. Approximation Algorithms for Discounted Graph Search with Norm Objectives. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 27:1-27:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{griesbach_et_al:LIPIcs.APPROX/RANDOM.2026.27,
author = {Griesbach, Svenja M. and Hommelsheim, Felix and Klimm, Max},
title = {{Approximation Algorithms for Discounted Graph Search with Norm Objectives}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {27:1--27:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.27},
URN = {urn:nbn:de:0030-drops-277445},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.27},
annote = {Keywords: Approximation Algorithm, Expanding Search, Pathwise Search, Search Problem, Graph Exploration, Traveling Repairperson Problem, Minimum Latency Problem}
}
Document
APPROX
Threshold Rounding and Bounded-Degree Boolean MAX 2-CSP
Abstract
We describe an Ω̃(1/d⁴)-improvement over threshold rounding schemes for a broad class of Boolean MAX 2-CSP instances in which every variable appears in at most d constraints. In the case of MAX 2-SAT, we improve the ratio further and obtain an (β_⋆ + Ω̃(1/d²))-factor approximation algorithm for bounded-degree MAX 2-SAT instances, where β_⋆ is the UGC-optimal approximation ratio for MAX 2-SAT achieved by the LLZ algorithm [Lewin et al., 2002]. Our result generalizes an (α_GW + Ω̃(1/d²))-factor approximation algorithm for MAX CUT on graphs with degrees bounded by d, due to Hsieh and Kothari [Hsieh and Kothari, 2023]. Together with the state-of-the-art approximability results for MAX DI-CUT and MAX 2-AND [Brakensiek et al., 2023], our result suggests that similar improvements exist for bounded-degree instances of these problems as well.
Cite as
Suprovat Ghoshal, Neng Huang, Euiwoong Lee, Konstantin Makarychev, and Yury Makarychev. Threshold Rounding and Bounded-Degree Boolean MAX 2-CSP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ghoshal_et_al:LIPIcs.APPROX/RANDOM.2026.28,
author = {Ghoshal, Suprovat and Huang, Neng and Lee, Euiwoong and Makarychev, Konstantin and Makarychev, Yury},
title = {{Threshold Rounding and Bounded-Degree Boolean MAX 2-CSP}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {28:1--28:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.28},
URN = {urn:nbn:de:0030-drops-277451},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.28},
annote = {Keywords: MAX 2-SAT, Approximation Algorithms, Constraint Satisfaction Problems}
}
Document
APPROX
Incremental Consistent k-Center Clustering
Abstract
In the incremental consistent k-center clustering problem, we are given a sequence of adversarial point insertions and aim to maintain a k-center solution with small approximation ratio and small recourse. In this work, we explore the following question: what is the best approximation ratio of a polynomial-time algorithm with a worst-case recourse of 1? Our result improves upon the 6-approximation algorithm of Forster and Skarlatos [SODA '25], which itself improved over the 8-approximation algorithm of Charikar, Chekuri, Feder, and Motwani [STOC '97]. Moreover, we show that any incremental k-center algorithm that achieves an approximation ratio strictly less than √ 2 requires a worst-case recourse of k.
Cite as
Mara Grilnberger and Antonis Skarlatos. Incremental Consistent k-Center Clustering. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 29:1-29:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{grilnberger_et_al:LIPIcs.APPROX/RANDOM.2026.29,
author = {Grilnberger, Mara and Skarlatos, Antonis},
title = {{Incremental Consistent k-Center Clustering}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {29:1--29:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.29},
URN = {urn:nbn:de:0030-drops-277466},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.29},
annote = {Keywords: Consistent Clustering, k-Center, Dynamic Algorithms}
}
Document
APPROX
Optimal Stable Coresets for Geometric Median via Uniform Sampling
Abstract
The geometric median problem asks to find a point in ℝ^d that minimizes the sum of Euclidean distances to an input set. It is a classical problem in computational geometry and appears as a subroutine in numerous optimization tasks, many of which require the solution to satisfy additional structural constraints. A common approach to reduce the input size is to construct a coreset, which is a small weighted subset that faithfully represents the input for a specific optimization problem. Strong coresets preserve the cost of every candidate solution but require linear time to construct; weak coresets admit sublinear construction, in fact by uniform sampling, but only preserve near-optimal solutions, which is insufficient when the solution is constrained. To address this, we focus instead on the recently introduced intermediate notion of a stable coreset, which simultaneously handles all constrained variants. Currently, there is a large gap between the known sample sizes for stable and weak coresets.
Our main result is that a uniform sample of size O(ε^{-2} log 1/ε) is a stable (ε, O(ε))-coreset for the geometric median, with high constant probability, and this bound is tight up to the logarithmic factor. Our analysis adapts recent machinery of Carmel and Krauthgamer (ICLR 2026) for constructing stable coresets, which incurs an O(log d) factor. We show an iterative argument that progressively reduces the sample size, and eliminates this dependence on the dimension d. At a high level, this approach resembles the technique of iterative size reduction, which is applicable for strong coresets but not for weak coresets.
Cite as
Amir Carmel, Robert Krauthgamer, and Nir Petruschka. Optimal Stable Coresets for Geometric Median via Uniform Sampling. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{carmel_et_al:LIPIcs.APPROX/RANDOM.2026.30,
author = {Carmel, Amir and Krauthgamer, Robert and Petruschka, Nir},
title = {{Optimal Stable Coresets for Geometric Median via Uniform Sampling}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {30:1--30:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.30},
URN = {urn:nbn:de:0030-drops-277475},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.30},
annote = {Keywords: clustering, geometric median, coresets, stable coresets, uniform sampling}
}
Document
RANDOM
Unbounded-Width CSPs Are Untestable in a Sublinear Number of Queries
Abstract
The bounded-degree query model, introduced by Goldreich and Ron (Algorithmica, 2002), is a standard framework in graph property testing and sublinear-time algorithms. Many properties studied in this model, such as bipartiteness and 3-colorability of graphs, can be expressed as satisfiability of constraint satisfaction problems (CSPs). We prove that for the entire class of unbounded-width CSPs, testing satisfiability requires Ω(n) queries in the bounded-degree model. This result unifies and generalizes several previous lower bounds. In particular, it applies to all CSPs that are known to be NP-hard to solve, including k-colorability of 𝓁-uniform hypergraphs for any k,𝓁 ⩾ 2 with (k,𝓁) ≠ (2,2).
Our proof combines the techniques from Bogdanov, Obata, and Trevisan (FOCS, 2002), who established the first Ω(n) query lower bound for CSP testing in the bounded-degree model, with known results from universal algebra.
Cite as
Yumou Fei. Unbounded-Width CSPs Are Untestable in a Sublinear Number of Queries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 31:1-31:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{fei:LIPIcs.APPROX/RANDOM.2026.31,
author = {Fei, Yumou},
title = {{Unbounded-Width CSPs Are Untestable in a Sublinear Number of Queries}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {31:1--31:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.31},
URN = {urn:nbn:de:0030-drops-277489},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.31},
annote = {Keywords: constraint satisfaction problems, property testing}
}
Document
RANDOM
Tight Simulation of a Distribution Using Conditional Samples
Abstract
We present an algorithm for simulating a distribution using prefix conditional samples (Adar, Fischer and Levi, 2024), as well as "prefix-compatible" conditional models such as the interval model (Cannone, Ron and Servedio, 2015) and the subcube model (CRS15, Bhattacharyya and Chakraborty, 2018). The sample complexity is O(log² N/ε²) prefix conditional samples per query, which improves on the previously known Õ(log³ N/ε²) (Kumar, Meel and Pote, 2025). Moreover, our simulating distribution is O(ε²)-close to the input distribution with respect to the Kullback-Leibler divergence, which is stricter than the usual guarantee of being O(ε)-close with respect to the total-variation distance.
We show that our algorithm is tight with respect to the highly-related task of estimation: every algorithm that is able to estimate the mass of individual elements within (1 ± ε)-multiplicative error must make Ω(log²N/ε²) prefix conditional samples per element.
Cite as
Tomer Adar. Tight Simulation of a Distribution Using Conditional Samples. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{adar:LIPIcs.APPROX/RANDOM.2026.32,
author = {Adar, Tomer},
title = {{Tight Simulation of a Distribution Using Conditional Samples}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {32:1--32:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.32},
URN = {urn:nbn:de:0030-drops-277496},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.32},
annote = {Keywords: Distribution learning, Distribution simulation, Probability estimation}
}
Document
RANDOM
When Local and Non-Local Meet: Quadratic Improvement for Edge Estimation with Independent Set Queries
Abstract
We study the problem of estimating the number of edges in an unknown graph. We consider a hybrid model in which an algorithm may issue independent set, degree, and neighbor queries. We show that this model admits strictly more efficient edge estimation than either access type alone. Specifically, we give a randomized algorithm that outputs a (1±ε)-approximation of the number of edges using O(min(√m, √{n/√m})⋅(log n)/ε^{5/2}) queries, and prove a nearly matching lower bound. In contrast, prior work shows that in the local query model (Goldreich and Ron, Random Structures & Algorithms 2008) and in the independent set query model (Beame et al. ITCS 2018, Chen et al. SODA 2020), edge estimation requires Θ̃(n/√m) queries in the same parameter regimes. Our results therefore yield a quadratic improvement in the hybrid model, and no asymptotically better improvement is possible.
Cite as
Tomer Adar, Yahel Hotam, and Amit Levi. When Local and Non-Local Meet: Quadratic Improvement for Edge Estimation with Independent Set Queries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 33:1-33:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{adar_et_al:LIPIcs.APPROX/RANDOM.2026.33,
author = {Adar, Tomer and Hotam, Yahel and Levi, Amit},
title = {{When Local and Non-Local Meet: Quadratic Improvement for Edge Estimation with Independent Set Queries}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {33:1--33:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.33},
URN = {urn:nbn:de:0030-drops-277509},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.33},
annote = {Keywords: Edge count estimation, Degree oracle, Neighbor oracle, Independent-set oracle}
}
Document
RANDOM
Two-Sided Lossless Expanders in the Unbalanced Setting
Abstract
We present the first explicit construction of two-sided lossless expanders in the unbalanced setting (bipartite graphs that have polynomially many more nodes on the left than on the right). Prior to our work, all known explicit constructions in the unbalanced setting achieved only one-sided lossless expansion.
Specifically, we show that the one-sided lossless expanders constructed by Kalev and Ta-Shma (RANDOM'22) - that are based on multiplicity codes introduced by Kopparty, Saraf, and Yekhanin (STOC'11) - are, in fact, two-sided lossless expanders. Moreover, we show that our result is tight, thus completely characterizing the graph of Kalev and Ta-Shma.
