Volume
LIPIcs, Volume 353
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)
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- published at: 2025-09-15
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-397-3
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Complete Volume
LIPIcs, Volume 353, APPROX/RANDOM 2025, Complete Volume
Abstract
LIPIcs, Volume 353, APPROX/RANDOM 2025, Complete Volume
Cite as
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 1-1388, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@Proceedings{ene_et_al:LIPIcs.APPROX/RANDOM.2025,
title = {{LIPIcs, Volume 353, APPROX/RANDOM 2025, Complete Volume}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {1--1388},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025},
URN = {urn:nbn:de:0030-drops-247169},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025},
annote = {Keywords: LIPIcs, Volume 353, APPROX/RANDOM 2025, Complete Volume}
}
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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 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 0:i-0:xxiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ene_et_al:LIPIcs.APPROX/RANDOM.2025.0,
author = {Ene, Alina and Chattopadhyay, Eshan},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {0:i--0:xxiv},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.0},
URN = {urn:nbn:de:0030-drops-247158},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
APPROX
Approximation Schemes for Orienteering and Deadline TSP in Doubling Metrics
Abstract
In this paper we look at various extensions of the classic Traveling Salesman Problem (TSP) on graphs with bounded doubling dimension and bounded treewidth and present approximation schemes for them. Suppose we are given a weighted graph G = (V,E) with a start node s ∈ V, distances on the edges d:E → ℚ^+ and integer k. In k-stroll problem the goal is to find a path from s of minimum length that visits at least k vertices. In k-path we are given an additional end node t ∈ V and the path is supposed to go from s to t. The dual problem to k-stroll is the rooted orienteering in which instead of k we are given a budget B and the goal is to find a walk of length at most B starting at s that visits as many vertices as possible. In the point-to-point orienteering (P2P orienteering) we are given start and end nodes s,t and the walk is supposed to start at s and end at t. In the deadline TSP (which generalizes P2P orienteering) we are given a deadline D(v) for each v ∈ V and the goal is to find a walk starting at s that visits as many vertices as possible before their deadline (where the visit time of a node is the distance travelled from s to that node). The best approximation for rooted orienteering (or P2P orienteering) is (2+ε)-approximation [Chekuri et al., 2012] and O(log n)-approximation for deadline TSP [Nikhil Bansal et al., 2004]. For Euclidean metrics of fixed dimension, Chen and Har-Peled present [Chen and Har-Peled, 2008] a PTAS for rooted orienteering. There is no known approximation scheme for deadline TSP for any metric (not even trees). Our main result is the first approximation scheme for deadline TSP on metrics with bounded doubling dimension (which includes Euclidean metrics). To do so we first we present a quasi-polynomial time approximation scheme for k-path and P2P orienteering on such metrics. More specifically, if G is a metric with doubling dimension κ and aspect ratio Δ, we present a (1+ε)-approximation that runs in time n^{O((logΔ/ε) ^{2κ+1})}. Building upon these, we obtain an approximation scheme for deadline TSP when the distances and deadlines are integer which runs in time n^{O((log Δ/ε) ^{2κ+2})}. The same approach also implies a bicriteria (1+ε,1+ε)-approximation for deadline TSP for when distances and deadlines are in ℚ^+. For graphs with bounded treewidth ω we show how to solve k-path and P2P orienteering exactly in polynomial time and a (1+ε)-approximation for deadline TSP in time n^O((ωlogΔ/ε)²).
Cite as
Kinter Ren and Mohammad R. Salavatipour. Approximation Schemes for Orienteering and Deadline TSP in Doubling Metrics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 1:1-1:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ren_et_al:LIPIcs.APPROX/RANDOM.2025.1,
author = {Ren, Kinter and Salavatipour, Mohammad R.},
title = {{Approximation Schemes for Orienteering and Deadline TSP in Doubling Metrics}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {1:1--1:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.1},
URN = {urn:nbn:de:0030-drops-243678},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.1},
annote = {Keywords: Deadline Traveling Salesman Problem, Orienteering, Doubling Metrics, Approximation algorithm}
}
Document
APPROX
Covering Simple Orthogonal Polygons with Rectangles
Abstract
We study the problem of Covering Orthogonal Polygons with Rectangles, focusing on three variants: covering the interior, the boundary, and the corners. While previous work provided constant-factor approximation algorithms for these problems, significant improvements had not been achieved for over two decades. The main contribution of this work is the development of a Polynomial Time Approximation Scheme (PTAS) for both the Boundary Cover and Corner Cover problems on simple polygons, using a local search algorithm. Our work advances the state of the art, improving upon the previous best-known 4-approximation for the Boundary Cover and 2-approximation for the Corner Cover problems.
The technical core of our work lies in proving the existence of planar support graphs for certain geometric hypergraphs defined by the polygon and its containment-maximal rectangles. This structural insight enables the application of the local search framework to achieve the PTAS results. We also demonstrate the limitations of this approach by constructing instances where local search fails for the Interior Cover and certain dual problems, such as the Maximum Antirectangle and Hitting Set problems. Additionally, the methods yield a PTAS for a special case of the Discrete Independent Set problem for rectangles. These results not only settle longstanding open questions but also introduce new techniques that may be of independent interest within computational geometry.
Cite as
Aniket Basu Roy. Covering Simple Orthogonal Polygons with Rectangles. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 2:1-2:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{basuroy:LIPIcs.APPROX/RANDOM.2025.2,
author = {Basu Roy, Aniket},
title = {{Covering Simple Orthogonal Polygons with Rectangles}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {2:1--2:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.2},
URN = {urn:nbn:de:0030-drops-243686},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.2},
annote = {Keywords: Polygon Covering, Approximation Algorithms, Orthogonal Polygons, Rectangles, Local Search, Planar Supports}
}
Document
APPROX
Maximum And- vs. Even-SAT
Abstract
A multiset of literals, called a clause, is strongly satisfied by an assignment if no literal evaluates to false. Finding an assignment that maximises the number of strongly satisfied clauses is NP-hard. We present a simple algorithm that finds, given a multiset of clauses that admits an assignment that strongly satisfies ρ of the clauses, an assignment in which at least ρ of the clauses are weakly satisfied, in the sense that an even number of literals evaluate to false.
In particular, this implies an efficient algorithm for finding an undirected cut of value ρ in a graph G given that a directed cut of value ρ in G is promised to exist. A similar argument also gives an efficient algorithm for finding an acyclic subgraph of G with ρ edges under the same promise.
Cite as
Tamio-Vesa Nakajima and Stanislav Živný. Maximum And- vs. Even-SAT. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 3:1-3:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{nakajima_et_al:LIPIcs.APPROX/RANDOM.2025.3,
author = {Nakajima, Tamio-Vesa and \v{Z}ivn\'{y}, Stanislav},
title = {{Maximum And- vs. Even-SAT}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {3:1--3:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.3},
URN = {urn:nbn:de:0030-drops-243696},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.3},
annote = {Keywords: approximation, promise constraint satisfaction, max and, max even, max cut, max dicut, max acyclic}
}
Document
APPROX
Streaming Algorithms for Network Design
Abstract
We consider the Survivable Network Design problem (SNDP) in the single-pass insertion-only streaming model. The input to SNDP is an edge-weighted graph G = (V, E) and an integer connectivity requirement r(uv) for each u, v ∈ V. The objective is to find a minimum-weight subgraph H ⊆ G such that, for every pair of vertices u, v ∈ V, u and v are r(uv)-edge/vertex-connected. Recent work by [Ce Jin et al., 2024] obtained approximation algorithms for edge-connectivity augmentation, and via that, also derived algorithms for edge-connectivity SNDP (EC-SNDP). In this work we consider vertex-connectivity setting (VC-SNDP) and obtain several results for it as well as improved results for EC-SNDP.
- We provide a general framework for solving connectivity problems including SNDP and others in streaming; this is based on a connection to fault-tolerant spanners. For VC-SNDP we provide an O(tk)-approximation in Õ(k^{1-1/t}n^{1 + 1/t}) space, where k is the maximum connectivity requirement, assuming an exact algorithm at the end of the stream. Using a refined LP-based analysis, we provide an O(β t)-approximation where β is the integrality gap of the natural cut-based LP relaxation. These are the first approximation algorithms in the streaming model for VC-SNDP. When applied to the EC-SNDP, our framework provides an O(t)-approximation in Õ(k^{1/2-1/(2t)}n^{1 + 1/t} + kn) space, improving the O(t log k)-approximation of [Ce Jin et al., 2024] using Õ(kn^{1+1/t}) space; this also extends to element-connectivity SNDP.
- We consider vertex connectivity-augmentation in the link-arrival model. The input is a k-vertex-connected spanning subgraph G, and additional weighted links L arrive in the stream; the goal is to store the min-weight set of links such that G ∪ L is (k+1)-vertex-connected. We obtain constant-factor approximations in near-linear space for k = 1, 2. Our result for k = 2 is based on using the SPQR tree, a novel application for this well-known representation of 2-connected graphs.
Cite as
Chandra Chekuri, Rhea Jain, Sepideh Mahabadi, and Ali Vakilian. Streaming Algorithms for Network Design. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 4:1-4:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chekuri_et_al:LIPIcs.APPROX/RANDOM.2025.4,
author = {Chekuri, Chandra and Jain, Rhea and Mahabadi, Sepideh and Vakilian, Ali},
title = {{Streaming Algorithms for Network Design}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {4:1--4:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.4},
URN = {urn:nbn:de:0030-drops-243709},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.4},
annote = {Keywords: Streaming Algorithms, Survivable Network Design, Fault-Tolerant Spanners}
}
Document
APPROX
Min-CSPs on Complete Instances II: Polylogarithmic Approximation for Min-NAE-3-SAT
Abstract
This paper studies complete k-Constraint Satisfaction Problems (CSPs), where an n-variable instance has exactly one nontrivial constraint for each subset of k variables, i.e., it has binom(n,k) constraints. A recent work started a systematic study of complete k-CSPs [Anand, Lee, Sharma, SODA'25], and showed a quasi-polynomial time algorithm that decides if there is an assignment satisfying all the constraints of any complete Boolean-alphabet k-CSP, algorithmically separating complete instances from dense instances.
The tractability of this decision problem is necessary for any nontrivial (multiplicative) approximation for the minimization version, whose goal is to minimize the number of violated constraints. The same paper raised the question of whether it is possible to obtain nontrivial approximation algorithms for complete Min-k-CSPs with k ≥ 3.
In this work, we make progress in this direction and show a quasi-polynomial time polylog(n)-approximation to Min-NAE-3-SAT on complete instances, which asks to minimize the number of 3-clauses where all the three literals equal the same bit. To the best of our knowledge, this is the first known example of a CSP whose decision version is NP-Hard in general (and dense) instances while admitting a polylog(n)-approximation in complete instances. Our algorithm presents a new iterative framework for rounding a solution from the Sherali-Adams hierarchy, where each iteration interleaves the two well-known rounding tools: the conditioning procedure, in order to "almost fix" many variables, and the thresholding procedure, in order to "completely fix" them.
Finally, we improve the running time of the decision algorithms of Anand, Lee, and Sharma and show a simple algorithm that decides any complete Boolean-alphabet k-CSP in polynomial time.
Cite as
Aditya Anand, Euiwoong Lee, Davide Mazzali, and Amatya Sharma. Min-CSPs on Complete Instances II: Polylogarithmic Approximation for Min-NAE-3-SAT. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{anand_et_al:LIPIcs.APPROX/RANDOM.2025.5,
author = {Anand, Aditya and Lee, Euiwoong and Mazzali, Davide and Sharma, Amatya},
title = {{Min-CSPs on Complete Instances II: Polylogarithmic Approximation for Min-NAE-3-SAT}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {5:1--5:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.5},
URN = {urn:nbn:de:0030-drops-243712},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.5},
annote = {Keywords: Constraint Satisfiability Problems, Approximation Algorithms, Sherali Adams}
}
Document
APPROX
QSETH Strikes Again: Finer Quantum Lower Bounds for Lattice Problem, Strong Simulation, Hitting Set Problem, and More
Abstract
Despite the wide range of problems for which quantum computers offer a computational advantage over their classical counterparts, there are also many problems for which the best known quantum algorithm provides a speedup that is only quadratic, or even subquadratic. Such a situation could also be desirable if we don't want quantum computers to solve certain problems fast - say problems relevant to post-quantum cryptography. When searching for algorithms and when analyzing the security of cryptographic schemes, we would like to have evidence that these problems are difficult to solve on quantum computers; but how do we assess the exact complexity of these problems?
For most problems, there are no known ways to directly prove time lower bounds, however it can still be possible to relate the hardness of disparate problems to show conditional lower bounds. This approach has been popular in the classical community, and is being actively developed for the quantum case [Aaronson et al., 2020; Buhrman et al., 2021; Harry Buhrman et al., 2022; Andris Ambainis et al., 2022].
In this paper, by the use of the QSETH framework [Buhrman et al., 2021] we are able to understand the quantum complexity of a few natural variants of CNFSAT, such as parity-CNFSAT or counting-CNFSAT, and also are able to comment on the non-trivial complexity of approximate versions of counting-CNFSAT. Without considering such variants, the best quantum lower bounds will always be quadratically lower than the equivalent classical bounds, because of Grover’s algorithm; however, we are able to show that quantum algorithms will likely not attain even a quadratic speedup for many problems. These results have implications for the complexity of (variations of) lattice problems, the strong simulation and hitting set problems, and more. In the process, we explore the QSETH framework in greater detail and present a useful guide on how to effectively use the QSETH framework.
Cite as
Yanlin Chen, Yilei Chen, Rajendra Kumar, Subhasree Patro, and Florian Speelman. QSETH Strikes Again: Finer Quantum Lower Bounds for Lattice Problem, Strong Simulation, Hitting Set Problem, and More. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 6:1-6:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2025.6,
author = {Chen, Yanlin and Chen, Yilei and Kumar, Rajendra and Patro, Subhasree and Speelman, Florian},
title = {{QSETH Strikes Again: Finer Quantum Lower Bounds for Lattice Problem, Strong Simulation, Hitting Set Problem, and More}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {6:1--6:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.6},
URN = {urn:nbn:de:0030-drops-243723},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.6},
annote = {Keywords: Quantum conditional lower bounds, Fine-grained complexity, Lattice problems, Quantum strong simulation, Hitting set problem, QSETH}
}
Document
APPROX
Budget and Profit Approximations for Spanning Tree Interdiction
Abstract
We give polynomial time logarithmic approximation guarantees for the budget minimization, as well as for the profit maximization versions of minimum spanning tree interdiction. In this problem, the goal is to remove some edges of an undirected graph with edge weights and edge costs, so as to increase the weight of a minimum spanning tree. In the budget minimization version, the goal is to minimize the total cost of the removed edges, while achieving a desired increase Δ in the weight of the minimum spanning tree. An alternative objective within the same framework is to maximize the profit of interdiction, namely the increase in the weight of the minimum spanning tree, subject to a budget constraint. There are known polynomial time O(1) approximation guarantees for a similar objective (maximizing the total cost of the tree, rather than the increase). However, the guarantee does not seem to apply to the increase in cost. Moreover, the same techniques do not seem to apply to the budget version.