Cite as
Eshan Chattopadhyay, Mohit Gurumukhani, Noam Ringach, and Yunya Zhao. Two-Sided Lossless Expanders in the Unbalanced Setting. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chattopadhyay_et_al:LIPIcs.APPROX/RANDOM.2026.34,
author = {Chattopadhyay, Eshan and Gurumukhani, Mohit and Ringach, Noam and Zhao, Yunya},
title = {{Two-Sided Lossless Expanders in the Unbalanced Setting}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {34:1--34:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.34},
URN = {urn:nbn:de:0030-drops-277517},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.34},
annote = {Keywords: Pseudorandomness, lossless expanders, multiplicity codes, condensers}
}
Document
RANDOM
Towards Tight Bounds for Testing k-Colorability
Abstract
Determining the sample complexity for testing k-colorability is perhaps the most well studied problem in property testing. It was (implicitly) introduced almost 50 years ago by Bollobás, Erdős, Simonovits and Szemerédi, who used the regularity lemma in order to give a tower-type bound for this problem. This bound has been successively improved in a long list of works, bringing the state-of-the-art bounds to lie between Ω(1/ε) and O((k/ε)log²(1/ε)). We obtain the following new results:
ii) Our first main result is an improved O((k/ε)log(1/ε)) upper bound, bringing the sample complexity closer to "truly" linear in ε. To prove this result, we improve upon a variant of the container method, introduced recently in the breakthrough paper of Blais and Seth.
iii) Perhaps the most important gap in our understanding of this problem is that while the best known upper bound increases with k, the lower bound is independent of k. Our second main result fills this gap by providing a new Ω((log k)/ε) lower bound, improving upon a result of Alon and Krivelevich from 2002. We conjecture that this is the true sample complexity of k-colorability.
Cite as
Nick Kushnir and Asaf Shapira. Towards Tight Bounds for Testing k-Colorability. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kushnir_et_al:LIPIcs.APPROX/RANDOM.2026.35,
author = {Kushnir, Nick and Shapira, Asaf},
title = {{Towards Tight Bounds for Testing k-Colorability}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {35:1--35:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.35},
URN = {urn:nbn:de:0030-drops-277523},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.35},
annote = {Keywords: k-colorability, property testing, sample complexity, graph property testing, random graphs}
}
Document
RANDOM
Distributed Gaussian Mean Testing Under Communication Constraints: Messages, Samples, and Coins
Abstract
We revisit the problem of Gaussian mean testing in a distributed, communication constrained setting, where each of n users independently observes samples from an unknown d-dimensional spherical Gaussian distribution 𝒢(μ,𝕀_d), and can communicate up to 𝓁 bits to a central referee. The referee’s goal is then to distinguish between cases (i) ‖μ‖₂ = 0 versus (ii) ‖μ‖₂ ≥ ε. This problem has been considered in the private- and public-coin settings, when each user holds exactly one sample, or more generally when each holds exactly m samples. In this work, we significantly generalize the question in three directions: when the users only share a small number s of random bits, when each user holds a different number of samples m_k, and when each user can send a different number of bits 𝓁_k to the referee.
Cite as
Clément L. Canonne and Nimitt. Distributed Gaussian Mean Testing Under Communication Constraints: Messages, Samples, and Coins. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{canonne_et_al:LIPIcs.APPROX/RANDOM.2026.36,
author = {Canonne, Cl\'{e}ment L. and Nimitt},
title = {{Distributed Gaussian Mean Testing Under Communication Constraints: Messages, Samples, and Coins}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {36:1--36:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.36},
URN = {urn:nbn:de:0030-drops-277536},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.36},
annote = {Keywords: Distribution Testing, Property Testing, Distributed Algorithms, Communication Constraints}
}
Document
RANDOM
Hard-To-Sample Distributions from Robust Extractors
Abstract
We provide a unified method for constructing explicit distributions which are difficult for restricted models of computation to generate. Our constructions are based on a new notion of robust extractors, which are extractors that remain sound even when a small number of points violate the min-entropy constraint. Using such objects, we show that for a broad range of sampling models (e.g., low-depth circuits, small-space sources, etc.), every output of the model has distance 1 - o(1) from our target distribution, qualitatively recovering essentially all previously known hardness results. Our work extends that of Viola (SICOMP '14), who developed an earlier unified framework based on traditional extractors to rule out sampling with very small error.
As a further application of our technique, we leverage a recent extractor construction of Chattopadhyay, Goodman, and Gurumukhani (ITCS '24) to present the first explicit distribution with distance 1 - o(1) from the output of any low-degree 𝔽₂-polynomial source. We note that a similar bound was obtained concurrently and independently by Khodabandeh and Shinkar (ECCC '26). We also describe a potential avenue toward proving a similar hardness result for AC⁰[⊕] circuits.
Cite as
Farzan Byramji, Daniel M. Kane, Jackson Morris, and Anthony Ostuni. Hard-To-Sample Distributions from Robust Extractors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 37:1-37:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{byramji_et_al:LIPIcs.APPROX/RANDOM.2026.37,
author = {Byramji, Farzan and Kane, Daniel M. and Morris, Jackson and Ostuni, Anthony},
title = {{Hard-To-Sample Distributions from Robust Extractors}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {37:1--37:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.37},
URN = {urn:nbn:de:0030-drops-277543},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.37},
annote = {Keywords: sampling, extractor, low-degree polynomials}
}
Document
RANDOM
Almost All Graphs Are Vertex-Minor Universal
Abstract
Answering a question of Claudet, we prove that the uniformly random graph G∼ 𝔾(n, 1/2) is Ω(√n)-vertex-minor universal with high probability. That is, for some constant α≈ 0.911, any graph on any α√ n specified vertices of G can be obtained as a vertex-minor of G. This has direct implications for quantum communications networks: an n-vertex k-vertex-minor universal graph corresponds to an n-qubit k-stabilizer universal graph state, which has the property that one can induce any stabilizer state on any k qubits using only local operations and classical communications.
We further employ our methods in two other contexts. We obtain a bipartite pivot-minor version of our main result, and we use it to derive a universality statement for minors in random binary matroids. We also introduce the vertex-minor Ramsey number R_{vm}(k) to be the smallest value n such that every n-vertex graph contains an independent set of size k as a vertex-minor. Supported by our main result, we conjecture that R_{vm}(k) is polynomial in k. We prove Ω(k²) ≤ R_{vm}(k) ≤ 2^k - 1.
Cite as
Ruben Ascoli, Bryce Frederickson, Sarah Frederickson, Caleb McFarland, and Logan Post. Almost All Graphs Are Vertex-Minor Universal. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ascoli_et_al:LIPIcs.APPROX/RANDOM.2026.38,
author = {Ascoli, Ruben and Frederickson, Bryce and Frederickson, Sarah and McFarland, Caleb and Post, Logan},
title = {{Almost All Graphs Are Vertex-Minor Universal}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {38:1--38:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.38},
URN = {urn:nbn:de:0030-drops-277559},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.38},
annote = {Keywords: vertex-minors, random graphs, quantum networks, graph states}
}
Document
RANDOM
Improved Error Reduction for Weighted PRGs
Abstract
We devise an error-reduction procedure that transforms a PRG for length-n, width-w read-once branching programs with error 1/poly(n) and seed length s₀, over any alphabet, into a weighted PRG with seed length s₀ + O(log 1/ε + log log ((log w)/log n)) ⋅ log w). Using this reduction, we improve upon the state-of-the-art weighted PRG constructions of Hoza (RANDOM 2021) and Cheng and Wu (SODA 2026), achieving optimal dependence on the program’s arity while matching the best known bounds in all other parameters.
Our motivation for obtaining optimal dependence on the arity stems from a result of Cheng and Hoza (CCC 2020, ToC 2022), who showed that a PRG with optimal arity and error dependence yields a PRG with seed length O(log^{3/2} n) (for, say, constant width), thereby breaking the long-standing log-squared barrier.
Cite as
Ben Chen, Gil Cohen, Dean Doron, Yuval Khaskelberg, and Amnon Ta-Shma. Improved Error Reduction for Weighted PRGs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 39:1-39:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2026.39,
author = {Chen, Ben and Cohen, Gil and Doron, Dean and Khaskelberg, Yuval and Ta-Shma, Amnon},
title = {{Improved Error Reduction for Weighted PRGs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {39:1--39:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.39},
URN = {urn:nbn:de:0030-drops-277562},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.39},
annote = {Keywords: Space-bounded computation, pseudorandom generators}
}
Document
RANDOM
Output-Sparse Matrix Multiplication Using Compressed Sensing
Abstract
We give two algorithms for output-sparse matrix multiplication (OSMM), the problem of multiplying two n × n matrices A, B when their product AB is promised to have at most O(n^δ) many non-zero entries for a given value δ ∈ [0, 2]. We then show how to speed up these algorithms in the fully sparse matrix multiplication (FSMM) setting, where the input matrices A, B are themselves sparse. All of our algorithms work over arbitrary rings.
Our first, deterministic algorithm for OSMM works via a two-pass reduction to compressed sensing. It runs in roughly n^{ω(δ/2, 1, 1)} time, where ω(⋅, ⋅, ⋅) is the rectangular matrix multiplication exponent. This substantially improves on prior deterministic algorithms for output-sparse matrix multiplication.
Our second, randomized algorithm for OSMM works via a reduction to compressed sensing and a variant of matrix multiplication verification, and runs in roughly n^{ω(δ-1, 1, 1)} time. This algorithm and its extension to the fully sparse setting have running times that match those of the (randomized) algorithms for OSMM and FSMM, respectively, in recent work of Abboud, Bringmann, Fischer, and Künnemann (SODA, 2024). Our algorithm is quite simple when taking a suitable compressed sensing scheme as a black box.
Cite as
Huck Bennett, Karthik Gajulapalli, Alexander Golovnev, and Evelyn Warton. Output-Sparse Matrix Multiplication Using Compressed Sensing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bennett_et_al:LIPIcs.APPROX/RANDOM.2026.40,
author = {Bennett, Huck and Gajulapalli, Karthik and Golovnev, Alexander and Warton, Evelyn},
title = {{Output-Sparse Matrix Multiplication Using Compressed Sensing}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {40:1--40:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.40},
URN = {urn:nbn:de:0030-drops-277570},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.40},
annote = {Keywords: Matrix Multiplication, Sparse Matrices, Compressed Sensing}
}
Document
RANDOM
From Decision to Random Certificates: Exponential Separation for Edge Estimation with Independent Set Queries
Abstract
We study the problem of estimating the number of edges in an undirected, unweighted graph using sublinear query access. We consider a query model that preserves the structure of Independent Set (IS) queries, but augments their output with a random certificate: given a vertex subset, the oracle returns a uniformly random edge from the induced subgraph if one exists, and returns null otherwise.
Using this access, we give a randomized algorithm that outputs a (1 ± ε)-approximation to the number of edges with constant success probability using Õ(log² m) queries. This implies an exponential separation from both standard IS queries and global random edge-sampling models: estimating the number of edges using standard IS queries require Θ̃(min {√m, n/√m}) queries, while direct random edge-sample access requires Θ̃(√m) samples. Beyond separation in query complexity, our algorithm is output-sensitive: its query complexity is polylogarithmic in the number of edges in the graph. This aligns with the classical objective in group testing, where one seeks algorithms that are both worst-case optimal and instance-adaptive.
Conceptually, our model connects group testing, the decision-versus-counting dichotomy, graph property testing, and the "power of a random certificate", and can be viewed as a structured form of conditional sampling of edges in graphs.