Our approximation guarantees are motivated by studying the question of minimizing the cost of increasing the minimum spanning tree by any amount. We show that in contrast to the budget and profit problems, this version of interdiction is polynomial time-solvable, and we give an efficient algorithm for solving it. The solution motivates a graph-theoretic relaxation of the NP-hard interdiction problem. The gain in minimum spanning tree weight, as a function of the set of removed edges, is super-modular. Thus, the budget problem is an instance of minimizing a linear function subject to a super-modular covering constraint. We use the graph-theoretic relaxation to design and analyze a batch greedy-based algorithm.
Cite as
Rafail Ostrovsky, Yuval Rabani, and Yoav Siman Tov. Budget and Profit Approximations for Spanning Tree Interdiction. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ostrovsky_et_al:LIPIcs.APPROX/RANDOM.2025.7,
author = {Ostrovsky, Rafail and Rabani, Yuval and Siman Tov, Yoav},
title = {{Budget and Profit Approximations for Spanning Tree Interdiction}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {7:1--7:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.7},
URN = {urn:nbn:de:0030-drops-243733},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.7},
annote = {Keywords: minimum spanning tree, spanning tree interdiction, combinatorial approximation algorithms, partial cut}
}
Document
APPROX
Multipass Linear Sketches for Geometric LP-Type Problems
Abstract
LP-type problems such as the Minimum Enclosing Ball (MEB), Linear Support Vector Machine (SVM), Linear Programming (LP), and Semidefinite Programming (SDP) are fundamental combinatorial optimization problems, with many important applications in machine learning applications such as classification, bioinformatics, and noisy learning. We study LP-type problems in several streaming and distributed big data models, giving ε-approximation linear sketching algorithms with a focus on the high accuracy regime with low dimensionality d, that is, when d < (1/ε)^0.999. Our main result is an O(ds) pass algorithm with O(s(√d/ε)^{3d/s}) ⋅ poly(d, log (1/ε)) space complexity in words, for any parameter s ∈ [1, d log (1/ε)], to solve ε-approximate LP-type problems of O(d) combinatorial and VC dimension. Notably, by taking s = d log (1/ε), we achieve space complexity polynomial in d and polylogarithmic in 1/ε, presenting exponential improvements in 1/ε over current algorithms. We complement our results by showing lower bounds of (1/ε)^Ω(d) for any 1-pass algorithm solving the (1 + ε)-approximation MEB and linear SVM problems, further motivating our multi-pass approach.
Cite as
N. Efe Çekirge, William Gay, and David P. Woodruff. Multipass Linear Sketches for Geometric LP-Type Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 8:1-8:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{cekirge_et_al:LIPIcs.APPROX/RANDOM.2025.8,
author = {\c{C}ekirge, N. Efe and Gay, William and Woodruff, David P.},
title = {{Multipass Linear Sketches for Geometric LP-Type Problems}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {8:1--8:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.8},
URN = {urn:nbn:de:0030-drops-243741},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.8},
annote = {Keywords: Streaming, sketching, LP-type problems}
}
Document
APPROX
Optimal Competitive Ratio for Optimization Problems with Congestion Effects
Abstract
In this work we study online optimization problems with congestion effects. These are problems where tasks arrive online and a decision maker is required to allocate them on the fly to available resources in order to minimize the cost suffered, which grows with the amount of resources used. This class of problems corresponds to the online counterpart of well-known studied problems, including optimization problems with diseconomies of scale [Konstantin Makarychev and Maxim Sviridenko, 2018], minimum cost in congestion games [Gairing and Paccagnan, 2023], and load balancing problems [Baruch Awerbuch et al., 1995].
Within this setting, our work settles the problem of designing online algorithms with optimal competitive ratio, i.e., algorithms whose incurred cost is as close as possible to that of an oracle with complete knowledge of the future instance ahead of time. We provide three contributions underpinning this result. First, we show that no online algorithm can achieve a competitive ratio below a given factor depending solely on the resource costs. Second, we show that, when guided by carefully modified cost functions, the greedy algorithm achieves a competitive ratio matching this lower bound and thus is optimal. Finally, we show how to compute such modified cost functions in polynomial time.
Cite as
Miriam Fischer, Dario Paccagnan, and Cosimo Vinci. Optimal Competitive Ratio for Optimization Problems with Congestion Effects. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 9:1-9:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{fischer_et_al:LIPIcs.APPROX/RANDOM.2025.9,
author = {Fischer, Miriam and Paccagnan, Dario and Vinci, Cosimo},
title = {{Optimal Competitive Ratio for Optimization Problems with Congestion Effects}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {9:1--9:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.9},
URN = {urn:nbn:de:0030-drops-243754},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.9},
annote = {Keywords: Online Algorithms, Competitive Ratio, Algorithmic Game Theory, Greedy Algorithms, Congestion Games}
}
Document
APPROX
Improved Approximation Guarantees for Advertisement Placement
Abstract
The advertisement placement problem involves selecting and scheduling ads within a timeline that has capacity constraints to maximize profit. Each task is characterized by its height, width, and profit, and must be fully scheduled across multiple time slots. This problem models practical scenarios such as internet advertising and energy management, and it also generalizes classical combinatorial optimization problems like the knapsack and bin packing problems.
We present a simple (2+ε)-approximation algorithm for any ε > 0, which improves upon the state-of-the-art 3+ε factor established by Freund and Naor twenty years ago. Our approach combines rounding techniques with dynamic programming and an efficient extension of list scheduling. Furthermore, we enhance this method with linear programming techniques to provide an almost optimal (1+ε)-approximation algorithm under resource augmentation, which allows for a slight increase in time slot capacities.
Cite as
Waldo Gálvez, Roberto Oliva, and Victor Verdugo. Improved Approximation Guarantees for Advertisement Placement. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{galvez_et_al:LIPIcs.APPROX/RANDOM.2025.10,
author = {G\'{a}lvez, Waldo and Oliva, Roberto and Verdugo, Victor},
title = {{Improved Approximation Guarantees for Advertisement Placement}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {10:1--10:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.10},
URN = {urn:nbn:de:0030-drops-243762},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.10},
annote = {Keywords: Advertisement Placement, Two-dimensional Packing, Geometric Knapsack, Resource Allocation}
}
Document
APPROX
On Finding Randomly Planted Cliques in Arbitrary Graphs
Abstract
We study a planted clique model introduced by Feige [Uriel Feige, 2021] where a complete graph of size c⋅ n is planted uniformly at random in an arbitrary n-vertex graph. We give a simple deterministic algorithm that, in almost linear time, recovers a clique of size (c/3)^O(1/c) ⋅ n as long as the original graph has maximum degree at most (1-p)n for some fixed p > 0. The proof hinges on showing that the degrees of the final graph are correlated with the planted clique, in a way similar to (but more intricate than) the classical G(n,1/2)+K_√n planted clique model. Our algorithm suggests a separation from the worst-case model, where, assuming the Unique Games Conjecture, no polynomial algorithm can find cliques of size Ω(n) for every fixed c > 0, even if the input graph has maximum degree (1-p)n. Our techniques extend beyond the planted clique model. For example, when the planted graph is a balanced biclique, we recover a balanced biclique of size larger than the best guarantees known for the worst case.
Cite as
Francesco Agrimonti, Marco Bressan, and Tommaso d'Orsi. On Finding Randomly Planted Cliques in Arbitrary Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{agrimonti_et_al:LIPIcs.APPROX/RANDOM.2025.11,
author = {Agrimonti, Francesco and Bressan, Marco and d'Orsi, Tommaso},
title = {{On Finding Randomly Planted Cliques in Arbitrary Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {11:1--11:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.11},
URN = {urn:nbn:de:0030-drops-243774},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.11},
annote = {Keywords: Computational Complexity, Planted Clique, Semi-random, Unique Games Conjecture, Approximation Algorithms}
}
Document
APPROX
Relational Approximations for Subspace Primitives
Abstract
We explore fundamental geometric computations on point sets that are given to the algorithm implicitly. In particular, we are given a database which is a collection of tables with numerical values, and the geometric computation is to be performed on the join of the tables. Explicitly computing this join takes time exponential in the size of the tables. We are therefore interested in geometric problems that can be solved by algorithms whose running time is a polynomial in the size of the tables. Such relational algorithms are typically not able to explicitly compute the join.
To avoid the NP-completeness bottleneck, researchers assume that the tables have a tractable combinatorial structure, like being acyclic. Even with this assumption, simple geometric computations turn out to be non-trivial and sometimes intractable. In this article, we study the problem of computing the maximum distance of a point in the join to a given subspace, and develop approximation algorithms for this NP-hard problem.
Cite as
Xiang Liu and Kasturi Varadarajan. Relational Approximations for Subspace Primitives. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{liu_et_al:LIPIcs.APPROX/RANDOM.2025.12,
author = {Liu, Xiang and Varadarajan, Kasturi},
title = {{Relational Approximations for Subspace Primitives}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {12:1--12:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.12},
URN = {urn:nbn:de:0030-drops-243781},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.12},
annote = {Keywords: relational algorithm, Euclidean distance, subspace approximation}
}
Document
APPROX
Max-Cut with Multiple Cardinality Constraints
Abstract
We study the classic Max-Cut problem under multiple cardinality constraints, which we refer to as the Constrained Max-Cut problem. Given a graph G = (V, E), a partition of the vertices into c disjoint parts V₁, …, V_c, and cardinality parameters k₁, …, k_c, the goal is to select a set S ⊆ V such that |S ∩ V_i| = k_i for each i ∈ [c], maximizing the total weight of edges crossing S (i.e., edges with exactly one endpoint in S).
By designing an approximate kernel for Constrained Max-Cut and building on the correlation rounding technique of Raghavendra and Tan (2012), we present a (0.858 - ε)-approximation algorithm for the problem when c = O(1). The algorithm runs in time O(min{k/ε, n}^poly(c/ε) + poly(n)), where k = ∑_{i∈[c]} k_i and n = |V|. This improves upon the (1/2 + ε₀)-approximation of Feige and Langberg (2001) for Max-Cut_k (the special case when c = 1, k₁ = k), and generalizes the (0.858 - ε)-approximation of Raghavendra and Tan (2012), which only applies when min{k,n-k} = Ω(n) and does not handle multiple constraints.
We also establish that, for general values of c, it is NP-hard to determine whether a feasible solution exists that cuts all edges. Finally, we present a 1/2-approximation algorithm for Max-Cut under an arbitrary matroid constraint.
Cite as
Yury Makarychev, Madhusudhan Reddy Pittu, and Ali Vakilian. Max-Cut with Multiple Cardinality Constraints. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 13:1-13:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{makarychev_et_al:LIPIcs.APPROX/RANDOM.2025.13,
author = {Makarychev, Yury and Pittu, Madhusudhan Reddy and Vakilian, Ali},
title = {{Max-Cut with Multiple Cardinality Constraints}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {13:1--13:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.13},
URN = {urn:nbn:de:0030-drops-243790},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.13},
annote = {Keywords: Maxcut, Semi-definite Programming, Sum of Squares Hierarchy}
}
Document
APPROX
Improved Lower Bounds on Multiflow-Multicut Gaps
Abstract
Given a set of source-sink pairs, the maximum multiflow problem asks for the maximum total amount of flow that can be feasibly routed between them. The minimum multicut, a dual problem to multiflow, seeks the minimum-cost set of edges whose removal disconnects all the source-sink pairs. It is easy to see that the value of the minimum multicut is at least that of the maximum multiflow, and their ratio is called the multiflow-multicut gap. The classical max-flow min-cut theorem states that when there is only one source-sink pair, the gap is exactly one. However, in general, it is well known that this gap can be arbitrarily large. In this paper, we study this gap for classes of planar graphs and establish improved lower bound results. In particular, we show that this gap is at least 20/9 for the class of planar graphs, improving upon the decades-old lower bound of 2. More importantly, we develop new techniques for proving such a lower bound, which may be useful in other settings as well.
Cite as
Sina Kalantarzadeh and Nikhil Kumar. Improved Lower Bounds on Multiflow-Multicut Gaps. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kalantarzadeh_et_al:LIPIcs.APPROX/RANDOM.2025.14,
author = {Kalantarzadeh, Sina and Kumar, Nikhil},
title = {{Improved Lower Bounds on Multiflow-Multicut Gaps}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {14:1--14:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.14},
URN = {urn:nbn:de:0030-drops-243803},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.14},
annote = {Keywords: Approximation Algorithms, Randomized Algorithms, Linear Programming, Graph Algorithms, Scheduling, Multicut, Multiflow}
}
Document
APPROX
A Polynomial-Time Approximation Algorithm for Complete Interval Minors
Abstract
As shown by Robertson and Seymour, deciding whether the complete graph K_t is a minor of an input graph G is a fixed parameter tractable problem when parameterized by t. From the approximation viewpoint, a substantial gap remains: there is no PTAS for finding the largest complete minor unless P = NP, whereas the best known result is a polytime O(√ n)-approximation algorithm by Alon, Lingas and Wahlén.
We investigate the complexity of finding K_t as interval minor in ordered graphs (i.e. graphs with a linear order on the vertices, in which intervals are contracted to form minors). Our main result is a polytime f(t)-approximation algorithm, where f is triply exponential in t but independent of n. The algorithm is based on delayed decompositions and shows that ordered graphs without a K_t interval minor can be constructed via a bounded number of three operations: closure under substitutions, edge union, and concatenation of a stable set. As a byproduct, graphs avoiding K_t as an interval minor have bounded chromatic number.
Cite as
Romain Bourneuf, Julien Cocquet, Chaoliang Tang, and Stéphan Thomassé. A Polynomial-Time Approximation Algorithm for Complete Interval Minors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 15:1-15:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bourneuf_et_al:LIPIcs.APPROX/RANDOM.2025.15,
author = {Bourneuf, Romain and Cocquet, Julien and Tang, Chaoliang and Thomass\'{e}, St\'{e}phan},
title = {{A Polynomial-Time Approximation Algorithm for Complete Interval Minors}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {15:1--15:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.15},
URN = {urn:nbn:de:0030-drops-243814},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.15},
annote = {Keywords: Approximation algorithm, Ordered graphs, Interval minors, Delayed decompositions}
}
Document
APPROX
Sparsest Cut and Eigenvalue Multiplicities on Low Degree Abelian Cayley Graphs
Abstract
Whether or not the Sparsest Cut problem admits an efficient O(1)-approximation algorithm is a fundamental algorithmic question with connections to geometry and the Unique Games Conjecture.
Revisiting spectral algorithms for Sparsest Cut, we present a novel, simple algorithm that combines eigenspace enumeration with a new algorithm for the Cut Improvement problem. The runtime of our algorithm is parametrized by a quantity that we call the solution dimension SD_ε(G): the smallest k such that the subspace spanned by the first k Laplacian eigenvectors contains all but ε fraction of a sparsest cut.
Our algorithm matches the guarantees of prior methods based on the threshold-rank paradigm, while also extending beyond them. To illustrate this, we study its performance on low degree Cayley graphs over Abelian groups - canonical examples of graphs with poor expansion properties.
We prove that low degree Abelian Cayley graphs have small solution dimension, yielding an algorithm that computes a (1+ε)-approximation to the uniform Sparsest Cut of a degree-d Cayley graph over an Abelian group of size n in time n^O(1) ⋅ exp{(d/ε)^O(d)}. Along the way to bounding the solution dimension of Abelian Cayley graphs, we analyze their sparse cuts and spectra, proving that the collection of O(1)-approximate sparsest cuts has an ε-net of size exp{(d/ε)^O(d)} and that the multiplicity of λ₂ is bounded by 2^O(d). The latter bound is tight and improves on a previous bound of 2^O(d²) by Lee and Makarychev.
Cite as
Tommaso d'Orsi, Chris Jones, Jake Ruotolo, Salil Vadhan, and Jiyu Zhang. Sparsest Cut and Eigenvalue Multiplicities on Low Degree Abelian Cayley Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 16:1-16:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{dorsi_et_al:LIPIcs.APPROX/RANDOM.2025.16,
author = {d'Orsi, Tommaso and Jones, Chris and Ruotolo, Jake and Vadhan, Salil and Zhang, Jiyu},
title = {{Sparsest Cut and Eigenvalue Multiplicities on Low Degree Abelian Cayley Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {16:1--16:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.16},
URN = {urn:nbn:de:0030-drops-243827},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.16},
annote = {Keywords: Sparsest Cut, Spectral Graph Theory, Cayley Graphs, Approximation Algorithms}
}
Document
APPROX
Spectral Refutations of Semirandom k-LIN over Larger Fields
Abstract
We study the problem of strongly refuting semirandom k-LIN(𝔽) instances: systems of k-sparse inhomogeneous linear equations over a finite field 𝔽. For the case of 𝔽 = 𝔽₂, this is the well-studied problem of refuting semirandom instances of k-XOR, where the works of [Venkatesan Guruswami et al., 2022; Jun-Ting Hsieh et al., 2023] establish a tight trade-off between runtime and clause density for refutation: for any choice of a parameter 𝓁, they give an n^{O(𝓁)}-time algorithm to certify that there is no assignment that can satisfy more than 1/2 + ε-fraction of constraints in a semirandom k-XOR instance, provided that the instance has O(n)⋅(n/𝓁)^{k/2 - 1} log n/ε⁴ constraints, and the work of [Pravesh K. Kothari et al., 2017] provides good evidence that this tight up to a polylog(n) factor via lower bounds for the Sum-of-Squares hierarchy. However, for larger fields, the only known results for this problem are established via black-box reductions to the case of 𝔽₂, resulting in a |𝔽|^{3k} gap between the current best upper and lower bounds.
In this paper, we give an algorithm for refuting semirandom k-LIN(𝔽) instances with the "correct" dependence on the field size |𝔽|. For any choice of a parameter 𝓁, our algorithm runs in (|𝔽|)^O(𝓁)-time and strongly refutes semirandom k-LIN(𝔽) instances with at least O(n) ⋅ (|𝔽^*| n/𝓁) ^{k/2 - 1} log(n|𝔽^*|)/ε⁴ constraints. We give good evidence that this dependence on the field size |𝔽| is optimal by proving a lower bound for the Sum-of-Squares hierarchy that matches this threshold up to a polylog(n |𝔽^*|) factor. Our results also extend beyond finite fields to the more general case of ℤ_m and arbitrary finite Abelian groups. Our key technical innovation is a generalization of the "𝔽₂ Kikuchi matrices" of [Alexander S. Wein et al., 2019; Venkatesan Guruswami et al., 2022] to larger fields, and finite Abelian groups more generally.
Cite as
Nicholas Kocurek and Peter Manohar. Spectral Refutations of Semirandom k-LIN over Larger Fields. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kocurek_et_al:LIPIcs.APPROX/RANDOM.2025.17,
author = {Kocurek, Nicholas and Manohar, Peter},
title = {{Spectral Refutations of Semirandom k-LIN over Larger Fields}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {17:1--17:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.17},
URN = {urn:nbn:de:0030-drops-243834},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.17},
annote = {Keywords: Spectral Algorithms, CSP Refutation, Kikuchi Matrices}
}
Document
APPROX
A Randomized Rounding Approach for DAG Edge Deletion
Abstract
In the DAG Edge Deletion problem, we are given an edge-weighted directed acyclic graph and a parameter k, and the goal is to delete the minimum weight set of edges so that the resulting graph has no paths of length k. This problem, which has applications to scheduling, was introduced in 2015 by Kenkre, Pandit, Purohit, and Saket. They gave a k-approximation and showed that it is UGC-Hard to approximate better than ⌊0.5k⌋ for any constant k ≥ 4 using a work of Svensson from 2012. The approximation ratio was improved to 2/3(k+1) by Klein and Wexler in 2016.
In this work, we introduce a randomized rounding framework based on distributions over vertex labels in [0,1]. The most natural distribution is to sample labels independently from the uniform distribution over [0,1]. We show this leads to a (2-√2)(k+1) ≈ 0.585(k+1)-approximation. By using a modified (but still independent) label distribution, we obtain a 0.549(k+1)-approximation for the problem, as well as show that no independent distribution over labels can improve our analysis to below 0.542(k+1). Finally, we show a 0.5(k+1)-approximation for bipartite graphs and for instances with structured LP solutions. Whether this ratio can be obtained in general is open.
Cite as
Sina Kalantarzadeh, Nathan Klein, and Victor Reis. A Randomized Rounding Approach for DAG Edge Deletion. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kalantarzadeh_et_al:LIPIcs.APPROX/RANDOM.2025.18,
author = {Kalantarzadeh, Sina and Klein, Nathan and Reis, Victor},
title = {{A Randomized Rounding Approach for DAG Edge Deletion}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {18:1--18:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.18},
URN = {urn:nbn:de:0030-drops-243840},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.18},
annote = {Keywords: Approximation Algorithms, Randomized Algorithms, Linear Programming, Graph Algorithms, Scheduling}
}
Document
APPROX
On the Constant-Factor Approximability of Minimum Cost Constraint Satisfaction Problems
Abstract
We study minimum cost constraint satisfaction problems (MinCostCSP) through the algebraic lens. We show that for any constraint language Γ which has the dual discriminator operation as a polymorphism, there exists a |D|-approximation algorithm for MinCostCSP(Γ) where D is the domain. Complementing our algorithmic result, we show that any constraint language Γ where MinCostCSP(Γ) admits a constant-factor approximation must have a near-unanimity (NU) polymorphism unless P = NP, extending a similar result by Dalmau et al. on MinCSPs. These results imply a dichotomy of constant-factor approximability for constraint languages that contain all permutation relations (a natural generalization for Boolean CSPs that allow variable negation): either MinCostCSP(Γ) has an NU polymorphism and is |D|-approximable, or it does not have any NU polymorphism and is NP-hard to approximate within any constant factor. Finally, we present a constraint language which has a majority polymorphism, but is nonetheless NP-hard to approximate within any constant factor assuming the Unique Games Conjecture, showing that the condition of having an NU polymorphism is in general not sufficient unless UGC fails.
Cite as
Ian DeHaan, Neng Huang, and Euiwoong Lee. On the Constant-Factor Approximability of Minimum Cost Constraint Satisfaction Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{dehaan_et_al:LIPIcs.APPROX/RANDOM.2025.19,
author = {DeHaan, Ian and Huang, Neng and Lee, Euiwoong},
title = {{On the Constant-Factor Approximability of Minimum Cost Constraint Satisfaction Problems}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {19:1--19:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.19},
URN = {urn:nbn:de:0030-drops-243851},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.19},
annote = {Keywords: Constraint satisfaction problems, approximation algorithms, polymorphisms}
}
Document
APPROX
Approximating Maximum Cut on Interval Graphs and Split Graphs Beyond Goemans-Williamson
Abstract
We present a polynomial-time (α_{GW} + ε)-approximation algorithm for the Maximum Cut problem on interval graphs and split graphs, where α_{GW} ≈ 0.878 is the approximation guarantee of the Goemans-Williamson algorithm and ε > 10^{-34} is a fixed constant. To attain this, we give an improved analysis of a slight modification of the Goemans-Williamson algorithm for graphs in which triangles can be packed into a constant fraction of their edges. We then pair this analysis with structural results showing that both interval graphs and split graphs either have such a triangle packing or have maximum cut close to their number of edges. We also show that, subject to the Small Set Expansion Hypothesis, there exists a constant c > 0 such that there is no polyomial-time (1 - c)-approximation for Maximum Cut on split graphs.
Cite as
Jungho Ahn, Ian DeHaan, Eun Jung Kim, and Euiwoong Lee. Approximating Maximum Cut on Interval Graphs and Split Graphs Beyond Goemans-Williamson. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 20:1-20:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ahn_et_al:LIPIcs.APPROX/RANDOM.2025.20,
author = {Ahn, Jungho and DeHaan, Ian and Kim, Eun Jung and Lee, Euiwoong},
title = {{Approximating Maximum Cut on Interval Graphs and Split Graphs Beyond Goemans-Williamson}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {20:1--20:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.20},
URN = {urn:nbn:de:0030-drops-243869},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.20},
annote = {Keywords: Maximum cut, graph theory, interval graphs, split graphs}
}
Document
APPROX
Dual Charging for Half-Integral TSP
Abstract
In this extended abstract, we show that the max entropy algorithm is a randomized 1.49776 approximation for half-integral TSP, improving upon the previous known bound of 1.49993 from Karlin et al. This also improves upon the best-known approximation for half-integral TSP due to Gupta et al. Our improvement results from using the dual, instead of the primal, to analyze the expected cost of the matching. We believe this method of analysis could lead to a simpler proof that max entropy is a better-than-3/2 approximation in the general case.
Cite as
Nathan Klein and Mehrshad Taziki. Dual Charging for Half-Integral TSP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 21:1-21:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{klein_et_al:LIPIcs.APPROX/RANDOM.2025.21,
author = {Klein, Nathan and Taziki, Mehrshad},
title = {{Dual Charging for Half-Integral TSP}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {21:1--21:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.21},
URN = {urn:nbn:de:0030-drops-243879},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.21},
annote = {Keywords: Approximation Algorithms, Graph Algorithms, Randomized Rounding, Linear Programming}
}
Document
APPROX
Directed Buy-At-Bulk Spanners
Abstract
We present a framework that unifies directed buy-at-bulk network design and directed spanner problems, namely, buy-at-bulk spanners. The goal is to find a minimum-cost routing solution for network design problems that captures economies at scale, while satisfying demands and distance constraints for terminal pairs. A more restricted version of this problem was shown to be O(2^{log^{1-ε} n})-hard to approximate, where n is the number of vertices, under a standard complexity assumption, by Elkin and Peleg (Theory of Computing Systems, 2007). Our results for buy-at-bulk spanners are the following.
- When the edge lengths are integral with magnitude polynomial in n we present:
1) An Õ(n^{4/5 + ε})-approximation polynomial-time randomized algorithm for uniform demands.
2) An Õ(k^{1/2 + ε})-approximation polynomial-time randomized algorithm for general demands, where k is the number of terminal pairs. This can be improved to an Õ(k^{ε})-approximation algorithm for the single-source problem. The same approximation ratios hold in the online setting.
- When the edge lengths are rational and well-conditioned, we present an Õ(k^{1/2 + ε})-approximation polynomial-time randomized algorithm that may slightly violate the distance constraints. The result can be improved to an Õ(k^ε)-approximation algorithm for the single-source problem. The same approximation ratios hold for the online setting when the condition number is given in advance.
To the best of our knowledge, these are the first sublinear factor approximation algorithms for directed buy-at-bulk spanners. We allow the edge lengths to be negative and the demands to be non-unit, unlike the previous literature. Our approximation ratios match the state-of-the-art ratios in special cases, namely, buy-at-bulk network design by Antonakopoulos (WAOA, 2010) and (online) weighted spanners by Grigorescu, Kumar, and Lin (APPROX 2023). Furthermore, we improve the competitive ratio for online buy-at-bulk by Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP, 2018) by a factor of log R, where R is the ratio between the maximum demand and the minimum demand.
Cite as
Elena Grigorescu, Nithish Kumar, and Young-San Lin. Directed Buy-At-Bulk Spanners. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 22:1-22:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{grigorescu_et_al:LIPIcs.APPROX/RANDOM.2025.22,
author = {Grigorescu, Elena and Kumar, Nithish and Lin, Young-San},
title = {{Directed Buy-At-Bulk Spanners}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {22:1--22:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.22},
URN = {urn:nbn:de:0030-drops-243885},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.22},
annote = {Keywords: buy-at-bulk spanners, minimum density junction tree, resource constrained shortest path}
}
Document
APPROX
Improved FPT Approximation for Sum of Radii Clustering with Mergeable Constraints
Abstract
In this work, we study k-min-sum-of-radii (k-MSR) clustering under mergeable constraints. k-MSR seeks to group data points using a set of up to k balls, such that the sum of the radii of the balls is minimized. A clustering constraint is called mergeable if merging two clusters satisfying the constraint, results in a cluster that also satisfies the constraint. Many popularly studied constraints are mergeable, including fairness constraints and lower bound constraints.
In our work, we design a (4+ε)-approximation for k-MSR under any given mergeable constraint with runtime 2^{O(k/(ε)⋅log²k/ε)} n⁴, i.e., fixed-parameter tractable in k for constant ε. Our result directly improves upon the FPT (6+ε)-approximation by Carta et al. [Carta et al., 2024]. We also provide a hardness result that excludes the exact solvability of k-MSR under any given mergeable constraint in time f(k)n^o(k), assuming ETH is true.
Cite as
Sayan Bandyapadhyay and Tianzhi Chen. Improved FPT Approximation for Sum of Radii Clustering with Mergeable Constraints. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bandyapadhyay_et_al:LIPIcs.APPROX/RANDOM.2025.23,
author = {Bandyapadhyay, Sayan and Chen, Tianzhi},
title = {{Improved FPT Approximation for Sum of Radii Clustering with Mergeable Constraints}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {23:1--23:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.23},
URN = {urn:nbn:de:0030-drops-243894},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.23},
annote = {Keywords: sum-of-radii clustering, mergeable constraints, approximation algorithm}
}
Document
APPROX
Improved Approximation Algorithms for the EPR Hamiltonian
Abstract
The EPR Hamiltonian is a family of 2-local quantum Hamiltonians introduced by King [King, 2023]. We introduce a polynomial time (1+√5)/4≈0.809-approximation algorithm for the problem of computing the ground energy of the EPR Hamiltonian, improving upon the previous state of the art of 0.72 [Jorquera et al., 2024].
Cite as
Nathan Ju and Ansh Nagda. Improved Approximation Algorithms for the EPR Hamiltonian. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 24:1-24:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ju_et_al:LIPIcs.APPROX/RANDOM.2025.24,
author = {Ju, Nathan and Nagda, Ansh},
title = {{Improved Approximation Algorithms for the EPR Hamiltonian}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {24:1--24:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.24},
URN = {urn:nbn:de:0030-drops-243909},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.24},
annote = {Keywords: Approximation Algorithms, Quantum Local Hamiltonian}
}
Document
APPROX
Covering a Few Submodular Constraints and Applications
Abstract
We consider the problem of covering multiple submodular constraints. Given a finite ground set N, a cost function c: N → ℝ_+, r monotone submodular functions f_1,f_2,…,f_r over N and requirements b_1,b_2,…,b_r the goal is to find a minimum cost subset S ⊆ N such that f_i(S) ≥ b_i for 1 ≤ i ≤ r. When r = 1 this is the well-known Submodular Set Cover problem. Previous work [Chekuri et al., 2022] considered the setting when r is large and developed bi-criteria approximation algorithms, and approximation algorithms for the important special case when each f_i is a weighted coverage function. These are fairly general models and capture several concrete and interesting problems as special cases. The approximation ratios for these problem are at least Ω(log r) which is unavoidable when r is part of the input. In this paper, motivated by some recent applications, we consider the problem when r is a fixed constant and obtain two main results. When the f_i are weighted coverage functions from a deletion-closed set system we obtain a (1+ε)(e/(e-1))(1+β)-approximation where β is the approximation ratio for the underlying set cover instances via the natural LP. Second, for covering multiple submodular constraints we obtain a randomized bi-criteria approximation algorithm that for any given integer α ≥ 1 outputs a set S such that f_i(S) ≥ (1-1/e^α-ε)b_i for each i ∈ [r] and 𝔼[c(S)] ≤ (1+ε)α ⋅ OPT. These results show that one can obtain nearly as good an approximation for any fixed r as what one would achieve for r = 1. We also demonstrate applications of our results to implicit covering problems such as fair facility location.