Cite as
Debarshi Chanda, Buddha Dev Das, Arijit Ghosh, and Gopinath Mishra. From Decision to Random Certificates: Exponential Separation for Edge Estimation with Independent Set Queries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 41:1-41:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chanda_et_al:LIPIcs.APPROX/RANDOM.2026.41,
author = {Chanda, Debarshi and Das, Buddha Dev and Ghosh, Arijit and Mishra, Gopinath},
title = {{From Decision to Random Certificates: Exponential Separation for Edge Estimation with Independent Set Queries}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {41:1--41:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.41},
URN = {urn:nbn:de:0030-drops-277584},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.41},
annote = {Keywords: Property Testing, Edge Estimation}
}
Document
RANDOM
Glauber Dynamics for Random Field Ising Models on Bounded Degree Graphs
Abstract
We study the ferromagnetic random field Ising model (RFIM) on a graph G = (V,E) having maximal degree Δ, where the external field at each vertex is an i.i.d. random variable. When the random field distribution is sufficiently anti-concentrated, we prove that with high probability over the quenched randomness of the external field, the Glauber dynamics of this RFIM mixes in polynomial time as a consequence of a Poincaré inequality. This model is relevant to the Griffiths phase where the correlations decay exponentially fast in expectation over the quenched random field, but contraction does not hold point-wise due to the existence of weak fields that lead to low-temperature behavior. Previously, fast mixing of Glauber dynamics under large disorder was only proven on the integer lattice, and for RFIM on general graphs, only a sampling algorithm based on self-avoiding walks was known.
Cite as
Yi Han. Glauber Dynamics for Random Field Ising Models on Bounded Degree Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{han:LIPIcs.APPROX/RANDOM.2026.42,
author = {Han, Yi},
title = {{Glauber Dynamics for Random Field Ising Models on Bounded Degree Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {42:1--42:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.42},
URN = {urn:nbn:de:0030-drops-277590},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.42},
annote = {Keywords: Random field Ising model, Glauber dynamics, Poincar\'{e} inequality, Markov chain mixing, bounded degree graphs}
}
Document
RANDOM
Markov Chains with Rewinding
Abstract
Motivated by techniques developed in recent progress on lower bounds for sublinear time algorithms (Behnezhad, Roghani and Rubinstein, STOC 2023, FOCS 2023, and STOC 2024) we introduce and study a new class of randomized algorithmic processes that we call "Markov Chains with Rewinding". In this setting an agent/algorithm interacts with a (partially observable) Markovian random evolution by periodically/strategically rewinding the Markov chain to previous states. Depending on the application this may lead the evolution to desired states faster, or allow the agent to efficiently learn or test properties of the underlying Markov chain that may be infeasible or inefficient with passive observation.
We study the task of identifying the initial state in a given partially observable Markov chain. Analysis of this question in specific Markov chains is the central ingredient in the above cited works and we aim to systematize the analysis in our work. Our first result is that any pair of states distinguishable with any rewinding strategy can also be distinguished with a non-adaptive rewinding strategy (i.e., one whose rewinding choices are predetermined before observing any outcomes of the chain). Therefore, while rewinding strategies can be shown to be strictly more powerful than passive strategies (i.e., those that do not rewind back to previous states), adaptivity does not give additional power to a rewinding strategy in the absence of efficiency considerations.
The difference becomes apparent however when we introduce a natural efficiency measure, namely the query complexity (i.e., the number of observations they need to identify distinguishable states). Our second main contribution is to quantify this efficiency gap. We present a non-adaptive rewinding strategy whose query complexity is within a polynomial of that of the optimal (adaptive) strategy, and show that such a polynomial loss is necessary in general.
Cite as
Amir Azarmehr, Soheil Behnezhad, Alma Ghafari, and Madhu Sudan. Markov Chains with Rewinding. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 43:1-43:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{azarmehr_et_al:LIPIcs.APPROX/RANDOM.2026.43,
author = {Azarmehr, Amir and Behnezhad, Soheil and Ghafari, Alma and Sudan, Madhu},
title = {{Markov Chains with Rewinding}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {43:1--43:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.43},
URN = {urn:nbn:de:0030-drops-277603},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.43},
annote = {Keywords: Partially observable Markov chains, rewinding algorithm, sublinear algorithms}
}
Document
RANDOM
One-Way Functions and Polynomial-Time Dimension
Abstract
The theory of randomness in computation has developed around several viewpoints on randomness and information, including statistical tests, compression schemes, betting strategies, and efficiently samplable sources. At the level of computability, several of these viewpoints turn out to be equivalent. A fundamental question is how robust these viewpoints remain when the underlying algorithms are subject to computational resource bounds. This paper studies this question for two polynomial-time notions of information density for infinite binary sequences. Polynomial-time dimension, denoted dim_P, quantifies information density using polynomial-time betting strategies called s-gales. Polynomial-time Kolmogorov complexity rate, denoted 𝒦_poly, gives a compression-based notion of polynomial-time information density using polynomial-time descriptions. Hitchcock and Vinodchandran (CCC 2004) showed that dim_P(X) ≥ 𝒦_poly(X) for every sequence X, and asked whether equality always holds. This question was later also posed by Stull.
Our main result proves a duality between the non-robustness of these polynomial-time information-density notions and the existence of one-way functions. Assuming one-way functions exist, we construct a polynomial-time samplable distribution over infinite sequences such that, with probability 1, the sampled sequence X satisfies dim_P(X) > 𝒦_poly(X) by a uniform gap. Conversely, we show that if some polynomial-time samplable distribution yields such an almost-sure uniform separation, then infinitely-often one-way functions exist. Thus, separations between these two polynomial-time information-density notions over efficiently samplable sources lie at the same frontier as one-way functions, with the reverse direction yielding infinitely-often one-way functions. This duality gives a negative answer, assuming one-way functions, to the open question posed by Hitchcock, Vinodchandran, and Stull. Furthermore, we show that there are individual sequences witnessing separations between dim_P and 𝒦_poly, and that the gap can be made arbitrarily close to 1. We also establish analogous bounds for strong polynomial-time dimension and asymptotic upper polynomial-time Kolmogorov complexity rates. The main technical challenge is to connect finite-length pseudorandomness assumptions with asymptotic information-density measures on infinite sequences. Our proof addresses this by developing several new constructions and arguments involving probabilistic tools such as the Borel-Cantelli lemma, Kolmogorov’s inequality for martingales, and techniques for approximating probabilities of efficiently samplable distributions. The work shows that the question of non-robustness for polynomial-time information-density notions, which is prima facie different from standard questions about randomness, pseudorandomness, and cryptography, is in fact intimately related to the existence of one-way functions.
Cite as
Satyadev Nandakumar, Subin Pulari, Akhil S, and Suronjona Sarma. One-Way Functions and Polynomial-Time Dimension. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 44:1-44:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{nandakumar_et_al:LIPIcs.APPROX/RANDOM.2026.44,
author = {Nandakumar, Satyadev and Pulari, Subin and S, Akhil and Sarma, Suronjona},
title = {{One-Way Functions and Polynomial-Time Dimension}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {44:1--44:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.44},
URN = {urn:nbn:de:0030-drops-277619},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.44},
annote = {Keywords: Polynomial-time dimension, One-way functions, Resource bounded randomness, Kolmogorov complexity, Polynomial-time martingales}
}
Document
RANDOM
Recovering Planted Colorings in Sublinear Time
Abstract
We study the problem of recovering a planted k-coloring in sublinear time. Given an expander G with a planted coloring, the goal is to efficiently construct a small-space data structure that allows consistent color queries: given a vertex v, the algorithm quickly returns the color of v according to the planted solution.
We work in the adversarial planted coloring model of David and Feige [STOC 2016], where an adversary chooses a d-regular spectral λ-expander G on n vertices and plants a balanced k-coloring by partitioning the vertices into k equal parts and deleting all edges within each part. This model generalizes the earlier random graph models studied by Blum and Spencer [J. Algorithms 1995] and Alon and Kahale [STOC 1994].
We give the first sublinear-time algorithm for recovering planted colorings in this model. In the adjacency list model, our algorithm has preprocessing time and space Õ(n^{1/2 + O(1/log(d/λ))}), and produces a data structure that answers color queries in time Õ(n^{1/2 + O(1/log(d/λ))}). With a high constant probability, the resulting labeling agrees with the planted coloring on all but an O(√{λ/d}) fraction of vertices, up to a permutation of the k colors.
Our algorithm gives sublinear time inner product access to the bottom eigenspace of the normalized adjacency matrix by using random walks, which allows us to adapt the classical spectral approach of Alon and Kahale in sublinear time.
Cite as
Weronika Wrzos-Kaminska. Recovering Planted Colorings in Sublinear Time. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 45:1-45:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{wrzoskaminska:LIPIcs.APPROX/RANDOM.2026.45,
author = {Wrzos-Kaminska, Weronika},
title = {{Recovering Planted Colorings in Sublinear Time}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {45:1--45:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.45},
URN = {urn:nbn:de:0030-drops-277627},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.45},
annote = {Keywords: sublinear algorithms, spectral algorithms, graph coloring}
}
Document
RANDOM
One-Sided-Error Parameterized Reductions for the Minimum Distance and Shortest Vector Problems
Abstract
It is notoriously difficult to obtain deterministic reductions for the Minimum Distance Problem (MDP) and the Shortest Vector Problem (SVP). Under two-sided-error randomized reductions, Bennett, Cheraghchi, Guruswami, and Ribeiro (STOC 2023) proved parameterized hardness of approximation for these problems. We partially derandomize their reductions and present one-sided-error randomized reductions: MDP is W[1]-hard to approximate within an arbitrary constant factor under FPT many-one one-sided-error randomized reductions; For every fixed p ≥ 1, SVP in the 𝓁_p norm is W[1]-hard to approximate within an arbitrary constant factor below 2^{1/p}.
We demonstrate the usefulness of one-sided-error randomized reductions by showing that they can be conditionally derandomized when the target problem has an OR function. Under a standard hardness-vs-randomness assumption, namely a plausible lower-bound assumption against nondeterministic circuits, we prove a general theorem formalizing this derandomization. Here, an OR function combines several instances into one instance that preserves their disjunction. We construct such OR functions for the relevant MDP and SVP gap problems, and thereby obtain deterministic W[1]-hardness for approximating MDP over every fixed finite field within every constant factor, and for approximating SVP in 𝓁_p norms for every fixed integer p within every factor below 2^{1/p}. Applying the same framework to Micciancio’s one-sided-error randomized reduction (ToC 2012) yields, under the same circuit lower-bound assumption, deterministic polynomial-time NP-hardness of approximating Euclidean SVP within every constant factor.