Cite as
Tanvi Bajpai, Chandra Chekuri, and Pooja Kulkarni. Covering a Few Submodular Constraints and Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{bajpai_et_al:LIPIcs.APPROX/RANDOM.2025.25,
author = {Bajpai, Tanvi and Chekuri, Chandra and Kulkarni, Pooja},
title = {{Covering a Few Submodular Constraints and Applications}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {25:1--25:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.25},
URN = {urn:nbn:de:0030-drops-243917},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.25},
annote = {Keywords: covering, linear programming, rounding, fairness}
}
Document
APPROX
Non-Adaptive Evaluation of k-of- n Functions: Tight Gap and a Unit-Cost PTAS
Abstract
We consider the Stochastic Boolean Function Evaluation (SBFE) problem in the well-studied case of k-of-n functions: There are independent Boolean random variables x_1,… ,x_n where each variable i has a known probability p_i of taking value 1, and a known cost c_i that can be paid to find out its value. The value of the function is 1 iff there are at least k 1s among the variables. The goal is to efficiently compute a strategy that, at minimum expected cost, tests the variables until the function value is determined. While an elegant polynomial-time exact algorithm is known when tests can be made adaptively, we focus on the non-adaptive variant, for which much less is known.
First, we show a clean and tight lower bound of 2 on the adaptivity gap, i.e., the worst-case multiplicative loss in the objective function caused by disallowing adaptivity, of the problem. This improves the tight lower bound of 3/2 for the unit-cost variant.
Second, we give a PTAS for computing the best non-adaptive strategy in the unit-cost case, the first PTAS for an SBFE problem. At the core, our scheme establishes a novel notion of two-sided dominance (w.r.t. the optimal solution) by guessing so-called milestone tests for a set of carefully chosen buckets of tests. To turn this technique into a polynomial-time algorithm, we use a decomposition approach paired with a random-shift argument.
Cite as
Mads Anker Nielsen, Lars Rohwedder, and Kevin Schewior. Non-Adaptive Evaluation of k-of- n Functions: Tight Gap and a Unit-Cost PTAS. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{nielsen_et_al:LIPIcs.APPROX/RANDOM.2025.26,
author = {Nielsen, Mads Anker and Rohwedder, Lars and Schewior, Kevin},
title = {{Non-Adaptive Evaluation of k-of- n Functions: Tight Gap and a Unit-Cost PTAS}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {26:1--26:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.26},
URN = {urn:nbn:de:0030-drops-243920},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.26},
annote = {Keywords: Approximation scheme, Boolean functions, stochastic combinatorial optimization, stochastic function evaluation, sequential testing, adaptivity}
}
Document
APPROX
Triangles Improve 0.878 Approximation for Maxcut
Abstract
Maxcut is a fundamental problem in graph algorithms, extensively studied for its theoretical and practical significance. The goal is to partition the vertex set of a graph G = (V, E) into disjoint subsets S and V⧵S so as to maximize the number of edges crossing the cut (S,V⧵S). The seminal work of Goemans and Williamson [Goemans and Williamson, 1995] introduced a semidefinite programming (SDP) based algorithm achieving a α_{GW} ≈ 0.87856-approximation for general graphs, guaranteed to be optimal under the Unique Games Conjecture [Khot, 2002; Khot et al., 2007].
We revisit the Goemans–Williamson SDP and prove that the standard Maxcut SDP achieves a (α_{GW} + Ω(1))-approximation whenever the input graph contains Ω(|E|) edge-disjoint triangles. Our analysis builds on classical rounding techniques studied in [Goemans and Williamson, 1995; Zwick, 1999] and introduces a refined understanding of the SDP solution structure in regimes where the previous guarantees are tight. Our result identifies a simple combinatorial property that may be satisfied by many natural graph classes.
As applications, we show that unit ball graphs and graphs satisfying a spectral transitivity condition (as studied in [Gupta et al., 2016; Basu et al., 2024]) meet our structural criterion, and therefore we get better than α_{GW} approximation guarantees for them. Our algorithm runs in nearly linear time 𝒪̃(|E|), offering a more practical alternative to the PTAS of [Jansen et al., 2005] for unit ball graphs, which has exponential dependence on the approximation parameter.
Cite as
Fredie George, Anand Louis, and Rameesh Paul. Triangles Improve 0.878 Approximation for Maxcut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 27:1-27:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{george_et_al:LIPIcs.APPROX/RANDOM.2025.27,
author = {George, Fredie and Louis, Anand and Paul, Rameesh},
title = {{Triangles Improve 0.878 Approximation for Maxcut}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {27:1--27:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.27},
URN = {urn:nbn:de:0030-drops-243931},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.27},
annote = {Keywords: Approximation Algorithms, Maxcut, Semidefinite Programming, Edge-disjoint Triangles, Unit Ball Graphs, Spectral Triadic Graphs}
}
Document
RANDOM
Implications of Better PRGs for Permutation Branching Programs
Abstract
We study the challenge of derandomizing constant-width standard-order read-once branching programs (ROBPs). Let c ∈ [1, 2) be any constant. We prove that if there are explicit pseudorandom generators (PRGs) for width-6 length-n permutation ROBPs with error 1/n and seed length Õ(log^c n), then there are explicit hitting set generators (HSGs) for width-4 length-n ROBPs with threshold 1/polylog(n) and seed length Õ(log^c n).
For context, there are known explicit PRGs that fool constant-width permutation ROBPs with error ε and seed length O(log(n)⋅log(1/ε)) (Koucký, Nimbhorkar, and Pudlák STOC 2011; De CCC 2011; Steinke ECCC 2012). When ε = 1/n, there are known constructions of weighted pseudorandom generators (WPRGs) that fool polynomial-width permutation ROBPs with seed length Õ(log^{3/2} n) (Pyne and Vadhan CCC 2021; Chen, Hoza, Lyu, Tal, and Wu FOCS 2023; Chattopadhyay and Liao ITCS 2024), but unweighted PRGs with seed length o(log² n) remain elusive. Meanwhile, for width-4 ROBPs, there are no known explicit PRGs, WPRGs, or HSGs with seed length o(log²n).
Our reduction can be divided into two parts. First, we show that explicit low-error PRGs for width-6 permutation ROBPs with seed length Õ(log^c n) would imply explicit low-error PRGs for width-3 ROBPs with seed length Õ(log^c n). This would improve Meka, Reingold, and Tal’s PRG (STOC 2019), which has seed length o(log²n) only when the error parameter is relatively large. Second, we show that for any w, n, s, and ε, an explicit PRG for width-w ROBPs with error 0.01/n and seed length s would imply an explicit ε-HSG for width-(w + 1) ROBPs with seed length O(s + log(n)⋅log(1/ε)).
Cite as
Dean Doron and William M. Hoza. Implications of Better PRGs for Permutation Branching Programs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2025.28,
author = {Doron, Dean and Hoza, William M.},
title = {{Implications of Better PRGs for Permutation Branching Programs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {28:1--28:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.28},
URN = {urn:nbn:de:0030-drops-243946},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.28},
annote = {Keywords: hitting set generators, pseudorandom generators, read-once branching programs}
}
Document
RANDOM
A Simplified Reduction for Error Correcting Matrix Multiplication Algorithms
Abstract
We study the problem of transforming an algorithm for matrix multiplication, whose output has a small fraction of the entries correct into a matrix multiplication algorithm, whose output is fully correct for all inputs. In this work, we provide a new and simple way to transform an average-case algorithm that takes two matrices A,B ∈ 𝔽_p^{n×n} for a prime p, and outputs a matrix that agrees with the matrix product AB on a 1/p + ε fraction of entries on average for a small ε > 0, into a worst-case algorithm that correctly computes the matrix product for all possible inputs.
Our reduction employs list-decodable codes to transform an average-case algorithm into an algorithm with one-sided error, which are known to admit efficient reductions from the work of Gola, Shinkar, and Singh [Gola et al., 2024]. Our reduction is more concise and straightforward compared to the recent work of Hirahara and Shimizu [Hirahara and Shimizu, 2025], and improves the overhead in the running time incurred during the reduction.
Cite as
Igor Shinkar and Harsimran Singh. A Simplified Reduction for Error Correcting Matrix Multiplication Algorithms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{shinkar_et_al:LIPIcs.APPROX/RANDOM.2025.29,
author = {Shinkar, Igor and Singh, Harsimran},
title = {{A Simplified Reduction for Error Correcting Matrix Multiplication Algorithms}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {29:1--29:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.29},
URN = {urn:nbn:de:0030-drops-243953},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.29},
annote = {Keywords: Matrix Multiplication, Reductions, Worst case to average case reductions}
}
Document
RANDOM
Algorithmic Contiguity from Low-Degree Conjecture and Applications in Correlated Random Graphs
Abstract
In this paper, assuming a natural strengthening of the low-degree conjecture, we provide evidence of computational hardness for two problems: (1) the (partial) matching recovery problem in the sparse correlated Erdős-Rényi graphs G(n,q;ρ) when the edge-density q = n^{-1+o(1)} and the correlation ρ < √{α} lies below the Otter’s threshold, this resolves a remaining problem in [Jian Ding et al., 2023]; (2) the detection problem between a pair of correlated sparse stochastic block model S(n,λ/n;k,ε;s) and a pair of independent stochastic block models S(n,λs/n;k,ε) when ε² λ s < 1 lies below the Kesten-Stigum (KS) threshold and s < √α lies below the Otter’s threshold, this resolves a remaining problem in [Guanyi Chen et al., 2024].
One of the main ingredient in our proof is to derive certain forms of algorithmic contiguity between two probability measures based on bounds on their low-degree advantage. To be more precise, consider the high-dimensional hypothesis testing problem between two probability measures ℙ and ℚ based on the sample Y. We show that if the low-degree advantage Adv_{≤D}(dℙ/dℚ) = O(1), then (assuming the low-degree conjecture) there is no efficient algorithm A such that ℚ(A(Y) = 0) = 1-o(1) and ℙ(A(Y) = 1) = Ω(1). This framework provides a useful tool for performing reductions between different inference tasks.
Cite as
Zhangsong Li. Algorithmic Contiguity from Low-Degree Conjecture and Applications in Correlated Random Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 30:1-30:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{li:LIPIcs.APPROX/RANDOM.2025.30,
author = {Li, Zhangsong},
title = {{Algorithmic Contiguity from Low-Degree Conjecture and Applications in Correlated Random Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {30:1--30:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.30},
URN = {urn:nbn:de:0030-drops-243965},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.30},
annote = {Keywords: Algorithmic Contiguity, Low-degree Conjecture, Correlated Random Graphs}
}
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RANDOM
New Statistical and Computational Results for Learning Junta Distributions
Abstract
We study the problem of learning junta distributions on {±1}ⁿ, where a distribution is a k-junta if its probability mass function depends on a subset of at most k variables. We make two main contributions:
- We show that learning k-junta distributions is computationally equivalent to learning k-parity functions with noise (LPN), a landmark problem in computational learning theory.
- We design an algorithm for learning junta distributions whose statistical complexity is optimal, up to polylogarithmic factors. Computationally, our algorithm matches the complexity of previous (non-sample-optimal) algorithms. Combined, our two contributions imply that our algorithm cannot be significantly improved, statistically or computationally, barring a breakthrough for LPN.
Cite as
Lorenzo Beretta. New Statistical and Computational Results for Learning Junta Distributions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 31:1-31:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{beretta:LIPIcs.APPROX/RANDOM.2025.31,
author = {Beretta, Lorenzo},
title = {{New Statistical and Computational Results for Learning Junta Distributions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {31:1--31:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.31},
URN = {urn:nbn:de:0030-drops-243978},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.31},
annote = {Keywords: Junta Distributions, Learning Parities with Noise}
}
Document
RANDOM
Quantum Property Testing in Sparse Directed Graphs
Abstract
We initiate the study of quantum property testing in sparse directed graphs, and more particularly in the unidirectional model, where the algorithm is allowed to query only the outgoing edges of a vertex. In the classical unidirectional model, the problem of testing k-star-freeness, and more generally k-source-subgraph-freeness, is almost maximally hard for large k. We prove that this problem has almost quadratic advantage in the quantum setting. Moreover, we show that this advantage is nearly tight, by showing a quantum lower bound using the method of dual polynomials on an intermediate problem for a new, property testing version of the k-collision problem that was not studied before.
To illustrate that not all problems in graph property testing admit such a quantum speedup, we consider the problem of 3-colorability in the related undirected bounded-degree model, when graphs are now undirected. This problem is maximally hard to test classically, and we show that also quantumly it requires a linear number of queries.
Cite as
Simon Apers, Frédéric Magniez, Sayantan Sen, and Dániel Szabó. Quantum Property Testing in Sparse Directed Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 32:1-32:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{apers_et_al:LIPIcs.APPROX/RANDOM.2025.32,
author = {Apers, Simon and Magniez, Fr\'{e}d\'{e}ric and Sen, Sayantan and Szab\'{o}, D\'{a}niel},
title = {{Quantum Property Testing in Sparse Directed Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {32:1--32:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.32},
URN = {urn:nbn:de:0030-drops-243987},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.32},
annote = {Keywords: property testing, quantum computing, bounded-degree directed graphs, dual polynomial method, collision finding}
}
Document
RANDOM
Bit-Fixing Extractors for Almost-Logarithmic Entropy
Abstract
An oblivious bit-fixing source is a distribution over {0,1}ⁿ, where k bits are uniform and independent and the rest n-k are fixed a priori to some constant value. Extracting (close to) true randomness from an oblivious bit-fixing source has been studied since the 1980s, with applications in cryptography and complexity theory.
We construct explicit extractors for oblivious bit-fixing source that support k = Õ(log n), outputting almost all the entropy with low error. The previous state-of-the-art construction that outputs many bits is due to Rao [Rao, CCC '09], and requires entropy k ≥ log^{c} n for some large constant c. The two key components in our constructions are new low-error affine condensers for poly-logarithmic entropies (that we achieve using techniques from the nonmalleable extractors literature), and a dual use of linear condensers for OBF sources.
Cite as
Dean Doron and Ori Fridman. Bit-Fixing Extractors for Almost-Logarithmic Entropy. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2025.33,
author = {Doron, Dean and Fridman, Ori},
title = {{Bit-Fixing Extractors for Almost-Logarithmic Entropy}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {33:1--33:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.33},
URN = {urn:nbn:de:0030-drops-243994},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.33},
annote = {Keywords: Seedless extractors, oblivious bit-fixing sources}
}
Document
RANDOM
Eigenvalue Bounds for Symmetric Markov Chains on Multislices with Applications
Abstract
We consider random walks on "balanced multislices" of any "grid" that respects the "symmetries" of the grid, and show that a broad class of such walks are good spectral expanders. (A grid is a set of points of the form 𝒮ⁿ for finite 𝒮, and a balanced multi-slice is the subset that contains an equal number of coordinates taking every value in 𝒮. A walk respects symmetries if the probability of going from u = (u_1,…,u_n) to v = (v_1,…,v_n) is invariant under simultaneous permutations of the coordinates of u and v.) Our main theorem shows that, under some technical conditions, every such walk where a single step leads to an almost 𝒪(1)-wise independent distribution on the next state, conditioned on the previous state, satisfies a non-trivially small singular value bound.
We give two applications of our theorem to error-correcting codes: (1) We give an analog of the Ore-DeMillo-Lipton-Schwartz-Zippel lemma for polynomials, and junta-sums, over balanced multislices. (2) We also give a local list-correction algorithm for d-junta-sums mapping an arbitrary grid 𝒮ⁿ to an Abelian group, correcting from a near-optimal (1/|𝒮|^d - ε) fraction of errors for every ε > 0, where a d-junta-sum is a sum of (arbitrarily many) d-juntas (and a d-junta is a function that depends on only d of the n variables).
Our proofs are obtained by exploring the representation theory of the symmetric group and merging it with some careful spectral analysis.
Cite as
Prashanth Amireddy, Amik Raj Behera, Srikanth Srinivasan, and Madhu Sudan. Eigenvalue Bounds for Symmetric Markov Chains on Multislices with Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 34:1-34:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{amireddy_et_al:LIPIcs.APPROX/RANDOM.2025.34,
author = {Amireddy, Prashanth and Behera, Amik Raj and Srinivasan, Srikanth and Sudan, Madhu},
title = {{Eigenvalue Bounds for Symmetric Markov Chains on Multislices with Applications}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {34:1--34:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.34},
URN = {urn:nbn:de:0030-drops-244004},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.34},
annote = {Keywords: Markov Chains, Random Walk, Multislices, Representation Theory of Symmetric Group, Local Correction, Low-degree Polynomials, Polynomial Distance Lemma}
}
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RANDOM
Fooling Near-Maximal Decision Trees
Abstract
For any constant α > 0, we construct an explicit pseudorandom generator (PRG) that fools n-variate decision trees of size m with error ε and seed length (1 + α) ⋅ log₂ m + O(log(1/ε) + log log n). For context, one can achieve seed length (2 + o(1)) ⋅ log₂ m + O(log(1/ε) + log log n) using well-known constructions and analyses of small-bias distributions, but such a seed length is trivial when m ≥ 2^{n/2}. Our approach is to develop a new variant of the classic concept of almost k-wise independence, which might be of independent interest. We say that a distribution X over {0, 1}ⁿ is k-wise ε-probably uniform if every Boolean function f that depends on only k variables satisfies 𝔼[f(X)] ≥ (1 - ε) ⋅ 𝔼[f]. We show how to sample a k-wise ε-probably uniform distribution using a seed of length (1 + α) ⋅ k + O(log(1/ε) + log log n).
Meanwhile, we also show how to construct a set H ⊆ 𝔽₂ⁿ such that every feasible system of k linear equations in n variables over 𝔽₂ has a solution in H. The cardinality of H and the time complexity of enumerating H are at most 2^{k + o(k) + polylog n}, whereas small-bias distributions would give a bound of 2^{2k + O(log(n/k))}.
By combining our new constructions with work by Chen and Kabanets (TCS 2016), we obtain nontrivial PRGs and hitting sets for linear-size Boolean circuits. Specifically, we get an explicit PRG with seed length (1 - Ω(1)) ⋅ n that fools circuits of size 2.99 ⋅ n over the U₂ basis, and we get a hitting set with time complexity 2^{(1 - Ω(1)) ⋅ n} for circuits of size 2.49 ⋅ n over the B₂ basis.
Cite as
William M. Hoza and Zelin Lv. Fooling Near-Maximal Decision Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 35:1-35:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{hoza_et_al:LIPIcs.APPROX/RANDOM.2025.35,
author = {Hoza, William M. and Lv, Zelin},
title = {{Fooling Near-Maximal Decision Trees}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {35:1--35:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.35},
URN = {urn:nbn:de:0030-drops-244019},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.35},
annote = {Keywords: almost k-wise independence, decision trees, pseudorandom generators}
}
Document
RANDOM
Simplifying Armoni’s PRG
Abstract
We propose a simple variant of the INW pseudo-random generator, where blocks have varying lengths, and prove it gives the same parameters as the more complicated construction of Armoni’s PRG. This shows there is no need for the specialized PRGs of Nisan and Zuckerman and Armoni, and they can be obtained as simple variants of INW.
For the construction to work we need space-efficient extractors with tiny entropy loss. We use the extractors from [Chattopadhyay and Liao, 2020] instead of [Guruswami et al., 2009] taking advantage of the very high min-entropy regime we work with. We remark that using these extractors has the additional benefit of making the dependence on the branching program alphabet Σ correct.
Cite as
Ben Chen and Amnon Ta-Shma. Simplifying Armoni’s PRG. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 36:1-36:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2025.36,
author = {Chen, Ben and Ta-Shma, Amnon},
title = {{Simplifying Armoni’s PRG}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {36:1--36:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.36},
URN = {urn:nbn:de:0030-drops-244024},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.36},
annote = {Keywords: PRG, ROBP, read-once, random, psuedorandom, armoni, derandomization}
}
Document
RANDOM
Permanental Rank vs Determinantal Rank of Random Matrices over Finite Fields
Abstract
This paper is motivated by basic complexity and probability questions about permanents of random matrices over small finite fields, and in particular, about properties separating the permanent and the determinant.
Let q be a fixed odd prime, and let k ≤ n both be growing. For a uniformly random n × k matrix A over 𝔽_q, we study the probability that all k × k submatrices of A have zero permanent; namely that A does not have full permanental rank.
When k = n, this is simply the probability that a random square matrix over 𝔽_q has zero permanent, which we do not understand. We believe that the probability in this case is 1/q + o(1), which would be in contrast to the case of the determinant, where the answer is 1/q + Ω_q(1).
Our main result is that when k is O(√n), the probability that a random n × k matrix does not have full permanental rank is essentially the same as the probability that the matrix has a 0 column, namely (1 +o(1)) k/qⁿ. In contrast, for determinantal (standard) rank the analogous probability is Θ(q^k/q^n).
At the core of our result are some basic linear algebraic properties of the permanent that distinguish it from the determinant.
Cite as
Fatemeh Ghasemi, Gal Gross, and Swastik Kopparty. Permanental Rank vs Determinantal Rank of Random Matrices over Finite Fields. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ghasemi_et_al:LIPIcs.APPROX/RANDOM.2025.37,
author = {Ghasemi, Fatemeh and Gross, Gal and Kopparty, Swastik},
title = {{Permanental Rank vs Determinantal Rank of Random Matrices over Finite Fields}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {37:1--37:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.37},
URN = {urn:nbn:de:0030-drops-244037},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.37},
annote = {Keywords: Permanent, random matrices over a finite field}
}
Document
RANDOM
Low-Degree Polynomials Are Good Extractors
Abstract
We prove that random low-degree polynomials (over 𝔽₂) are unbiased, in an extremely general sense. That is, we show that random low-degree polynomials are good randomness extractors for a wide class of distributions. Prior to our work, such results were only known for the small families of (1) uniform sources, (2) affine sources, and (3) local sources. We significantly generalize these results, and prove the following.
1) Low-degree polynomials extract from small families. We show that a random low-degree polynomial is a good low-error extractor for any small family of sources. In particular, we improve the positive result of Alrabiah, Chattopadhyay, Goodman, Li, and Ribeiro (ICALP 2022) for local sources, and give new results for polynomial and variety sources via a single unified approach.
2) Low-degree polynomials extract from sumset sources. We show that a random low-degree polynomial is a good extractor for sumset sources, which are the most general large family of sources (capturing independent sources, interleaved sources, small-space sources, and more). Formally, for any even d, we show that a random degree d polynomial is an ε-error extractor for n-bit sumset sources with min-entropy k = O(d(n/ε²)^{2/d}). This is nearly tight in the polynomial error regime.
Our results on sumset extractors imply new complexity separations for linear ROBPs, and the tools that go into its proof may be of independent interest. The two main tools we use are a new structural result on sumset-punctured Reed-Muller codes, paired with a novel type of reduction between extractors. Using the new structural result, we obtain new limits on the power of sumset extractors, strengthening and generalizing the impossibility results of Chattopadhyay, Goodman, and Gurumukhani (ITCS 2024).
Cite as
Omar Alrabiah, Jesse Goodman, Jonathan Mosheiff, and João Ribeiro. Low-Degree Polynomials Are Good Extractors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 38:1-38:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{alrabiah_et_al:LIPIcs.APPROX/RANDOM.2025.38,
author = {Alrabiah, Omar and Goodman, Jesse and Mosheiff, Jonathan and Ribeiro, Jo\~{a}o},
title = {{Low-Degree Polynomials Are Good Extractors}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {38:1--38:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.38},
URN = {urn:nbn:de:0030-drops-244048},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.38},
annote = {Keywords: randomness extractors, low-degree polynomials, local sources, polynomial sources, variety sources, sumset sources, sumset extractors, Reed-Muller codes, lower bounds}
}
Document
RANDOM
Consumable Data via Quantum Communication
Abstract
Classical data can be copied and re-used for computation, with adverse consequences economically and in terms of data privacy. Motivated by this, we formulate problems in one-way communication complexity where Alice holds some data x and Bob holds m inputs y_1, …, y_m. They want to compute m instances of a bipartite relation R(⋅,⋅) on every pair (x, y_1), …, (x, y_m). We call this the asymmetric direct sum question for one-way communication. We give examples where the quantum communication complexity of such problems scales polynomially with m, while the classical communication complexity depends at most logarithmically on m. Thus, for such problems, data behaves like a consumable resource that is effectively destroyed upon use when the owner stores and transmits it as quantum states, but not when transmitted classically. We show an application to a strategic data-selling game, and discuss other potential economic implications.
Cite as
Dar Gilboa, Siddhartha Jain, and Jarrod R. McClean. Consumable Data via Quantum Communication. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 39:1-39:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{gilboa_et_al:LIPIcs.APPROX/RANDOM.2025.39,
author = {Gilboa, Dar and Jain, Siddhartha and McClean, Jarrod R.},
title = {{Consumable Data via Quantum Communication}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {39:1--39:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.39},
URN = {urn:nbn:de:0030-drops-244059},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.39},
annote = {Keywords: quantum communication, one-time programs, data markets}
}
Document
RANDOM
Sublinear Space Graph Algorithms in the Continual Release Model
Abstract
The graph continual release model of differential privacy seeks to produce differentially private solutions to graph problems under a stream of edge updates where new private solutions are released after each update. Thus far, previously known edge-differentially private algorithms for most graph problems including densest subgraph and matchings in the continual release setting only output real-value estimates (not vertex subset solutions) and do not use sublinear space. Instead, they rely on computing exact graph statistics on the input [Hendrik Fichtenberger et al., 2021; Shuang Song et al., 2018]. In this paper, we leverage sparsification to address the above shortcomings for edge-insertion streams. Our edge-differentially private algorithms use sublinear space with respect to the number of edges in the graph while some also achieve sublinear space in the number of vertices in the graph. In addition, for the densest subgraph problem, we also output edge-differentially private vertex subset solutions; no previous graph algorithms in the continual release model output such subsets.
We make novel use of assorted sparsification techniques from the non-private streaming and static graph algorithms literature to achieve new results in the sublinear space, continual release setting. This includes algorithms for densest subgraph, maximum matching, as well as the first continual release k-core decomposition algorithm. We also develop a novel sparse level data structure for k-core decomposition that may be of independent interest. To complement our insertion-only algorithms, we conclude with polynomial additive error lower bounds for edge-privacy in the fully dynamic setting, where only logarithmic lower bounds were previously known.
Cite as
Alessandro Epasto, Quanquan C. Liu, Tamalika Mukherjee, and Felix Zhou. Sublinear Space Graph Algorithms in the Continual Release Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 40:1-40:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{epasto_et_al:LIPIcs.APPROX/RANDOM.2025.40,
author = {Epasto, Alessandro and Liu, Quanquan C. and Mukherjee, Tamalika and Zhou, Felix},
title = {{Sublinear Space Graph Algorithms in the Continual Release Model}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {40:1--40:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.40},
URN = {urn:nbn:de:0030-drops-244064},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.40},
annote = {Keywords: Differential Privacy, Continual Release, Densest Subgraph, k-Core Decomposition, Maximum Matching}
}
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RANDOM
What Is the Minimum Number of Random Bits Required for Computability and Efficiency in Anonymous Networks?
Abstract
Angluin (STOC'80) and Yamashita and Kameda (PODC'88) show that some useful distributed tasks are impossible (for deterministic algorithms) in a general network if nodes do not possess unique identifiers. However, any task decidable in the non-distributed context, can be solved deterministically if the network has a unique leader. Alternatively, much research has been devoted to randomized distributed algorithms in anonymous networks. We present tight upper and lower bounds for the fundamental question: How much randomness is necessary and sufficient to solve Leader Election (LE) in anonymous networks, i.e., to transform an anonymous network into a non-anonymous one? We prove that at least one random bit per node is required in some cases. Surprisingly, a single random bit is also enough, for a total of n bits, where n is the number of nodes. However, the time complexity of our (total of) n random bits algorithm for general networks turned out to be impractically high. Hence, we also developed time-efficient algorithms for the very symmetric graphs of cliques and cycles, paying only an additional cost of o(n) random bits.
The primary steps of our algorithms are of independent interest. At first glance, it seems that using one random bit per node, any algorithm can distinguish only two sets of nodes: those with 0 and those with 1. Our algorithms manage to partition the nodes into more than two sets with high probability. In some sense, they perform the task of a "distributed pseudorandom generator", for example, one of our algorithms turns n bits, one per node, into n unique (with high probability) numbers. Even though a complete graph looks very symmetric, the algorithms explore interesting asymmetries inherent in any n permutations (of n values each), if each describes the assignment (by the adversary) of ports in a node to edges leading to neighbors.
Finally, we show how to transform any randomized algorithm that generates xn+o(n) random bits in total to one where each node generates at most x+1 bits. Our results apply to both synchronous and asynchronous networks.
Cite as
Dariusz R. Kowalski, Piotr Krysta, and Shay Kutten. What Is the Minimum Number of Random Bits Required for Computability and Efficiency in Anonymous Networks?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 41:1-41:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kowalski_et_al:LIPIcs.APPROX/RANDOM.2025.41,
author = {Kowalski, Dariusz R. and Krysta, Piotr and Kutten, Shay},
title = {{What Is the Minimum Number of Random Bits Required for Computability and Efficiency in Anonymous Networks?}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {41:1--41:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.41},
URN = {urn:nbn:de:0030-drops-244071},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.41},
annote = {Keywords: Distributed computability, Anonymous Networks, Randomness, Leader Election}
}
Document
RANDOM
On the Spectral Expansion of Monotone Subsets of the Hypercube
Abstract
We study the spectral gap of subgraphs of the hypercube induced by monotone subsets of vertices. For a monotone subset A ⊆ {0,1}ⁿ of density μ(A), the previous best lower bound on the spectral gap, due to Cohen [Cohen, 2016], was γ ≳ μ(A)/n², improving upon the earlier bound γ ≳ μ(A)²/n² established by Ding and Mossel [Ding and Mossel, 2014]. In this paper, we prove the optimal lower bound γ ≳ μ(A)/n. As a corollary, we improve the mixing time upper bound of the random walk on constant-density monotone sets from O(n³), as shown by Ding and Mossel, to O(n²). Along the way, we develop two new inequalities that may be of independent interest: (1) a directed L²-Poincaré inequality on the hypercube, and (2) an "approximate" FKG inequality for monotone sets.