Cite as
Shuichi Hirahara and Kazuki Ogitsuka. One-Sided-Error Parameterized Reductions for the Minimum Distance and Shortest Vector Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 46:1-46:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hirahara_et_al:LIPIcs.APPROX/RANDOM.2026.46,
author = {Hirahara, Shuichi and Ogitsuka, Kazuki},
title = {{One-Sided-Error Parameterized Reductions for the Minimum Distance and Shortest Vector Problems}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {46:1--46:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.46},
URN = {urn:nbn:de:0030-drops-277637},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.46},
annote = {Keywords: Codes, Lattices, Minimum Distance Problem, Shortest Vector Problem, Parameterized complexity, derandomization}
}
Document
RANDOM
Sequential Sweeps and High Dimensional Expansion
Abstract
It is well known that the spectral gap of the down-up walk over an n-partite simplicial complex (also known as Glauber dynamics) cannot be better than O(1/n) due to natural obstructions such as coboundaries. We study an alternative random walk over partite simplicial complexes known as the sequential sweep or the systematic scan Glauber dynamics: Whereas the down-up walk at each step selects a random coordinate and updates it based on the remaining coordinates, the sequential sweep goes through each of the coordinates one by one in a deterministic order and applies the same update operation. It is natural, thus, to compare n-steps of the down-up walk with a single step of the sequential sweep. Interestingly, while the spectral gap of the n-th power of the down-up walk is still bounded from above by a constant, under a strong enough local spectral assumption (in the sense of Gur, Lifschitz, Liu, STOC 2022) we can show that the spectral gap of this walk can be arbitrarily close to 1. We also study other isoperimetric inequalities for these walks, and show that under the assumptions of local entropy contraction (related to the considerations of Gur, Lifschitz, Liu), these walks satisfy an entropy contraction inequality. Concretely, we generalize a result of Lubetzky, Lubotzky, and Parzanchevski (Journal of the EMS) about the rapid mixing of sequential sweep in Ramanujan complexes to suitable high dimensional expanders.
Cite as
Vedat Levi Alev and Ori Parzanchevski. Sequential Sweeps and High Dimensional Expansion. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 47:1-47:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{alev_et_al:LIPIcs.APPROX/RANDOM.2026.47,
author = {Alev, Vedat Levi and Parzanchevski, Ori},
title = {{Sequential Sweeps and High Dimensional Expansion}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {47:1--47:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.47},
URN = {urn:nbn:de:0030-drops-277642},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.47},
annote = {Keywords: Random walks, high dimensional expanders, Ramanujan complexes, systematic scan, Glauber dynamics}
}
Document
RANDOM
An Elementary Proof of the First LP Bound on the Rate of Binary Codes
Abstract
The asymptotic rate vs. distance problem is a long-standing fundamental problem in coding theory. The best upper bound to date was given in 1977, and has since received numerous proofs and interpretations. Here we provide a new, elementary proof of this bound that is based on counting walks in the Hamming cube.
Cite as
Nati Linial and Elyassaf Loyfer. An Elementary Proof of the First LP Bound on the Rate of Binary Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 48:1-48:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{linial_et_al:LIPIcs.APPROX/RANDOM.2026.48,
author = {Linial, Nati and Loyfer, Elyassaf},
title = {{An Elementary Proof of the First LP Bound on the Rate of Binary Codes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {48:1--48:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.48},
URN = {urn:nbn:de:0030-drops-277656},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.48},
annote = {Keywords: Coding theory, code bounds, convex optimization, linear progamming}
}
Document
RANDOM
Ultra-Sparse Expanders and the Free Method
Abstract
In this paper we ask how much expansion one can retain with almost no edges beyond connectivity. Concretely, for graphs of average degree 2+ε, what is the "Ramanujan bound" - how does spectral expansion scale with ε? We compare five ultra–sparse graph models - including the configuration model, subdivision of regular expanders, and the union of a cycle with a partial matching - and analyze each under the normalized or unnormalized notions of expansion. In the normalized setting, we prove bounds that are essentially optimal, determining the correct asymptotic dependence on ε. Our results extend to expansion in general irregular graphs.
For some models we prove rigorous bounds - primarily via finite free probability - while others remain beyond our current techniques. To bridge this gap, we introduce the Free Method, which produces quantitative predictions without proving existence - analogous to the probabilistic method, which certifies existence without providing an explicit construction. These predictions align with experiments. We expect the free method to be useful more broadly in graph-theoretic settings.
Cite as
Gil Cohen and Gal Maor. Ultra-Sparse Expanders and the Free Method. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 49:1-49:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cohen_et_al:LIPIcs.APPROX/RANDOM.2026.49,
author = {Cohen, Gil and Maor, Gal},
title = {{Ultra-Sparse Expanders and the Free Method}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {49:1--49:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.49},
URN = {urn:nbn:de:0030-drops-277669},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.49},
annote = {Keywords: Spectral expanders, free probability}
}
Document
RANDOM
Entropy Equivalence Testing
Abstract
We introduce the problem of entropy equivalence testing for probability distributions, a relaxation of the well-studied closeness testing problem, where the distribution testing algorithm is now only required to distinguish, given samples from two unknown distributions p,q and a parameter ε ∈ (0,1/2], between p = q and |H(p)-H(q)| ⩾ ε (where H denotes the Shannon entropy). We provide a time- and sample-efficient algorithm for this task, showing that the optimal sample complexity for this task can be significantly lower than that of closeness testing. As an application, we leverage this result to provide the first non-trivial testing algorithm for (standard) closeness of low-degree Bayesian networks, which significantly improves on either the sample or time complexity of a baseline based on full learning.
Cite as
Clément L. Canonne, Yash Pote, Jonathan Scarlett, and Joy Qiping Yang. Entropy Equivalence Testing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 50:1-50:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{canonne_et_al:LIPIcs.APPROX/RANDOM.2026.50,
author = {Canonne, Cl\'{e}ment L. and Pote, Yash and Scarlett, Jonathan and Yang, Joy Qiping},
title = {{Entropy Equivalence Testing}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {50:1--50:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.50},
URN = {urn:nbn:de:0030-drops-277676},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.50},
annote = {Keywords: Entropy, distribution testing, sublinear algorithm, Bayesian network}
}
Document
RANDOM
On Computing Total Variation Distance Between Mixtures of Product Distributions
Abstract
We study the problem of approximating the total variation distance between two mixtures of product distributions over an n-dimensional discrete domain. Given two mixtures ℙ and ℚ with k₁ and k₂ product distributions over [q]ⁿ, respectively, we give a randomized algorithm that approximates d_TV(ℙ,ℚ) within a multiplicative error of (1±ε) in time poly((nq)^{k₁+k₂}, 1/ε). We also study the special case of mixtures of Boolean subcubes over {0,1}ⁿ. For this class, we give a deterministic algorithm that exactly computes the total variation distance in time poly(n, 2^O(k₁+k₂)), and show that exact computation is #𝖯-hard when k₁+k₂ = Θ(n).
Cite as
Weiming Feng, Yucheng Fu, Minji Yang, and Anqi Zhang. On Computing Total Variation Distance Between Mixtures of Product Distributions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 51:1-51:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{feng_et_al:LIPIcs.APPROX/RANDOM.2026.51,
author = {Feng, Weiming and Fu, Yucheng and Yang, Minji and Zhang, Anqi},
title = {{On Computing Total Variation Distance Between Mixtures of Product Distributions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {51:1--51:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.51},
URN = {urn:nbn:de:0030-drops-277689},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.51},
annote = {Keywords: Randomized algorithm, Total variation distance}
}
Document
RANDOM
Testing k-Submodularity
Abstract
We initiate the study of property testing for k-submodular functions, a higher-dimensional analogue of submodular functions defined on partial partitions of a ground set. While k-submodularity retains the diminishing-returns flavor of ordinary submodularity, it also introduces a pairwise monotonicity constraint comparing competing assignments of the same element. This additional local structure makes the testing problem qualitatively different from the classical case.
Our results show a sharp contrast between distance regimes. In the 𝓁_p regime for p ≥ 1, we prove that every bounded k-submodular function is close to a junta on the hypergrid. Combined with an implicit-learning tester for hypergrid domains, this yields a constant-query tester for k-submodularity. In the Hamming distance regime, k-submodularity admits two qualitatively different local witnesses - violated squares for diminishing marginal gains, and violated triangles for pairwise-monotonicity failures - and the latter has no counterpart at k = 1. We prove density theorems for both witness types via repair on filters and ideals of partial partitions, yielding non-adaptive, one-sided sub-exponential-query testers for the two component properties of k-submodularity. We then exhibit a configuration in which the two repair directions are forced into opposition on a shared vertex, identifying a structural barrier to combining these into a tester for the full property.
Finally, for bounded-range functions, we give an adaptive tester for monotone k-submodularity via a pseudo-DNF representation and learning on the hypergrid. Several of the structural and learning tools developed here may be useful for testing other properties over product domains.
Cite as
Themistoklis Haris and Diptaksho Palit. Testing k-Submodularity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 52:1-52:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{haris_et_al:LIPIcs.APPROX/RANDOM.2026.52,
author = {Haris, Themistoklis and Palit, Diptaksho},
title = {{Testing k-Submodularity}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {52:1--52:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.52},
URN = {urn:nbn:de:0030-drops-277694},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.52},
annote = {Keywords: property testing, sublinear algorithms, submodular functions}
}
Document
RANDOM
Symmetric Distributions from Shallow Circuits
Abstract
We characterize the symmetric distributions that can be (approximately) generated by shallow Boolean circuits. More precisely, let f: {0,1}^m → {0,1}ⁿ be a Boolean function where each output bit depends on at most d input bits. Suppose the output distribution of f evaluated on uniformly random input bits is close in total variation distance to a symmetric distribution 𝒟 over {0,1}ⁿ. Then 𝒟 must be close to a mixture of the uniform distribution over n-bit strings of even Hamming weight, the uniform distribution over n-bit strings of odd Hamming weight, and γ-biased product distributions for γ an integer multiple of 2^{-d}. Moreover, the mixing weights are determined by low-degree, sparse 𝔽₂-polynomials. This extends the previous classification for generating symmetric distributions that are also uniform over their support.
Cite as
Daniel M. Kane, Anthony Ostuni, and Kewen Wu. Symmetric Distributions from Shallow Circuits. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 53:1-53:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kane_et_al:LIPIcs.APPROX/RANDOM.2026.53,
author = {Kane, Daniel M. and Ostuni, Anthony and Wu, Kewen},
title = {{Symmetric Distributions from Shallow Circuits}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {53:1--53:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.53},
URN = {urn:nbn:de:0030-drops-277707},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.53},
annote = {Keywords: Sampling, distributions, locality, circuit complexity}
}
Document
RANDOM
Load Balancing Under Adaptive Bin Deletions
Abstract
We analyze a balls-and-bins game against an adaptive adversary that sequentially deletes bins. Starting with n balls distributed across n bins, the adversary deletes a bin in each step, forcing the algorithm to redistribute its balls to surviving bins. We prove that after n/2 rounds, uniform random redistribution yields optimal O(n) recourse and O((log n)/(log log n)) maximum load. Furthermore, we show that applying the "power of two choices" reduces the maximum load to O(log log n) while maintaining linear recourse.
We also consider a variation of this game where the balls from the deleted bin are partitioned evenly among d ≪ n random bins rather than being redistributed independently. We demonstrate that keeping the balls together (d = 1), which gives small maximum load and recourse against an oblivious adversary, fails against an adaptive adversary. Nevertheless, we show that splitting the balls into just two groups (d = 2) is sufficient to recover linear recourse and efficient load balancing in the adaptive setting.
Cite as
Haim Kaplan, Shay Sapir, and Uri Stemmer. Load Balancing Under Adaptive Bin Deletions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 54:1-54:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kaplan_et_al:LIPIcs.APPROX/RANDOM.2026.54,
author = {Kaplan, Haim and Sapir, Shay and Stemmer, Uri},
title = {{Load Balancing Under Adaptive Bin Deletions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {54:1--54:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.54},
URN = {urn:nbn:de:0030-drops-277711},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.54},
annote = {Keywords: Balls and Bins, Adaptive Adversary}
}
Document
RANDOM
Arboricity Matters in Triangle Counting with Random Edges
Abstract
Given a simple, unweighted, undirected graph G = (V,E) with |V| = n and |E| = m, and parameters 0 < ε, δ < 1, along with Degree, Neighbour, Pair and RandomEdge query access to G, we provide a query-based randomized algorithm to generate an estimate T̂ of the number of triangles T in G, such that T̂ ∈ [(1-ε)T , (1+ε)T] with probability at least 1-δ. The query complexity of our algorithm is Õ(m α log(1/δ)/{ε³T}), where α is the arboricity of G. Our work can be seen as a natural progression to the line of recent works [Eden et al., SIAM J Comp., 2017; Assadi et al., ITCS 2019; Eden et al., SODA 2020] that considered subgraph or triangle counting with or without the use of RandomEdge query. Of these works, Eden et al. [SODA 2020] considers the role of arboricity. Our work is the first to consider how RandomEdge query can leverage the structural property of arboricity. Furthermore, continuing in the line of work of Assadi et al. [APPROX/RANDOM 2022], we also provide a lower bound of Ω̃(m α log(1/δ)/{ε²T}) that matches the upper bound exactly on arboricity, δ, and almost on ε.
Cite as
Arijit Bishnu, Debarshi Chanda, and Gopinath Mishra. Arboricity Matters in Triangle Counting with Random Edges. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 55:1-55:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bishnu_et_al:LIPIcs.APPROX/RANDOM.2026.55,
author = {Bishnu, Arijit and Chanda, Debarshi and Mishra, Gopinath},
title = {{Arboricity Matters in Triangle Counting with Random Edges}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {55:1--55:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.55},
URN = {urn:nbn:de:0030-drops-277722},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.55},
annote = {Keywords: Triangle counting, Sublinear algorithms, Subgraph counting}
}
Document
RANDOM
An Improved Construction of Variety-Evasive Subspace Families
Abstract
We study the question of explicitly constructing variety-evasive subspace families, a pseudorandom primitive introduced by Guo (Computational Complexity 2024) that generalizes both hitting sets and lossless rank condensers. Roughly speaking, a variety-evasive subspace family ℋ is a collection of subspaces such that for every algebraic variety V in a fixed family ℱ, there is some subspace W ∈ ℋ that is in general position with respect to V.
We give an explicit construction of a subspace families that evade all degree-d varieties in an n-dimensional affine or projective space. Our construction improves on the size of the variety-evasive subspace families constructed by Guo and, for varieties of degree n^{1 + Ω(1)}, is polynomially close to Guo’s lower bound on the size of any such variety-evasive subspace family. Our variety-evasive subspace families rely on an improved construction of hitting sets for Chow forms of algebraic varieties.
Cite as
Robert Andrews and Abhibhav Garg. An Improved Construction of Variety-Evasive Subspace Families. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{andrews_et_al:LIPIcs.APPROX/RANDOM.2026.56,
author = {Andrews, Robert and Garg, Abhibhav},
title = {{An Improved Construction of Variety-Evasive Subspace Families}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {56:1--56:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.56},
URN = {urn:nbn:de:0030-drops-277738},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.56},
annote = {Keywords: algebraic complexity, pseudorandomness, varieties, variety-evasive subspaces, Chow forms}
}
Document
RANDOM
Testing Unate Distributions
Abstract
We initiate the study of unate distributions over {±1}ⁿ - a natural analogue of unate Boolean functions - by considering two basic testing problems that parallel well-studied questions for monotone distributions:
- Uniformity Testing of Unate Distributions: We show that Θ̃(n^{3/2}) samples are sufficient and necessary, in contrast to the Θ̃(n) sample complexity of the analogous problem for monotone distributions (Rubinfeld and Servedio, STOC 2005; Adamaszek, Czumaj, and Sohler, SODA 2010).
- Unateness Testing of Arbitrary Distributions: We give a tester that uses Õ(n^{3/2}) conditional samples in the subcube conditional model. On the other hand, every tester that draws conditional samples in a similar fashion, namely from O(1)-dimensional subcubes, must have an Ω̃(n^{2/3}) complexity. In the same model, the complexity of monotonicity testing was recently shown to be Θ̃(n) (Chakrabarty et al., STOC 2025). Our algorithms for both problems significantly outperform the naive approach of reducing to the monotone case, which would incur Ω(n²) sample complexity. Our uniformity tester relies on a subroutine that "weakly" learns the hidden orientations of a unate distribution, together with a new correlation bound for these estimates. Both tools may be of independent interest in studying monotonicity and unateness over {±}ⁿ.
Cite as
Daeho Lee, Shivam Nadimpalli, Mingda Qiao, and Ronitt Rubinfeld. Testing Unate Distributions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 57:1-57:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{lee_et_al:LIPIcs.APPROX/RANDOM.2026.57,
author = {Lee, Daeho and Nadimpalli, Shivam and Qiao, Mingda and Rubinfeld, Ronitt},
title = {{Testing Unate Distributions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {57:1--57:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.57},
URN = {urn:nbn:de:0030-drops-277741},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.57},
annote = {Keywords: Distribution testing, unate distributions, monotone distributions, uniformity testing, subcube conditioning}
}
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Homomorphism Testing with Resilience to Online Manipulations
Abstract
A central challenge in property testing is verifying algebraic structure with minimal access to data. A landmark result addressing this challenge, the linearity test of Blum, Luby, and Rubinfeld (JCSS `93), spurred a rich body of work on testing algebraic properties such as linearity and its generalizations to low-degree polynomials and group homomorphisms. However, classical tests for these properties assume unrestricted, noise-free access to the input function - an assumption that breaks down in adversarial or dynamic settings. To address this, Kalemaj, Raskhodnikova, and Varma (Theory of Computing `23) introduced the online manipulation model, where an adversary erases or corrupts query responses over time, based on the tester’s past queries.
We initiate the study of manipulation-resilient testing for group homomorphism in this online model. Our main result is an optimal tester that makes O(1/ε+log t) queries, where ε is the distance parameter and t is the number of function values the adversary can erase or corrupt per query. Our result recovers the celebrated O(1/ε) bound by Ben-Or, Coppersmith, Luby, and Rubinfeld (Random Struct. Algorithms `08) for homomorphism testing in the standard property testing model, albeit with a different tester. Our tester, Random Signs Test, lifts known manipulation-resilient linearity testers for 𝔽₂ⁿ → 𝔽₂ to general group domains and codomains by introducing more randomness: instead of verifying the homomorphism condition for a sum of random elements, it uses additions and subtractions of random elements, randomly selecting a sign for each element. We also obtain improved group-specific query bounds for key families of groups. Our results show that despite the challenges of online manipulation, group homomorphism - a fundamental algebraic property - is efficiently testable across a wide range of domains and codomains. Finally, we formalize a general framework for proving online resilience.
Cite as
Esty Kelman, Uri Meir, Debanuj Nayak, and Sofya Raskhodnikova. Homomorphism Testing with Resilience to Online Manipulations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 58:1-58:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kelman_et_al:LIPIcs.APPROX/RANDOM.2026.58,
author = {Kelman, Esty and Meir, Uri and Nayak, Debanuj and Raskhodnikova, Sofya},
title = {{Homomorphism Testing with Resilience to Online Manipulations}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {58:1--58:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.58},
URN = {urn:nbn:de:0030-drops-277753},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.58},
annote = {Keywords: Property Testing, Sublinear Algorithms, Online Manipulation Resilience, Group Theory}
}
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On the LSH Distortion of Ulam and Cayley Similarities
Abstract
Locality-sensitive hashing (LSH) has found widespread use as a fundamental primitive, particularly to accelerate nearest neighbor search. An LSH scheme for a similarity function S:𝒳 × 𝒳 → [0,1] is a distribution over hash functions on 𝒳 with the property that the probability of collision of any two elements x,y ∈ 𝒳 is exactly equal to S(x,y). However, not all similarity functions admit exact LSH schemes. The notion of LSH distortion measures how multiplicatively close a similarity function is to having an LSH scheme.
In this work, we study the LSH distortion of the Ulam and Cayley similarities, which are popular similarity measures on permutations of n elements. We show that the Ulam similarity admits a sublinear LSH distortion of O(n/√{log n}); we also prove a lower bound of Ω(n^{0.12}) on the best LSH distortion achievable. On the other hand, we show that the LSH distortion of the Cayley similarity is Θ(n).
Cite as
Flavio Chierichetti, Mirko Giacchini, Ravi Kumar, and Erasmo Tani. On the LSH Distortion of Ulam and Cayley Similarities. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 59:1-59:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chierichetti_et_al:LIPIcs.APPROX/RANDOM.2026.59,
author = {Chierichetti, Flavio and Giacchini, Mirko and Kumar, Ravi and Tani, Erasmo},
title = {{On the LSH Distortion of Ulam and Cayley Similarities}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {59:1--59:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.59},
URN = {urn:nbn:de:0030-drops-277765},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.59},
annote = {Keywords: Locality-sensitive Hashing, Ulam metric, Cayley Metric, Distortion, Permutations, Representation Theory}
}
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Fine-Grained Complexity via Quantum Natural Proofs
Abstract
Buhrman, Patro, and Speelman [Buhrman et al., 2021] presented a framework of conjectures that together form a quantum analogue of the strong exponential-time hypothesis and its variants. They called it the QSETH framework. In this paper, using a notion of quantum natural proofs (built from natural proofs introduced by Razborov and Rudich), we show how part of the QSETH conjecture that requires properties to be "compression oblivious" can in many cases be replaced by assuming the existence of quantum-secure pseudorandom functions, a standard hardness assumption. Combined with techniques from Fourier analysis of Boolean functions, we show that properties such as parity and majority are compression oblivious for certain circuit class Λ if subexponentially secure quantum pseudorandom functions exist in Λ, answering an open question in [Buhrman et al., 2021].