Cite as
Yumou Fei and Renato Ferreira Pinto Jr.. On the Spectral Expansion of Monotone Subsets of the Hypercube. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 42:1-42:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{fei_et_al:LIPIcs.APPROX/RANDOM.2025.42,
author = {Fei, Yumou and Ferreira Pinto Jr., Renato},
title = {{On the Spectral Expansion of Monotone Subsets of the Hypercube}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {42:1--42:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.42},
URN = {urn:nbn:de:0030-drops-244081},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.42},
annote = {Keywords: Random walks, mixing time, FKG inequality, Poincar\'{e} inequality, directed isoperimetry}
}
Document
RANDOM
Gabidulin Codes Achieve List Decoding Capacity with an Order-Optimal Column-To-Row Ratio
Abstract
In this paper, we show that random Gabidulin codes of block length n and rate R achieve the (average-radius) list decoding capacity of radius 1-R-ε in the rank metric with an order-optimal column-to-row ratio of O(ε). This extends the recent work of Guo, Xing, Yuan, and Zhang (FOCS 2024), improving their column-to-row ratio from O(ε/n) to O(ε). For completeness, we also establish a matching lower bound on the column-to-row ratio for capacity-achieving Gabidulin codes in the rank metric.
Our proof techniques build on the work of Guo and Zhang (FOCS 2023), who showed that randomly punctured Reed-Solomon codes over fields of quadratic size attain the generalized Singleton bound of Shangguan and Tamo (STOC 2020) in the Hamming metric. The proof of our lower bound follows the method of Alrabiah, Guruswami, and Li (SODA 2024) for codes in the Hamming metric.
Cite as
Zeyu Guo, Chaoping Xing, Chen Yuan, and Zihan Zhang. Gabidulin Codes Achieve List Decoding Capacity with an Order-Optimal Column-To-Row Ratio. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 43:1-43:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{guo_et_al:LIPIcs.APPROX/RANDOM.2025.43,
author = {Guo, Zeyu and Xing, Chaoping and Yuan, Chen and Zhang, Zihan},
title = {{Gabidulin Codes Achieve List Decoding Capacity with an Order-Optimal Column-To-Row Ratio}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {43:1--43:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.43},
URN = {urn:nbn:de:0030-drops-244095},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.43},
annote = {Keywords: coding theory, error-correcting codes, Gabidulin codes, rank-metric codes}
}
Document
RANDOM
Density Frankl–Rödl on the Sphere
Abstract
We establish a density variant of the Frankl–Rödl theorem on the sphere 𝕊^{n-1}, which concerns avoiding pairs of vectors with a specific distance, or equivalently, a prescribed inner product. In particular, we establish lower bounds on the probability that a randomly chosen pair of such vectors lies entirely within a measurable subset A ⊆ 𝕊^{n-1} of sufficiently large measure. Additionally, we prove a density version of spherical avoidance problems, which generalize from pairwise avoidance to broader configurations with prescribed pairwise inner products. Our framework encompasses a class of configurations we call inductive configurations, which include simplices with any prescribed inner product -1 < r < 1. As a consequence of our density statement, we show that all inductive configurations are sphere Ramsey.
Cite as
Venkatesan Guruswami and Shilun Li. Density Frankl–Rödl on the Sphere. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 44:1-44:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{guruswami_et_al:LIPIcs.APPROX/RANDOM.2025.44,
author = {Guruswami, Venkatesan and Li, Shilun},
title = {{Density Frankl–R\"{o}dl on the Sphere}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {44:1--44:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.44},
URN = {urn:nbn:de:0030-drops-244108},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.44},
annote = {Keywords: Frankl–R\"{o}dl, Sphere Ramsey, Sphere Avoidance, Reverse Hypercontractivity, Forbidden Angles}
}
Document
RANDOM
Local Computation Algorithms for Knapsack: Impossibility Results, and How to Avoid Them
Abstract
Local Computation Algorithms (LCA), as introduced by Rubinfeld, Tamir, Vardi, and Xie (2011), are a type of ultra-efficient algorithms which, given access to a (large) input for a given computational task, are required to provide fast query access to a consistent output solution, without maintaining a state between queries. This paradigm of computation in particular allows for hugely distributed algorithms, where independent instances of a given LCA provide consistent access to a common output solution.
The past decade has seen a significant amount of work on LCAs, by and large focusing on graph problems. In this paper, we initiate the study of Local Computation Algorithms for perhaps the archetypal combinatorial optimization problem, Knapsack. We first establish strong impossibility results, ruling out the existence of any non-trivial LCA for Knapsack as several of its relaxations. We then show how equipping the LCA with additional access to the Knapsack instance, namely, weighted item sampling, allows one to circumvent these impossibility results, and obtain sublinear-time and query LCAs. Our positive result draws on a connection to the recent notion of reproducibility for learning algorithms (Impagliazzo, Lei, Pitassi, and Sorrell, 2022), a connection we believe to be of independent interest for the design of LCAs.
Cite as
Clément L. Canonne, Yun Li, and Seeun William Umboh. Local Computation Algorithms for Knapsack: Impossibility Results, and How to Avoid Them. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 45:1-45:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{canonne_et_al:LIPIcs.APPROX/RANDOM.2025.45,
author = {Canonne, Cl\'{e}ment L. and Li, Yun and Umboh, Seeun William},
title = {{Local Computation Algorithms for Knapsack: Impossibility Results, and How to Avoid Them}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {45:1--45:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.45},
URN = {urn:nbn:de:0030-drops-244111},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.45},
annote = {Keywords: Local computation algorithms, Knapsack, algorithms, lower bounds}
}
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RANDOM
Efficient Parallel Ising Samplers via Localization Schemes
Abstract
We introduce efficient parallel algorithms for sampling from the Gibbs distribution and estimating the partition function of Ising models. These algorithms achieve parallel efficiency, with polylogarithmic depth and polynomial total work, and are applicable to Ising models in the following regimes: (1) Ferromagnetic Ising models with external fields; (2) Ising models with interaction matrix J of operator norm ‖J‖₂ < 1.
Our parallel Gibbs sampling approaches are based on localization schemes, which have proven highly effective in establishing rapid mixing of Gibbs sampling. In this work, we employ two such localization schemes to obtain efficient parallel Ising samplers: the field dynamics induced by negative-field localization, and restricted Gaussian dynamics induced by stochastic localization. This shows that localization schemes are powerful tools, not only for achieving rapid mixing but also for the efficient parallelization of Gibbs sampling.
Cite as
Xiaoyu Chen, Hongyang Liu, Yitong Yin, and Xinyuan Zhang. Efficient Parallel Ising Samplers via Localization Schemes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 46:1-46:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2025.46,
author = {Chen, Xiaoyu and Liu, Hongyang and Yin, Yitong and Zhang, Xinyuan},
title = {{Efficient Parallel Ising Samplers via Localization Schemes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {46:1--46:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.46},
URN = {urn:nbn:de:0030-drops-244129},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.46},
annote = {Keywords: Localization scheme, parallel sampling, Ising model}
}
Document
RANDOM
Time Lower Bounds for the Metropolis Process and Simulated Annealing
Abstract
The Metropolis process (MP) and Simulated Annealing (SA) are stochastic local search heuristics that are often used in solving combinatorial optimization problems. Despite significant interest, there are very few theoretical results regarding the quality of approximation obtained by MP and SA (with polynomially many iterations) for NP-hard optimization problems.
We provide rigorous lower bounds for MP and SA with respect to the classical maximum independent set problem when the algorithms are initialized from the empty set. We establish the existence of a family of graphs for which both MP and SA fail to find approximate solutions in polynomial time. More specifically, we show that for any ε ∈ (0,1) there are n-vertex graphs for which the probability SA (when limited to polynomially many iterations) will approximate the optimal solution within ratio Ω(1/n^{1-ε}) is exponentially small. Our lower bounds extend to graphs of constant average degree d, illustrating the failure of MP to achieve an approximation ratio of Ω(log(d)/d) in polynomial time. In some cases, our lower bounds apply even when the temperature is chosen adaptively. Finally, we prove exponential-time lower bounds when the inputs to these algorithms are bipartite graphs, and even trees, which are known to admit polynomial-time algorithms for the independent set problem.
Cite as
Zongchen Chen, Dan Mikulincer, Daniel Reichman, and Alexander S. Wein. Time Lower Bounds for the Metropolis Process and Simulated Annealing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 47:1-47:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2025.47,
author = {Chen, Zongchen and Mikulincer, Dan and Reichman, Daniel and Wein, Alexander S.},
title = {{Time Lower Bounds for the Metropolis Process and Simulated Annealing}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {47:1--47:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.47},
URN = {urn:nbn:de:0030-drops-244138},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.47},
annote = {Keywords: Metropolis Process, Simulated Annealing, Independent Set}
}
Document
RANDOM
Sharp Thresholds for the Overlap Gap Property: Ising p-Spin Glass and Random k-SAT
Abstract
The Ising p-spin glass and random k-SAT are two canonical examples of disordered systems that play a central role in understanding the link between geometric features of optimization landscapes and computational tractability. Both models exhibit hard regimes where all known polynomial-time algorithms fail and possess the multi Overlap Gap Property (m-OGP), an intricate geometrical property that rigorously rules out a broad class of algorithms exhibiting input stability.
We establish that, in both models, the symmetric m-OGP undergoes a sharp phase transition, and we pinpoint its exact threshold. For the Ising p-spin glass, our results hold for all sufficiently large p; for the random k-SAT, they apply to all k growing mildly with the number of Boolean variables. Notably, our findings yield qualitative insights into the power of OGP-based arguments. A particular consequence for the Ising p-spin glass is that the strength of the m-OGP in establishing algorithmic hardness grows without bound as m increases.
These are the first sharp threshold results for the m-OGP. Our analysis hinges on a judicious application of the second moment method, enhanced by concentration. While a direct second moment calculation fails, we overcome this via a refined approach that leverages an argument of Frieze [Frieze, 1990] and exploiting concentration properties of carefully constructed random variables.
Cite as
Eren C. Kızıldağ. Sharp Thresholds for the Overlap Gap Property: Ising p-Spin Glass and Random k-SAT. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kizildag:LIPIcs.APPROX/RANDOM.2025.48,
author = {K{\i}z{\i}lda\u{g}, Eren C.},
title = {{Sharp Thresholds for the Overlap Gap Property: Ising p-Spin Glass and Random k-SAT}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {48:1--48:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.48},
URN = {urn:nbn:de:0030-drops-244147},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.48},
annote = {Keywords: spin glasses, p-spin model, random constraint satisfaction problems, overlap gap property, phase transitions, computational complexity}
}
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RANDOM
Pseudorandomness of Expander Walks via Fourier Analysis on Groups
Abstract
A long line of work has studied the pseudorandomness properties of walks on expander graphs. A central goal is to measure how closely the distribution over n-length walks on an expander approximates the uniform distribution of n-independent elements. One approach to do so is to label the vertices of an expander with elements from an alphabet Σ, and study closeness of the mean of functions over Σⁿ, under these two distributions. We say expander walks ε-fool a function if the expander walk mean is ε-close to the true mean. There has been a sequence of works studying this question for various functions, such as the XOR function, the AND function, etc. We show that:
- The class of symmetric functions is O(|Σ|λ)-fooled by expander walks over any generic λ-expander, and any alphabet Σ . This generalizes the result of Cohen, Peri, Ta-Shma [STOC'21] which analyzes it for |Σ| = 2, and exponentially improves the previous bound of O(|Σ|^O(|Σ|) λ), by Golowich and Vadhan [CCC'22]. Moreover, if the expander is a Cayley graph over ℤ_|Σ|, we get a further improved bound of O(√{|Σ|} λ). Morever, when Σ is a finite group G, we show the following for functions over Gⁿ:
- The class of symmetric class functions is O({√|G|}/D λ}-fooled by expander walks over "structured" λ-expanders, if G is D-quasirandom.
- We show a lower bound of Ω(λ) for symmetric functions for any finite group G (even for "structured" λ-expanders).
- We study the Fourier spectrum of a class of non-symmetric functions arising from word maps, and show that they are exponentially fooled by expander walks.
Our proof employs Fourier analysis over general groups, which contrasts with earlier works that have studied either the case of ℤ₂ or ℤ. This enables us to get quantitatively better bounds even for unstructured sets.
Cite as
Fernando Granha Jeronimo, Tushant Mittal, and Sourya Roy. Pseudorandomness of Expander Walks via Fourier Analysis on Groups. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 49:1-49:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{jeronimo_et_al:LIPIcs.APPROX/RANDOM.2025.49,
author = {Jeronimo, Fernando Granha and Mittal, Tushant and Roy, Sourya},
title = {{Pseudorandomness of Expander Walks via Fourier Analysis on Groups}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {49:1--49:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.49},
URN = {urn:nbn:de:0030-drops-244157},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.49},
annote = {Keywords: Expander graphs, pseudorandomness}
}
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RANDOM
Shared Randomness in Locally Checkable Problems: The Role of Computational Assumptions
Abstract
Shared randomness is a valuable resource in distributed computing, allowing some form of coordination between processors without explicit communication. But what happens when the shared random string can affect the inputs to the system?
Consider the class of distributed graph problems where the correctness of solutions can be checked locally, known as Locally Checkable Labelings (LCL). LCL problems have been extensively studied in the LOCAL model, where nodes operate in synchronous rounds and have access only to local information. This has led to intriguing insights regarding the power of private randomness. E.g., for certain round complexity classes, derandomization does not incur an overhead (asymptotically).
This work considers a setting where the randomness is public. Recently, an LCL problem for which shared randomness can reduce the round complexity was discovered by Balliu et al. (ICALP 2025). This result applies to inputs set obliviously of the shared randomness, which may not always be a plausible assumption.
We define a model where the inputs can be adversarially chosen, even based on the shared randomness, which we now call preset public coins. We study LCL problems in the preset public coins model, under assumptions regarding the computational power of the adversary that selects the input. We show connections to hardness in the class TFNP. Our results are:
1) Assuming a hard-on-average problem in TFNP, we present an LCL problem that, in the preset public coins model, demonstrates a gap in the round complexity between polynomial-time and unbounded adversaries.
2) An LCL problem for which the error probability is significantly higher when facing unbounded adversaries implies a hard-on-average problem in TFNP/poly.