Cite as
Yanlin Chen, Yilei Chen, Rajendra Kumar, Subhasree Patro, and Florian Speelman. Fine-Grained Complexity via Quantum Natural Proofs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 60:1-60:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2026.60,
author = {Chen, Yanlin and Chen, Yilei and Kumar, Rajendra and Patro, Subhasree and Speelman, Florian},
title = {{Fine-Grained Complexity via Quantum Natural Proofs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {60:1--60:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.60},
URN = {urn:nbn:de:0030-drops-277771},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.60},
annote = {Keywords: Fine-grained complexity, Quantum natural proofs, QSETH, Pseudorandom functions}
}
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Locality of Curve-Decoding and Improved Proximity Gaps
Abstract
Proximity gaps are a property of error correcting codes that arise in the study of Interactive Oracle Proofs (IOPs) and Succinct Non-interactive Arguments of Zero Knowledge (SNARKs). Informally, we say that a code C ⊂ Σⁿ exhibits a proximity gap (with respect to degree-𝓁 curves) if for any degree-𝓁 curve u(x) ∈ Σⁿ, either every point on u(x) is close to C, or else most of them are far from C.
Recent work [Goyal and Guruswami, 2025] has established near-optimal proximity gaps for many families of codes, including subspace design codes, as well as random ensembles like random linear codes, Reed-Solomon codes with random evaluation points, and Gallager’s ensemble of LDPC codes. However, the parameters for these latter randomized ensembles are worse than the parameters for subspace design codes, and degrade as the degree 𝓁 increases.
In this work, we obtain improved proximity gaps for random ensembles of codes, including random linear codes, Reed-Solomon codes with random evaluation points, and Gallager’s ensemble. Quantitatively, our results for these random ensembles match the results that [Goyal and Guruswami, 2025] attained for subspace design codes. In fact, our techniques are a black-box transference from subspace design codes: Any progress on subspace design codes will automatically lead to analogous progress for these random ensembles.
To obtain our results, we extend the Local Coordinate-wise Linear (LCL) property framework developed in [Levi et al., 2025; Brakensiek et al., 2025] to a row-span constrained version. This allows us to cast curve-decodability - a property that implies proximity gaps - directly as an (row-span constrained) LCL property, and make use of that machinery. In contrast, because curve-decodability is not obviously a (vanilla) LCL property, prior work had worked with a proxy property instead, leading to the aforementioned parameter losses. In addition, we extend the framework to also show an equivalence theorem for Gallager’s ensemble of random LDPC codes and random linear codes for our row-span constrained LCL properties.
Cite as
Rohan Goyal, Venkatesan Guruswami, Yihang Sun, and Mary Wootters. Locality of Curve-Decoding and Improved Proximity Gaps. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 61:1-61:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{goyal_et_al:LIPIcs.APPROX/RANDOM.2026.61,
author = {Goyal, Rohan and Guruswami, Venkatesan and Sun, Yihang and Wootters, Mary},
title = {{Locality of Curve-Decoding and Improved Proximity Gaps}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {61:1--61:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.61},
URN = {urn:nbn:de:0030-drops-277784},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.61},
annote = {Keywords: Proximity gaps, random codes, curve decoding, local properties}
}
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Learning and Testing Convex Functions
Abstract
We consider the problems of learning and testing real-valued convex functions over Gaussian space. Despite the extensive study of function convexity across mathematics, statistics, and computer science, its learnability and testability have largely been examined only in discrete or restricted settings - typically with respect to the Hamming distance, which is ill-suited for real-valued functions.
In contrast, we study these problems in high dimensions under the standard Gaussian measure, assuming sample access to the function and a mild smoothness condition, namely Lipschitzness. A smoothness assumption is natural and, in fact, necessary even in one dimension: without it, convexity cannot be inferred from finitely many samples. As our main results, we give:
- Learning Convex Functions: An agnostic proper learning algorithm for Lipschitz convex functions that achieves error ε using n^O(1/ε²) samples, together with a complementary lower bound of n^poly(1/ε) samples in the correlational statistical query (CSQ) model.
- Testing Convex Functions: A tolerant (two-sided) tester for convexity of Lipschitz functions with the same sample complexity (as a corollary of our learning result), and a one-sided tester (which never rejects convex functions) using O(√n/ε)ⁿ samples.
Cite as
Renato Ferreira Pinto Jr., Cassandra Marcussen, Elchanan Mossel, and Shivam Nadimpalli. Learning and Testing Convex Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 62:1-62:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{ferreirapintojr._et_al:LIPIcs.APPROX/RANDOM.2026.62,
author = {Ferreira Pinto Jr., Renato and Marcussen, Cassandra and Mossel, Elchanan and Nadimpalli, Shivam},
title = {{Learning and Testing Convex Functions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {62:1--62:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.62},
URN = {urn:nbn:de:0030-drops-277798},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.62},
annote = {Keywords: Property testing, learning theory, Gaussian distribution, convex functions}
}
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Mixing and Cutoff for the Systematic Scan Dynamics of the Mean-Field Ferromagnetic Potts Model
Abstract
We study the mixing time of the systematic scan dynamics for the q-state ferromagnetic Potts model on the n-vertex complete graph, known as the mean-field model. This Markov chain updates vertices sequentially according to a fixed predetermined order, in contrast to the Glauber dynamics which updates a uniformly random vertex at each step. Systematic scan dynamics are attractive in practice as they often demonstrate strong empirical performance. However, their theoretical analysis remains far less developed than that of the Glauber dynamics.
We take a step toward addressing this imbalance by showing that for every q ≥ 2 and β < β_s, where β_s is the metastability threshold associated with the onset of slow mixing for the Glauber dynamics, the systematic scan dynamics for the ferromagnetic mean-field Potts model mixes in Θ(log n) scans or, equivalently, in Θ(nlog n) single site updates. We in fact prove a sharper result; namely, that there exists a constant c(β,q) > 0 such that the mixing time is c(β,q)log n + Θ(1), which implies that the Markov chain exhibits the cutoff phenomenon, with the total variation distance to the stationary distribution dropping abruptly from nearly 1 to nearly 0 within a narrow Θ(1) time window. This result is tight in β as well since the dynamics mixes exponentially slowly for β > β_s. To the best of our knowledge, this is the first general cutoff result for the systematic scan dynamics in the context of spin systems. The result may also be of independent interest in the theory of Markov chains, since the systematic scan dynamics is both global and non-reversible, two settings in which cutoff remains poorly understood.
Cite as
Antonio Blanca and Md Tahmidur Rafid. Mixing and Cutoff for the Systematic Scan Dynamics of the Mean-Field Ferromagnetic Potts Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 63:1-63:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{blanca_et_al:LIPIcs.APPROX/RANDOM.2026.63,
author = {Blanca, Antonio and Rafid, Md Tahmidur},
title = {{Mixing and Cutoff for the Systematic Scan Dynamics of the Mean-Field Ferromagnetic Potts Model}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {63:1--63:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.63},
URN = {urn:nbn:de:0030-drops-277801},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.63},
annote = {Keywords: Markov chain, mixing times, systematic scan, Potts model}
}
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Lifting Polynomial Complexity Measures Using Error-Correcting Codes
Abstract
We describe a method that lifts an arbitrary polynomial f with sparsity s to a polynomial that requires a read-once oblivious algebraic program of width s for every variable order. To do so, we introduce a technique for constructing a gadget based on erasure codes over finite fields, where each variable in f is substituted with a monomial that encodes a codeword into the exponents of the monomial. Our construction represents the first use of error-correcting codes in the context of lifting. As an application, we consider factor complexity, which studies how much the complexity of a polynomial can increase under factorization. Our result allows us to lift any gap in sparsity to the same gap in width in a generic manner.
Cite as
Ivan Hu and Dieter van Melkebeek. Lifting Polynomial Complexity Measures Using Error-Correcting Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 64:1-64:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hu_et_al:LIPIcs.APPROX/RANDOM.2026.64,
author = {Hu, Ivan and van Melkebeek, Dieter},
title = {{Lifting Polynomial Complexity Measures Using Error-Correcting Codes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {64:1--64:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.64},
URN = {urn:nbn:de:0030-drops-277813},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.64},
annote = {Keywords: lifting, lower bound, error-correcting code, read-once oblivious algebraic branching program, polynomial factorization}
}
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Property Testing of Computational Networks
Abstract
In this paper we initiate the study of property testing of weighted computational networks viewed as computational devices. Our goal is to design property testing algorithms that for a given computational network with oracle access to the weights of the network, accept (with probability at least 2/3) any network that computes a certain function (or a function with a certain property) and reject (with probability at least 2/3) any network that is far from computing the function (or any function with the given property). We parameterize the notion of being far and want to reject networks that are (ε,δ)-far, which means that one needs to change an ε-fraction of the description of the network to obtain a network that computes a function that differs in at most a δ-fraction of inputs from the desired function (or any function with a given property).
To exemplify our framework, we present a case study involving simple neural Boolean networks with ReLU activation function. As a highlight, we demonstrate that for such networks, any near-constant function is testable in query complexity independent of the network’s size. We also show that a similar result cannot be achieved in a natural generalization of the distribution-free model to our setting, and also in a related vanilla testing model. While the analysis of neural Boolean networks with ReLU activation function is rigorous and technically demanding, we consider it a testament to the robustness of our framework and the depth of our findings.
Cite as
Artur Czumaj and Christian Sohler. Property Testing of Computational Networks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 65:1-65:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{czumaj_et_al:LIPIcs.APPROX/RANDOM.2026.65,
author = {Czumaj, Artur and Sohler, Christian},
title = {{Property Testing of Computational Networks}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {65:1--65:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.65},
URN = {urn:nbn:de:0030-drops-277827},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.65},
annote = {Keywords: Property Testing, Testing Computational Networks}
}
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On the Advantage of Adaptivity for Sampling with Cell Probes
Abstract
We construct an explicit distribution 𝐃 over {0,1}^N that exhibits an essentially optimal separation between adaptive and non-adaptive cell-probe sampling. The distribution can be sampled exactly when each output bit is allowed two adaptive probes to an arbitrarily long sequence of independent uniform symbols from [N]. In contrast, any non-adaptive sampler requires Ω̃(N) non-adaptive cell probes to generate a distribution with total variation distance less than 1-o(1) from 𝐃. This provides a 2-vs-Ω̃(N) separation for sampling with adaptive versus non-adaptive cell probes, improving upon the 2-vs-Ω̃(log N) separation of Yu and Zhan (ITCS '24) and the (log N)^O(1)-vs-N^Ω(1) separation of Alekseev, Göös, Myasnikov, Riazanov, and Sokolov (STOC '26).
Cite as
Farzan Byramji, Daniel M. Kane, Jackson Morris, and Anthony Ostuni. On the Advantage of Adaptivity for Sampling with Cell Probes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 66:1-66:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{byramji_et_al:LIPIcs.APPROX/RANDOM.2026.66,
author = {Byramji, Farzan and Kane, Daniel M. and Morris, Jackson and Ostuni, Anthony},
title = {{On the Advantage of Adaptivity for Sampling with Cell Probes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {66:1--66:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.66},
URN = {urn:nbn:de:0030-drops-277839},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.66},
annote = {Keywords: sampling lower bound, cell probe model, adaptive sampling}
}
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Fast List Recovery of Univariate Multiplicity Codes
Abstract
Recent work gave near-linear time algorithms for list decoding Folded Reed-Solomon codes and univariate multiplicity codes up to capacity in their natural parameter regimes. Unlike most known list decoding algorithms, these techniques appeared inherently tied to list decoding, and it was unclear whether they could be extended to list recovery in near-linear time.
In this work, we resolve this question by giving Õ(n)-time algorithms for list recovery of Folded Reed-Solomon codes and univariate multiplicity codes up to capacity, where n is the block length. Our algorithms build on the lattice-based framework of the prior work, augmented with a new technical ingredient: the construction of suitably structured lattices over the univariate polynomial ring that capture the list recovery problem for these codes.
Cite as
Rohan Goyal, Prahladh Harsha, Mrinal Kumar, and Ashutosh Shankar. Fast List Recovery of Univariate Multiplicity Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 67:1-67:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{goyal_et_al:LIPIcs.APPROX/RANDOM.2026.67,
author = {Goyal, Rohan and Harsha, Prahladh and Kumar, Mrinal and Shankar, Ashutosh},
title = {{Fast List Recovery of Univariate Multiplicity Codes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {67:1--67:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.67},
URN = {urn:nbn:de:0030-drops-277840},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.67},
annote = {Keywords: list-recovery, multiplicity-codes, near-linear-algorithm}
}
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Algorithmic Phase Transition for Large Independent Sets in Dense Hypergraphs
Abstract
We study the algorithmic tractability of finding large independent sets in dense random hypergraphs. In the sparse regime, much of the natural algorithms can be formulated within either the local or the low-degree polynomial (LDP) framework, and a rich literature has subsequently identified nearly sharp algorithmic thresholds within these classes by exploiting their stability. In the dense setting, however, the algorithmic paradigms are fundamentally different: they are online and thus need not be stable. Perhaps more crucially, even for the classical Erdős-Rényi random graph G(n,p), LDPs are conjectured to fail in the "easy" regime accessible to online algorithms, thereby challenging their viability for dense models.
Our focus is on two models: (i) finding large independent sets in dense r-uniform Erdős-Rényi hypergraphs, where each size-r hyperedge is present independently with probability p, and (ii) the more challenging problem of finding large γ-balanced independent sets in dense r-uniform r-partite hypergraphs, where the vertex set is the disjoint union V_1 ⊔ ⋯ ⊔ V_r with |V_i| = n for all i, each hyperedge in V_1× ⋯ × V_r is present independently with probability p, and the i-th coordinate of γ ∈ ℚ^r specifies the proportion of vertices from V_i in the independent set. For both models, we pinpoint the size of the largest independent set and design online algorithms that achieve a multiplicative approximation factor of r^{1/(r-1)} in the uniform and (max_i γ_i)^{-1/(r-1)} in the r-partite model. Furthermore, we establish matching algorithmic lower bounds, showing that these computational gaps are sharp: no online algorithms can breach these gaps.
Our results provide a detailed landscape for dense hypergraphs, thereby completing the picture for dense models in a manner parallel to the sparse counterparts developed recently. Our main technical contribution is twofold: a novel staged and bucketed greedy algorithm and a stopping-time argument tailored to the hypergraph and multipartite structure for algorithmic hardness, both of which may be of independent interest. The algorithms and proof techniques in the dense regime differ substantially from those in the sparse, yet the resulting computational gaps are remarkably analogous, pointing to a form of universality. More conceptually, our results corroborate a hypothesis from statistical mechanics linking glassy equilibrium to computational hardness: optimization problems become far more intricate in the presence of global constraints.
Cite as
Abhishek Dhawan, Nhi U. Dinh, Eren C. Kızıldağ, Neeladri Maitra, and Bayram A. Şahin. Algorithmic Phase Transition for Large Independent Sets in Dense Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 68:1-68:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{dhawan_et_al:LIPIcs.APPROX/RANDOM.2026.68,
author = {Dhawan, Abhishek and Dinh, Nhi U. and K{\i}z{\i}lda\u{g}, Eren C. and Maitra, Neeladri and \c{S}ahin, Bayram A.},
title = {{Algorithmic Phase Transition for Large Independent Sets in Dense Hypergraphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {68:1--68:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.68},
URN = {urn:nbn:de:0030-drops-277851},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.68},
annote = {Keywords: independent sets, random hypergraphs, online algorithms, overlap gap property, statistical-computational gap, Erd\H{o}s-R\'{e}nyi hypergraph}
}
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RANDOM
QMA Lower Bounds for Batch Verification via Approximate Degree
Abstract
We study batch verification in QMA query and communication complexity, where the goal is to understand how the resources needed to verify m copies of a Boolean function f depend on m. We give a general technique for proving lower bounds on the witness-query tradeoff needed to batch verify a function f in terms of its approximate degree. Applying this technique to an explicit family of DNF formulas f, we show that attempting to save even a constant factor on the witness length of the baseline approach to batch verifying f necessitates a large polynomial increase in the query cost. We also obtain new lower bounds on the QMA query complexity of read-once CNF formulas and on the surjectivity and k-element distinctness functions. Our lower bounds also lift to give communication analogs of these results.
Cite as
Mark Bun, Mandar Juvekar, and Samuel King. QMA Lower Bounds for Batch Verification via Approximate Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 69:1-69:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{bun_et_al:LIPIcs.APPROX/RANDOM.2026.69,
author = {Bun, Mark and Juvekar, Mandar and King, Samuel},
title = {{QMA Lower Bounds for Batch Verification via Approximate Degree}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {69:1--69:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.69},
URN = {urn:nbn:de:0030-drops-277861},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.69},
annote = {Keywords: QMA, approximate degree, query complexity, communication complexity, quantum computation}
}
Document
RANDOM
Good Locally Testable Codes with Small Alphabet and Small Query Size
Abstract
Ben-Sasson, Goldreich and Sudan [Ben-Sasson et al., 2003] showed that a binary error correcting code admitting a 2-query tester cannot be good, i.e., it cannot have both linear distance and constant rate. They also showed that there are no good codes if the alphabet is a finite field 𝔽, the code is 𝔽-linear, and the 2-query tester is 𝔽-linear. We show that those are essentially the only limitations on the existence of good locally testable codes (LTCs). That is, there are good 2-query LTCs on any alphabet with more than 2 letters, and good 3-query LTCs with a binary alphabet. Similarly, there are good 3-query 𝔽-linear LTCs, and for every 𝔽-vector space V of dimension greater than 1, there are good 2-query LTCs with alphabet V whose tester is 𝔽-linear. This completely solves, for every q ≥ 2 and alphabet (resp. 𝔽-vector space) Σ, the question of whether there is a good q-query LTC (resp. 𝔽-LTC) with alphabet Σ. Our proof builds on the recent good 2-query 𝔽-LTCs of the first author and Kaufman [First and Kaufman, 2024], by establishing a general method for reducing the alphabet size of a good low-query LTC.
Cite as
Uriya A. First and Stav Lazarovici. Good Locally Testable Codes with Small Alphabet and Small Query Size. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 70:1-70:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{first_et_al:LIPIcs.APPROX/RANDOM.2026.70,
author = {First, Uriya A. and Lazarovici, Stav},
title = {{Good Locally Testable Codes with Small Alphabet and Small Query Size}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {70:1--70:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.70},
URN = {urn:nbn:de:0030-drops-277877},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.70},
annote = {Keywords: error correcting code, locally testable code, property testing}
}
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RANDOM
Parallel Sampling from the Ising p-Spin Model
Abstract
We study the parallel complexity of sampling from the high-temperature Ising mixed p-spin Gibbs measure, a canonical instance of a mean-field spin glass on the hypercube {±1}ⁿ. We propose two different algorithms for this problem, corresponding to two different regimes of accuracy.
Our first algorithm is a parallel implementation of a Markov chain known as block dynamics, combined with an approximate rejection sampling step that uses an Ising model in a novel way as a proposal distribution to approximate the quadratic interaction terms of the p-spin Hamiltonian. For any ε > 0, this algorithm runs in n^{1/3} polylog(n/ε) parallel time with poly(n/ε) work, and outputs a sample whose law is ε-close to the p-spin measure in total variation distance.
Our second algorithm uses Picard iterations to parallelize the Algorithmic Stochastic Localization (ASL) process of El Alaoui, Montanari, and Sellke (2025), and for any ε > ε_n, takes polylog(n/ε) parallel time and poly(n/ε) work to produce a sample that is ε-close to the p-spin measure in the normalized 2-Wasserstein metric. Here, ε_n > 0 is a threshold that goes to 0 as n → ∞. Our result constitutes a doubly exponential improvement in the ε dependence of the runtime and an exponential improvement in the ε dependence of the total work when compared to naïve ASL, whose runtime scales as exp(poly(1/ε)).
Cite as
Nima Anari, Aniket Das, and Alireza Haqi. Parallel Sampling from the Ising p-Spin Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 71:1-71:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{anari_et_al:LIPIcs.APPROX/RANDOM.2026.71,
author = {Anari, Nima and Das, Aniket and Haqi, Alireza},
title = {{Parallel Sampling from the Ising p-Spin Model}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {71:1--71:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.71},
URN = {urn:nbn:de:0030-drops-277889},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.71},
annote = {Keywords: spin glasses, parallel sampling, Glauber dynamics, stochastic localization}
}
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RANDOM
Quantum Algorithms for Path and Cycle Containment Problems
Abstract
The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph H is present in an input graph G, has been the subject of considerable study. This interest stems not only from the natural and well-motivated formulation of these problems, but also from a flurry of novel quantum algorithmic techniques that were developed specifically to solve them. Notably, even for relatively simple subgraphs, such as paths and cycles, a complete understanding of their query complexities remains elusive.
In this work, we consider several variants of path- and cycle-containment problems in the adjacency matrix model, where we search for paths or cycles of constant length k ∈ O(1). We compare the settings where the graphs are directed or undirected, where the goal is to detect or find the existence of a path/cycle, and where the path/cycle we're looking for has length exactly k, or at most k. We also consider several promise versions of these problems, where we know beforehand that the input graph has a certain structure. We characterize the relative difficulty of these variants of the path- and cycle-containment problems, by relating them to one another using randomized reductions, and grouping them into several equivalence classes.
When we restrict our attention to path-containment problems, this implies a dichotomy result. Some of the path-containment problems can be solved using a linear number of queries, and all the others are equivalent to one another (and additionally to several cycle-containment problems as well) under randomized reductions and up to constant multiplicative overhead. For the latter equivalence class, we prove a novel quantum-walk-based algorithm that achieves query complexity Õ(n^{3/2-α_k}), where α_k ∈ Θ(c^{-k}) and c = √{3+√17}/2 ≈ 1.33, beating the previous best upper bound O(n^{3/2}) on its query complexity. We also provide a conditional lower bound based on the graph-collision problem, which implies that this equivalence class does not admit linear-query quantum algorithms unless graph collision admits an O(√n) query algorithm.