Cite as
Adar Hadad and Moni Naor. Shared Randomness in Locally Checkable Problems: The Role of Computational Assumptions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 50:1-50:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{hadad_et_al:LIPIcs.APPROX/RANDOM.2025.50,
author = {Hadad, Adar and Naor, Moni},
title = {{Shared Randomness in Locally Checkable Problems: The Role of Computational Assumptions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {50:1--50:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.50},
URN = {urn:nbn:de:0030-drops-244161},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.50},
annote = {Keywords: Distributed Graph Algorithms, Common Random String, Cryptographic Hardness}
}
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RANDOM
Improved Mixing of Critical Hardcore Model
Abstract
The hardcore model is one of the most classic and widely studied examples of undirected graphical models. Given a graph G, the hardcore model describes a Gibbs distribution of λ-weighted independent sets of G. In the last two decades, a beautiful computational phase transition has been established at a precise threshold λ_c(Δ) where Δ denotes the maximum degree, where the task of sampling independent sets transitions from polynomial-time solvable to computationally intractable. We study the critical hardcore model where λ = λ_c(Δ) and show that the Glauber dynamics, a simple yet popular Markov chain algorithm, mixes in Õ(n^{7.44 + O(1/Δ)}) time on any n-vertex graph of maximum degree Δ ≥ 3, significantly improving the previous upper bound Õ(n^{12.88 + O(1/Δ)}) by the recent work [Chen et al., 2024]. The core property we establish in this work is that the critical hardcore model is O(√n)-spectrally independent, improving the trivial bound of n and matching the critical behavior of the Ising model. Our proof approach utilizes an online decision-making framework to study a site percolation model on the infinite (Δ-1)-ary tree, which can be interesting by itself.
Cite as
Zongchen Chen and Tianhui Jiang. Improved Mixing of Critical Hardcore Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 51:1-51:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2025.51,
author = {Chen, Zongchen and Jiang, Tianhui},
title = {{Improved Mixing of Critical Hardcore Model}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {51:1--51:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.51},
URN = {urn:nbn:de:0030-drops-244176},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.51},
annote = {Keywords: Hardcore model, Phase transition, Glauber dynamics, Spectral independence, Online decision making, Site percolation}
}
Document
RANDOM
Tarski Lower Bounds from Multi-Dimensional Herringbones
Abstract
Tarski’s theorem states that every monotone function from a complete lattice to itself has a fixed point. We analyze the query complexity of finding such a fixed point on the k-dimensional grid of side length n under the ≤ relation. In this setting, there is an unknown monotone function f: {0,1,…, n-1}^k → {0,1,…, n-1}^k and an algorithm must query a vertex v to learn f(v). The goal is to find a fixed point of f using as few oracle queries as possible.
We show that the randomized query complexity of this problem is Ω((k⋅log²n)/log k) for all n,k ≥ 2. This unifies and improves upon two prior results: a lower bound of Ω(log²n) from [Etessami et al., 2020] and a lower bound of Ω((k⋅log(n)/log(k)) from [Brânzei et al., 2024], respectively.
Cite as
Simina Brânzei, Reed C. Phillips, and Nicholas J. Recker. Tarski Lower Bounds from Multi-Dimensional Herringbones. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 52:1-52:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{branzei_et_al:LIPIcs.APPROX/RANDOM.2025.52,
author = {Br\^{a}nzei, Simina and Phillips, Reed C. and Recker, Nicholas J.},
title = {{Tarski Lower Bounds from Multi-Dimensional Herringbones}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {52:1--52:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.52},
URN = {urn:nbn:de:0030-drops-244186},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.52},
annote = {Keywords: Tarski’s theorem, monotone functions, lattices, fixed points, computational complexity, oracle model, query complexity, lower bounds}
}
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RANDOM
Near-Optimal List-Recovery of Linear Code Families
Abstract
We prove several results on linear codes achieving list-recovery capacity. We show that random linear codes achieve list-recovery capacity with constant output list size (independent of the alphabet size and length). That is, over alphabets of size at least 𝓁^Ω(1/ε), random linear codes of rate R are (1-R-ε, 𝓁, (𝓁/ε)^O(𝓁/ε))-list-recoverable for all R ∈ (0,1) and 𝓁. Together with a result of Levi, Mosheiff, and Shagrithaya, this implies that randomly punctured Reed-Solomon codes also achieve list-recovery capacity. We also prove that our output list size is near-optimal among all linear codes: all (1-R-ε, 𝓁, L)-list-recoverable linear codes must have L ≥ 𝓁^{Ω(R/ε)}.
Our simple upper bound combines the Zyablov-Pinsker argument with recent bounds from Kopparty, Ron-Zewi, Saraf, Wootters, and Tamo on the maximum intersection of a "list-recovery ball" and a low-dimensional subspace with large distance. Our lower bound is inspired by a recent lower bound of Chen and Zhang.
Cite as
Ray Li and Nikhil Shagrithaya. Near-Optimal List-Recovery of Linear Code Families. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{li_et_al:LIPIcs.APPROX/RANDOM.2025.53,
author = {Li, Ray and Shagrithaya, Nikhil},
title = {{Near-Optimal List-Recovery of Linear Code Families}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {53:1--53:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.53},
URN = {urn:nbn:de:0030-drops-244199},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.53},
annote = {Keywords: Error-Correcting Codes, Randomness, List-Recovery, Reed-Solomon Codes, Random Linear Codes}
}
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RANDOM
New Constructions of Pseudorandom Codes
Abstract
Introduced in [Christ and Gunn, 2024], pseudorandom error-correcting codes (PRCs) are a new cryptographic primitive with applications in watermarking generative AI models. These are codes where a collection of polynomially many codewords is computationally indistinguishable from random for an adversary that does not have the secret key, but anyone with the secret key is able to efficiently decode corrupted codewords. In this work, we examine the assumptions under which PRCs with robustness to a constant error rate exist.
1) We show that if both the planted hyperloop assumption introduced in [Andrej Bogdanov et al., 2023] and security of a version of Goldreich’s PRG hold, then there exist public-key PRCs for which no efficient adversary can distinguish a polynomial number of codewords from random with better than o(1) advantage.
2) We revisit the construction of [Christ and Gunn, 2024] and show that it can be based on a wider range of assumptions than presented in [Christ and Gunn, 2024]. To do this, we introduce a weakened version of the planted XOR assumption which we call the weak planted XOR assumption and which may be of independent interest.
3) We initiate the study of PRCs which are secure against space-bounded adversaries. We show how to construct secret-key PRCs of length O(n) which are unconditionally indistinguishable from random by poly(n) time, O(n^{1.5-ε}) space adversaries.
Cite as
Surendra Ghentiyala and Venkatesan Guruswami. New Constructions of Pseudorandom Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 54:1-54:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ghentiyala_et_al:LIPIcs.APPROX/RANDOM.2025.54,
author = {Ghentiyala, Surendra and Guruswami, Venkatesan},
title = {{New Constructions of Pseudorandom Codes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {54:1--54:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.54},
URN = {urn:nbn:de:0030-drops-244202},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.54},
annote = {Keywords: Error-correcting codes, Watermarking, Pseudorandomness}
}
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RANDOM
Lifting to Randomized Parity Decision Trees
Abstract
We prove a lifting theorem from randomized decision tree depth to randomized parity decision tree (PDT) size. We use the same property of the gadget, stifling, which was introduced by Chattopadhyay, Mande, Sanyal and Sherif [ITCS 23] to prove a lifting theorem for deterministic PDTs. Moreover, even the milder condition that the gadget has minimum parity certificate complexity at least 2 suffices for lifting to randomized PDT size.
To improve the dependence on the gadget g in the lower bounds for composed functions, we consider a related problem g_* whose inputs are certificates of g. It is implicit in the work of Chattopadhyay et al. that for any function f, lower bounds for the *-depth of f_* give lower bounds for the PDT size of f. We make this connection explicit in the deterministic case and show that it also holds for randomized PDTs. We then combine this with composition theorems for *-depth, which follow by adapting known composition theorems for decision trees. As a corollary, we get tight lifting theorems when the gadget is Indexing, Inner Product or Disjointness.
Cite as
Farzan Byramji and Russell Impagliazzo. Lifting to Randomized Parity Decision Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 55:1-55:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{byramji_et_al:LIPIcs.APPROX/RANDOM.2025.55,
author = {Byramji, Farzan and Impagliazzo, Russell},
title = {{Lifting to Randomized Parity Decision Trees}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {55:1--55:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.55},
URN = {urn:nbn:de:0030-drops-244213},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.55},
annote = {Keywords: Parity decision trees, composition}
}
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RANDOM
Efficient Polynomial Identity Testing over Nonassociative Algebras
Abstract
We design the first efficient polynomial identity testing algorithms over the nonassociative polynomial algebra. In particular, multiplication among the formal variables is commutative but it is not associative. This complements the strong lower bound results obtained over this algebra by Hrubeš, Yehudayoff, and Wigderson [Pavel Hrubes et al., 2010] and Fijalkow, Lagarde, Ohlmann, and Serre [Fijalkow et al., 2021] from the identity testing perspective. Our main results are the following:
- We construct nonassociative algebras (both commutative and noncommutative) which have no low degree identities. As a result, we obtain the first Amitsur-Levitzki type theorems [A. S. Amitsur and J. Levitzki, 1950] over nonassociative polynomial algebras. As a direct consequence, we obtain randomized polynomial-time black-box PIT algorithms for nonassociative polynomials which allow evaluation over such algebras.
- On the derandomization side, we give a deterministic polynomial-time identity testing algorithm for nonassociative polynomials given by arithmetic circuits in the white-box setting. Previously, such an algorithm was known with the additional restriction of noncommutativity [Vikraman Arvind et al., 2017].
- In the black-box setting, we construct a hitting set of quasipolynomial-size for nonassociative polynomials computed by arithmetic circuits of small depth. Understanding the black-box complexity of identity testing, even in the randomized setting, was open prior to our work.
Cite as
Partha Mukhopadhyay, C. Ramya, and Pratik Shastri. Efficient Polynomial Identity Testing over Nonassociative Algebras. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 56:1-56:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{mukhopadhyay_et_al:LIPIcs.APPROX/RANDOM.2025.56,
author = {Mukhopadhyay, Partha and C. Ramya and Shastri, Pratik},
title = {{Efficient Polynomial Identity Testing over Nonassociative Algebras}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {56:1--56:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.56},
URN = {urn:nbn:de:0030-drops-244224},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.56},
annote = {Keywords: Polynomial identity testing, nonassociative algebra, arithmetic circuits, black-box algorithms, white-box algorithms}
}
Document
RANDOM
List-Recovery of Random Linear Codes over Small Fields
Abstract
We study list-recoverability of random linear codes over small fields, both from errors and from erasures. We consider codes of rate ε-close to capacity, and aim to bound the dependence of the output list size L on ε, the input list size 𝓁, and the alphabet size q. Prior to our work, the best upper bound was L = q^O(𝓁/ε) (Zyablov and Pinsker, Prob. Per. Inf. 1981).
Previous work has identified cases in which linear codes provably perform worse than non-linear codes with respect to list-recovery. While there exist non-linear codes that achieve L = O(𝓁/ε), we know that L ≥ 𝓁^Ω(1/ε) is necessary for list recovery from erasures over fields of small characteristic, and for list recovery from errors over large alphabets.
We show that in other relevant regimes there is no significant price to pay for linearity, in the sense that we get the correct dependence on the gap-to-capacity ε and go beyond the Zyablov-Pinsker bound for the first time. Specifically, when q is constant and ε approaches zero,
- For list-recovery from erasures over prime fields, we show that L ≤ C₁/ε. By prior work, such a result cannot be obtained for low-characteristic fields.
- For list-recovery from errors over arbitrary fields, we prove that L ≤ C₂/ε. Above, C₁ and C₂ depend on the decoding radius, input list size, and field size. We provide concrete bounds on the constants above, and the upper bounds on L improve upon the Zyablov-Pinsker bound whenever q ≤ 2^{(1/ε)^c} for some small universal constant c > 0.
Cite as
Dean Doron, Jonathan Mosheiff, Nicolas Resch, and João Ribeiro. List-Recovery of Random Linear Codes over Small Fields. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 57:1-57:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2025.57,
author = {Doron, Dean and Mosheiff, Jonathan and Resch, Nicolas and Ribeiro, Jo\~{a}o},
title = {{List-Recovery of Random Linear Codes over Small Fields}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {57:1--57:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.57},
URN = {urn:nbn:de:0030-drops-244239},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.57},
annote = {Keywords: List recovery, random linear codes}
}
Document
RANDOM
Equality Is Far Weaker Than Constant-Cost Communication
Abstract
We exhibit an n-bit communication problem with a constant-cost randomized protocol but which requires n^Ω(1) deterministic (or even non-deterministic) queries to an Equality oracle. Therefore, even constant-cost randomized protocols cannot be efficiently "derandomized" using Equality oracles. This improves on several recent results and answers a question from the survey of Hatami and Hatami (SIGACT News 2024). It also gives a significantly simpler and quantitatively superior proof of the main result of Fang, Göös, Harms, and Hatami (STOC 2025), that constant-cost communication does not reduce to the k-Hamming Distance hierarchy.
Cite as
Mika Göös, Nathaniel Harms, and Artur Riazanov. Equality Is Far Weaker Than Constant-Cost Communication. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{goos_et_al:LIPIcs.APPROX/RANDOM.2025.58,
author = {G\"{o}\"{o}s, Mika and Harms, Nathaniel and Riazanov, Artur},
title = {{Equality Is Far Weaker Than Constant-Cost Communication}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {58:1--58:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.58},
URN = {urn:nbn:de:0030-drops-244246},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.58},
annote = {Keywords: Equality oracle, constant-cost communication, gamma-2 norm, spectral norm}
}
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RANDOM
Testing Tensor Products of Algebraic Codes
Abstract
Motivated by recent advances in locally testable codes and quantum LDPCs based on robust testability of tensor product codes, we explore the local testability of tensor products of (an abstraction of) algebraic geometry codes. Such codes are parameterized by, in addition to standard parameters such as block length n and dimension k, their genus g. We show that the tensor product of two algebraic geometry codes is robustly locally testable provided n = Ω((k+g)²). Apart from Reed-Solomon codes, this seems to be the first explicit family of two-wise tensor codes of high dual distance that is robustly locally testable by the natural test that measures the expected distance of a random row/column from the underlying code.
Cite as
Sumegha Garg, Madhu Sudan, and Gabriel Wu. Testing Tensor Products of Algebraic Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 59:1-59:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{garg_et_al:LIPIcs.APPROX/RANDOM.2025.59,
author = {Garg, Sumegha and Sudan, Madhu and Wu, Gabriel},
title = {{Testing Tensor Products of Algebraic Codes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {59:1--59:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.59},
URN = {urn:nbn:de:0030-drops-244254},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.59},
annote = {Keywords: Algebraic geometry codes, Robust testability, Tensor products of codes}
}
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RANDOM
Sink-Free Orientations: A Local Sampler with Applications
Abstract
For sink-free orientations in graphs of minimum degree at least 3, we show that there is a deterministic approximate counting algorithm that runs in time O((n^33/ε^32)log(n/ε)), a near-linear time sampling algorithm, and a randomised approximate counting algorithm that runs in time O((n/ε)²log(n/ε)), where n denotes the number of vertices of the input graph and 0 < ε < 1 is the desired accuracy. All three algorithms are based on a local implementation of the sink popping method (Cohn, Pemantle, and Propp, 2002) under the partial rejection sampling framework (Guo, Jerrum, and Liu, 2019).