Cite as
Arjan Cornelissen, Amin Shiraz Gilani, and Subhasree Patro. Quantum Algorithms for Path and Cycle Containment Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 72:1-72:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{cornelissen_et_al:LIPIcs.APPROX/RANDOM.2026.72,
author = {Cornelissen, Arjan and Gilani, Amin Shiraz and Patro, Subhasree},
title = {{Quantum Algorithms for Path and Cycle Containment Problems}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {72:1--72:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.72},
URN = {urn:nbn:de:0030-drops-277892},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.72},
annote = {Keywords: Quantum algorithms, query complexity, graph problems, fine-grained reductions}
}
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RANDOM
Testing the Independent Set Property in Hypergraphs
Abstract
The optimal sample complexity of testing if an n-vertex graph has an independent set of size ρ n, or is ε-far from having an independent set of size ρ n, was established to be Õ(ρ³/ε²), in a notable result by Blais and Seth (SICOMP 2025). In contrast, for q-uniform hypergraphs, there is a significant gap between the best known upper and lower bounds, and there has been no progress on the problem for the last two decades. In this work, we prove a new upper bound of Õ(qρ^{2q-3}/{ε²(q-2)!²}) on the sample complexity of testing the ρ-independent set property. The previous best known upper bound was Õ(2^q q! ρ^{2q}/ε³), due to Langberg (RANDOM 2004). This establishes the optimal dependence on ε and gives an exponential improvement in the dependence on q. We prove our result via a new application of the hypergraph container method.
Cite as
Elena Grigorescu, Shreya Nasa, and Cameron Seth. Testing the Independent Set Property in Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 73:1-73:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{grigorescu_et_al:LIPIcs.APPROX/RANDOM.2026.73,
author = {Grigorescu, Elena and Nasa, Shreya and Seth, Cameron},
title = {{Testing the Independent Set Property in Hypergraphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {73:1--73:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.73},
URN = {urn:nbn:de:0030-drops-277907},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.73},
annote = {Keywords: independent set, property testing, hypergraph container method, hypergraph}
}
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RANDOM
Toward a KKL Theorem for Any HDX
Abstract
The KKL Theorem, a seminal result in boolean function analysis, characterizes the structure of low-influence (non-expanding) functions on the hypercube. While recent years have seen breakthrough results across a variety of areas relying on analogs of the KKL Theorem beyond the cube (e.g. on product spaces, Grassmann graphs), further progress has been inhibited by our poor understanding of the phenomenon across more general domains. Motivated in this context, Bafna, Hopkins, Kaufman, and Lovett (STOC 2022) and Gur, Lifshitz, and Liu (STOC 2022) proved a generalized KKL-type Theorem for spectral high dimensional expanders (HDX). Their results, however, remain highly restricted due to strong quantitative expansion requirements on the underlying complex.
In this work, we introduce a simple local-to-global method for analyzing low influence functions on simplicial complexes. Using this method we prove a local-to-global KKL-type Theorem: any simplicial complex whose links satisfy a KKL-Theorem also satisfies such a result globally. Building on Gotlib and Kaufman (RANDOM 2023), we also prove a weaker dimension-dependent KKL-type Theorem for simplicial complexes with any non-trivial (two-sided) expansion. As concrete applications of our framework, we give the first characterization of non-expanding functions on "combinatorial" HDX such as dense clique complexes and a corresponding Kruskal-Katona Theorem, as well as a small-set expansion theorem for the Ramanujan Complexes of Lubotzky, Samuels, and Vishne (EJC '05).
Cite as
Max Hopkins. Toward a KKL Theorem for Any HDX. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 74:1-74:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{hopkins:LIPIcs.APPROX/RANDOM.2026.74,
author = {Hopkins, Max},
title = {{Toward a KKL Theorem for Any HDX}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {74:1--74:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.74},
URN = {urn:nbn:de:0030-drops-277916},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.74},
annote = {Keywords: High Dimensional Expanders, Boolean Function Analysis}
}
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RANDOM
Improved Local Computation of Edge Orientation
Abstract
In this paper, we study the problem of orienting the edges of a graph G so that every vertex has bounded out-degree in the local computation algorithms (LCA) model, as defined by Rubinfeld et al. (ICS 2011). More specifically, given a query e ∈ E our algorithm returns the orientation of e such that with high constant probability (namely, at least 0.9) the out-degree of each vertex is bounded by r where r is a parameter. We provide such an upper bound for any r = Ω(arb(G)⋅log n) where arb(G) denotes the arboricity of G (we note that such orientation exists only when r = Ω(arb(G))). Our query complexity is Õ(n⋅arb(G)/r²) in the worst case and O(1) on expectation (over the vertices and the randomness of the algorithm). This generalizes the upper bound by Mitrović-Rubinfeld-Singhal (ESA 2024) that provided a similar upper bound only when r = Ω((arb(G)²⋅n))^{1/3}.
For r = Ω(arb(G)⋅log n), our algorithm also improves their weaker upper bound of Õ(n/r) queries for the special case where the input graph is a tree (whose arboricity is 1).
Our algorithm assigns each vertex a level derived from locally sampled neighborhoods combined through a staggered multi-scale recursion, inspired by the recent arboricity-approximation framework of Dai–Ghaffari–Portmann (FOCS 2025). The main novelty of our approach is that it reconstructs levels consistently across the graph while simultaneously respecting the out-degree bound and maintaining locality of computation.
Cite as
Reut Levi and Bar Rushkin. Improved Local Computation of Edge Orientation. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 75:1-75:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{levi_et_al:LIPIcs.APPROX/RANDOM.2026.75,
author = {Levi, Reut and Rushkin, Bar},
title = {{Improved Local Computation of Edge Orientation}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {75:1--75:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.75},
URN = {urn:nbn:de:0030-drops-277929},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.75},
annote = {Keywords: Local Algorithms, Sublinear-time Algorithms, Edge Orientation, Bounded Arboricity}
}
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RANDOM
High Probability Streaming Lower Bounds for F₂ Estimation
Abstract
Estimating the second frequency moment (F₂) of an underlying frequency vector is a fundamental problem in the streaming model. While recent work by Braverman and Zamir [STOC 2025] resolved the space complexity for constant failure probability in the insertion-only model, the optimal dependence on the failure parameter δ remained open.
We close this gap by proving a tight high-probability lower bound of Ω(1/ε² log(1/δ) log(ε√n) / log(1/δ)) for (1±ε)-approximate F₂ estimation. The key challenge is the failure of prior multi-scale direct sum arguments under noise sensitivity. We introduce a noise-robust communication primitive, Exam Mostly Set Disjointness, and prove an Ω(m/t log(1/δ)) one-way lower bound. Embedding this into a multi-scale reduction yields the correct log(1/δ) dependence.
We also give two complementary algorithms under natural structure assumptions. For streams with frequency bound B, we design a subsampling method using continuous F₀ tracking that replaces a log n factor with log B. For k-sparse streams, we develop a two-stage sketch using approximate Morris counters, replacing log n with log k and achieving a further log log m dependence on stream length.
Cite as
William Swartworth, David P. Woodruff, and Samson Zhou. High Probability Streaming Lower Bounds for F₂ Estimation. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 76:1-76:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{swartworth_et_al:LIPIcs.APPROX/RANDOM.2026.76,
author = {Swartworth, William and Woodruff, David P. and Zhou, Samson},
title = {{High Probability Streaming Lower Bounds for F₂ Estimation}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {76:1--76:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.76},
URN = {urn:nbn:de:0030-drops-277936},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.76},
annote = {Keywords: streaming algorithms, lower bounds, moment estimation}
}
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RANDOM
Approximate Cauchy-Schwarz Inequality and Improved Bounds for Sherali-Adams Refutation of Semirandom CSPs
Abstract
We formulate an approximate Cauchy-Schwarz inequality and show that it is satisfied by solutions to the Sherali-Adams linear programming hierarchy (interpreted as "pseudo-distributions"). As a consequence, we resolve a question left open by the work of O'Donnell and Schramm [Ryan O'Donnell and Tselil Schramm, 2019] that they had explicitly attributed to the lack of such an inequality.
A Cauchy-Schwarz inequality is exactly satisfied by pseudo-distributions satisfying the constraints of the sum-of-squares semidefinite programming hierarchy and already has scores of applications. However, the proof there requires global positive semidefiniteness. Our approximate version, on the other hand, relies only on local positive semidefiniteness satisfied by the Sherali-Adams pseudo-distributions. Our formulation loses an additive error that scales with the L1 norm of the coefficients of the constituent polynomials, and this loss is asymptotically tight. Our proof is elementary and relies on a simple sampling argument.
As an application, we resolve a question left open in the work of O'Donnell and Schramm that gives a trade-off between constraint density and the Sherali-Adams degree for refuting random constraint satisfaction problems. Specifically, for odd arity CSPs, we show that the constraint density requirement for a given degree can be improved by a polynomial factor in n. Along the way, we observe that by a simple extension, the results in their work extend to a more general semirandom setting.
Cite as
Pravesh K. Kothari and Andrew D. Lin. Approximate Cauchy-Schwarz Inequality and Improved Bounds for Sherali-Adams Refutation of Semirandom CSPs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 77:1-77:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{kothari_et_al:LIPIcs.APPROX/RANDOM.2026.77,
author = {Kothari, Pravesh K. and Lin, Andrew D.},
title = {{Approximate Cauchy-Schwarz Inequality and Improved Bounds for Sherali-Adams Refutation of Semirandom CSPs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {77:1--77:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.77},
URN = {urn:nbn:de:0030-drops-277946},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.77},
annote = {Keywords: Sherali-Adams hierarchy, constraint satisfaction problems, semirandom CSPs, refutation, Cauchy-Schwarz inequality}
}
Document
RANDOM
Local Algorithms and the Failure of Log-Depth Quantum Advantage on Sparse Random CSPs
Abstract
We construct and analyze a message-passing algorithm for random constraint satisfaction problems (CSPs) at large clause density, generalizing work of El Alaoui, Montanari, and Sellke for Maximum Cut [Alaoui et al., 2023] through a connection between random CSPs and mean-field Ising spin glasses [Alaoui et al., 2021; Jones et al., 2023]. For CSPs with even predicates, the algorithm asymptotically solves a stochastic optimal control problem dual to an extended Parisi variational principle. This gives an optimal fraction of satisfied constraints among algorithms obstructed by the branching overlap gap property of Huang and Sellke [Huang and Sellke, 2025], notably including the Quantum Approximate Optimization Algorithm and all quantum circuits on a bounded-degree architecture of up to ε ⋅ log n depth [Chou et al., 2022].
Cite as
Antares Chen, Neng Huang, and Kunal Marwaha. Local Algorithms and the Failure of Log-Depth Quantum Advantage on Sparse Random CSPs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 78:1-78:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)
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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2026.78,
author = {Chen, Antares and Huang, Neng and Marwaha, Kunal},
title = {{Local Algorithms and the Failure of Log-Depth Quantum Advantage on Sparse Random CSPs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
pages = {78:1--78:26},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-449-9},
ISSN = {1868-8969},
year = {2026},
volume = {392},
editor = {Singh, Mohit and Gur, Tom},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.78},
URN = {urn:nbn:de:0030-drops-277952},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2026.78},
annote = {Keywords: Random CSP, message-passing algorithm}
}