Cite as
Konrad Anand, Graham Freifeld, Heng Guo, Chunyang Wang, and Jiaheng Wang. Sink-Free Orientations: A Local Sampler with Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 60:1-60:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{anand_et_al:LIPIcs.APPROX/RANDOM.2025.60,
author = {Anand, Konrad and Freifeld, Graham and Guo, Heng and Wang, Chunyang and Wang, Jiaheng},
title = {{Sink-Free Orientations: A Local Sampler with Applications}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {60:1--60:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.60},
URN = {urn:nbn:de:0030-drops-244267},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.60},
annote = {Keywords: Sink-free orientations, local sampling, deterministic counting}
}
Document
RANDOM
A Fast Coloring Oracle for Average Case Hypergraphs
Abstract
Hypergraph 2-colorability is one of the classical NP-hard problems. Person and Schacht [SODA'09] designed a deterministic algorithm whose expected running time is polynomial over a uniformly chosen 2-colorable 3-uniform hypergraph. Lee, Molla, and Nagle recently extended this to k-uniform hypergraphs for all k ≥ 3. Both papers relied heavily on the regularity lemma, hence their analysis was involved and their running time hid tower-type constants.
Our first result in this paper is a new simple and elementary deterministic 2-coloring algorithm that reproves the theorems of Person-Schacht and Lee-Molla-Nagle while avoiding the use of the regularity lemma. We also show how to turn our new algorithm into a randomized one with average expected running time of only O(n).
Our second and main result gives what we consider to be the ultimate evidence of just how easy it is to find a 2-coloring of an average 2-colorable hypergraph. We define a coloring oracle to be an algorithm which, given vertex v, assigns color red/blue to v while inspecting as few edges as possible, so that the answers to any sequence of queries to the oracle are consistent with a single legal 2-coloring of the input. Surprisingly, we show that there is a coloring oracle that, on average, can answer every vertex query in time O(1).
Cite as
Cassandra Marcussen, Edward Pyne, Ronitt Rubinfeld, Asaf Shapira, and Shlomo Tauber. A Fast Coloring Oracle for Average Case Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 61:1-61:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{marcussen_et_al:LIPIcs.APPROX/RANDOM.2025.61,
author = {Marcussen, Cassandra and Pyne, Edward and Rubinfeld, Ronitt and Shapira, Asaf and Tauber, Shlomo},
title = {{A Fast Coloring Oracle for Average Case Hypergraphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {61:1--61:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.61},
URN = {urn:nbn:de:0030-drops-244272},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.61},
annote = {Keywords: average-case algorithms, local computation algorithms, graph coloring}
}
Document
RANDOM
Avoiding Range via Turan-Type Bounds
Abstract
Given a circuit C : {0,1}^n → {0,1}^m from a circuit class 𝒞, with m > n, finding a y ∈ {0,1}^m such that ∀ x ∈ {0,1}ⁿ, C(x) ≠ y, is the range avoidance problem (denoted by C-Avoid). Deterministic polynomial time algorithms (even with access to NP oracles) solving this problem are known to imply explicit constructions of various pseudorandom objects like hard Boolean functions, linear codes, PRGs etc.
Deterministic polynomial time algorithms are known for NC⁰₂-Avoid when m > n, and for NC⁰₃-Avoid when m ≥ n²/log n, where NC⁰_k is the class of circuits with bounded fan-in which have constant depth and the output depends on at most k of the input bits. On the other hand, it is also known that NC⁰₃-Avoid when m = n+O(n^{2/3}) is at least as hard as explicit construction of rigid matrices. In fact, algorithms for solving range avoidance for even NC⁰₄ circuits imply new circuit lower bounds.
In this paper, we propose a new approach to solving the range avoidance problem via hypergraphs. We formulate the problem in terms of Turan-type problems in hypergraphs of the following kind: for a fixed k-uniform hypergraph H, what is the maximum number of edges that can exist in H_C, which does not have a sub-hypergraph isomorphic to H? We show the following:
- We first demonstrate the applicability of this approach by showing alternate proofs of some of the known results for the range avoidance problem using this framework.
- We then use our approach to show (using several different hypergraph structures for which Turan-type bounds are known in the literature) that there is a constant c such that Monotone-NC⁰₃-Avoid can be solved in deterministic polynomial time when m > cn².
- To improve the stretch constraint to linear, more precisely, to m > n, we show a new Turan-type theorem for a hypergraph structure (which we call the loose X_{2ℓ}-cycles). More specifically, we prove that any connected 3-uniform linear hypergraph with m > n edges must contain a loose X_{2ℓ} cycle. This may be of independent interest.
- Using this, we show that Monotone-NC⁰₃-Avoid can be solved in deterministic polynomial time when m > n, thus improving the known bounds of NC⁰₃-Avoid for the case of monotone circuits. In contrast, we note that efficient algorithms for solving Monotone-NC⁰₆-Avoid, already imply explicit constructions for rigid matrices.
- We also generalise our argument to solve the special case of range avoidance for NC⁰_k where each output function computed by the circuit is the majority function on its inputs, where m > n².
Cite as
Neha Kuntewar and Jayalal Sarma. Avoiding Range via Turan-Type Bounds. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 62:1-62:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{kuntewar_et_al:LIPIcs.APPROX/RANDOM.2025.62,
author = {Kuntewar, Neha and Sarma, Jayalal},
title = {{Avoiding Range via Turan-Type Bounds}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {62:1--62:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.62},
URN = {urn:nbn:de:0030-drops-244281},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.62},
annote = {Keywords: circuit lower bounds, explicit constructions, range avoidance, linear hypergraphs, Tur\'{a}n number of hypergraphs}
}
Document
RANDOM
Structured-Seed Local Pseudorandom Generators and Their Applications
Abstract
We introduce structured‑seed local pseudorandom generators (SSL-PRGs), pseudorandom generators whose seed is drawn from an efficiently sampleable, structured distribution rather than uniformly. This seemingly modest relaxation turns out to capture many known applications of local PRGs, yet it can be realized from a broader family of hardness assumptions. Our main technical contribution is a generic template for constructing SSL-PRGs that combines the following two ingredients: [i.]
1) noisy‑NC⁰ PRGs, computable by constant‑depth circuits fed with sparse noise, with
2) new local compression schemes for sparse vectors derived from combinatorial batch codes.
Instantiating the template under the sparse Learning‑Parity‑with‑Noise (LPN) assumption yields the first SSL-PRGs with polynomial stretch and constant locality from a subquadratic‑sample search hardness assumption; a mild strengthening of sparse‑LPN gives strong SSL-PRGs of arbitrary polynomial stretch. We further show that for all standard noise distributions, noisy‑local PRGs cannot be emulated by ordinary local PRGs, thereby separating the two notions.
Plugging SSL-PRGs into existing frameworks, we revisit the canonical applications of local PRGs and demonstrate that SSL-PRGs suffice for: (i) indistinguishability obfuscation, (ii) constant-overhead secure computation, (iii) compact homomorphic secret sharing, and (iv) deriving hardness results for PAC‑learning DNFs from sparse‑LPN.
Our work thus broadens the landscape of low‑depth pseudorandomness and anchors several primitives to a common, well‑motivated assumption.
Cite as
Benny Applebaum, Dung Bui, Geoffroy Couteau, and Nikolas Melissaris. Structured-Seed Local Pseudorandom Generators and Their Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 63:1-63:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{applebaum_et_al:LIPIcs.APPROX/RANDOM.2025.63,
author = {Applebaum, Benny and Bui, Dung and Couteau, Geoffroy and Melissaris, Nikolas},
title = {{Structured-Seed Local Pseudorandom Generators and Their Applications}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {63:1--63:26},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.63},
URN = {urn:nbn:de:0030-drops-244293},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.63},
annote = {Keywords: Pseudorandom Generator, Local Pseudorandom Generators, Secure Computation, Obfuscation, Hardness of Learning, Local Compression}
}
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RANDOM
Searching for Falsified Clause in Random (log{n})-CNFs Is Hard for Randomized Communication
Abstract
We show that for a randomly sampled unsatisfiable O(log n)-CNF over n variables the randomized two-party communication cost of finding a clause falsified by the given variable assignment is linear in n.
Cite as
Artur Riazanov, Anastasia Sofronova, Dmitry Sokolov, and Weiqiang Yuan. Searching for Falsified Clause in Random (log{n})-CNFs Is Hard for Randomized Communication. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 64:1-64:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{riazanov_et_al:LIPIcs.APPROX/RANDOM.2025.64,
author = {Riazanov, Artur and Sofronova, Anastasia and Sokolov, Dmitry and Yuan, Weiqiang},
title = {{Searching for Falsified Clause in Random (log\{n\})-CNFs Is Hard for Randomized Communication}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {64:1--64:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.64},
URN = {urn:nbn:de:0030-drops-244306},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.64},
annote = {Keywords: communication complexity, proof complexity, random CNF}
}
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RANDOM
Solving Linear Programs with Differential Privacy
Abstract
We study the problem of solving linear programs of the form Ax ≤ b, x ≥ 0 with differential privacy. For homogeneous LPs Ax ≥ 0, we give an efficient (ε,δ)-differentially private algorithm which with probability at least 1-β finds in polynomial time a solution that satisfies all but O(d²/ε log²(d/(δβ))√{log 1/ρ₀}) constraints, for problems with margin ρ₀ > 0. This improves the bound of O(d⁵/ε log^{1.5} 1/ρ₀ polylog(d,1/δ,1/β)) by [Kaplan-Mansour-Moran-Stemmer-Tur, STOC '25]. For general LPs Ax ≤ b, x ≥ 0 with potentially zero margin, we give an efficient (ε,δ)-differentially private algorithm that w.h.p drops O(d⁴/ε log^{2.5} d/δ √{log dU}) constraints, where U is an upper bound for the entries of A and b in absolute value. This improves the result by Kaplan et al. by at least a factor of d⁵. Our techniques build upon privatizing a rescaling perceptron algorithm by [Hoberg-Rothvoss, IPCO '17] and a more refined iterative procedure for identifying equality constraints by Kaplan et al.
Cite as
Alina Ene, Huy Le Nguyen, Ta Duy Nguyen, and Adrian Vladu. Solving Linear Programs with Differential Privacy. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{ene_et_al:LIPIcs.APPROX/RANDOM.2025.65,
author = {Ene, Alina and Le Nguyen, Huy and Nguyen, Ta Duy and Vladu, Adrian},
title = {{Solving Linear Programs with Differential Privacy}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {65:1--65:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.65},
URN = {urn:nbn:de:0030-drops-244315},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.65},
annote = {Keywords: Differential Privacy, Linear Programming}
}
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RANDOM
Testing Isomorphism of Boolean Functions over Finite Abelian Groups
Abstract
Let f and g be Boolean functions over a finite Abelian group 𝒢, where g is fully known and f is accessible via queries; that is, given any x ∈ 𝒢, we can obtain the value f(x). We study the problem of tolerant isomorphism testing: given parameters ε ≥ 0 and τ > 0, the goal is to determine, using as few queries as possible, whether there exists an automorphism σ of 𝒢 such that the fractional Hamming distance between f∘σ and g is at most ε, or whether for every automorphism σ, the distance is at least ε + τ.
We design an efficient tolerant property testing algorithm for this problem over finite Abelian groups with constant exponent. The exponent of a finite group refers to the largest order of any element in the group. The query complexity of our algorithm is polynomial in s and 1/τ, where s bounds the spectral norm of the function g, and τ is the tolerance parameter. In addition, we present an improved algorithm in the case where g is Fourier sparse, meaning that its Fourier expansion contains only a small number of nonzero coefficients.
Our approach draws on key ideas from Abelian group theory and Fourier analysis, including the annihilator of a subgroup, Pontryagin duality, and a pseudo inner product defined over finite Abelian groups. We believe that these techniques will be useful more broadly in the design of property testing algorithms.
Cite as
Swarnalipa Datta, Arijit Ghosh, Chandrima Kayal, Manaswi Paraashar, and Manmatha Roy. Testing Isomorphism of Boolean Functions over Finite Abelian Groups. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 66:1-66:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{datta_et_al:LIPIcs.APPROX/RANDOM.2025.66,
author = {Datta, Swarnalipa and Ghosh, Arijit and Kayal, Chandrima and Paraashar, Manaswi and Roy, Manmatha},
title = {{Testing Isomorphism of Boolean Functions over Finite Abelian Groups}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {66:1--66:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.66},
URN = {urn:nbn:de:0030-drops-244328},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.66},
annote = {Keywords: Analysis of Boolean functions, Abelian groups, Automorphism group, Function isomorphism, Spectral norm}
}
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RANDOM
On Sums of INW Pseudorandom Generators
Abstract
We study a new approach for constructing pseudorandom generators (PRGs) that fool constant-width standard-order read-once branching programs (ROBPs). Let X be the n-bit output distribution of the INW PRG (Impagliazzo, Nisan, and Wigderson, STOC 1994), instantiated using expansion parameter λ. We prove that the bitwise XOR of t independent copies of X fools width-w programs with error n^{log(w + 1)} ⋅ (λ⋅log n)^t. Notably, this error bound is meaningful even for relatively large values of λ such as λ = 1/O(log n).
Admittedly, our analysis does not yet imply any improvement in the bottom-line overall seed length required for fooling such programs - it just gives a new way of re-proving the well-known O(log² n) bound. Furthermore, we prove that this shortcoming is not an artifact of our analysis, but rather is an intrinsic limitation of our "XOR of INW" approach. That is, no matter how many copies of the INW generator we XOR together, and no matter how we set the expansion parameters, if the generator fools width-3 programs and the proof of correctness does not use any properties of the expander graphs except their spectral expansion, then we prove that the seed length of the generator is inevitably Ω(log² n).
Still, we hope that our work might be a step toward constructing near-optimal PRGs fooling constant-width ROBPs. We suggest that one could try running the INW PRG on t correlated seeds, sampled via another PRG, and taking the bitwise XOR of the outputs.
Cite as
William M. Hoza and Zelin Lv. On Sums of INW Pseudorandom Generators. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 67:1-67:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{hoza_et_al:LIPIcs.APPROX/RANDOM.2025.67,
author = {Hoza, William M. and Lv, Zelin},
title = {{On Sums of INW Pseudorandom Generators}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {67:1--67:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.67},
URN = {urn:nbn:de:0030-drops-244330},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.67},
annote = {Keywords: INW generator, pseudorandomness, space-bounded computation, XOR Lemmas}
}
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RANDOM
Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree
Abstract
We develop a new framework to prove the mixing or relaxation time for the Glauber dynamics on spin systems with unbounded degree. It works for general spin systems including both 2-spin and multi-spin systems. As applications for this approach:
- We prove the optimal O(n) relaxation time for the Glauber dynamics of random q-list-coloring on an n-vertices triangle-tree graph with maximum degree Δ such that q/Δ > α^⋆, where α^⋆ ≈ 1.763 is the unique positive solution of the equation α = exp(1/α). This improves the n^{1+o(1)} relaxation time for Glauber dynamics obtained by the previous work of Jain, Pham, and Vuong (2022). Besides, our framework can also give a near-linear time sampling algorithm under the same condition.
- We prove the optimal O(n) relaxation time and near-optimal Õ(n) mixing time for the Glauber dynamics on hardcore models with parameter λ in balanced bipartite graphs such that λ < λ_c(Δ_L) for the max degree Δ_L in left part and the max degree Δ_R of right part satisfies Δ_R = O(Δ_L). This improves the previous result by Chen, Liu, and Yin (2023). At the heart of our proof is the notion of coupling independence which allows us to consider multiple vertices as a huge single vertex with exponentially large domain and do a "coarse-grained" local-to-global argument on spin systems. The technique works for general (multi) spin systems and helps us obtain some new comparison results for Glauber dynamics.
Cite as
Xiaoyu Chen and Weiming Feng. Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 68:1-68:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)
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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2025.68,
author = {Chen, Xiaoyu and Feng, Weiming},
title = {{Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
pages = {68:1--68:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-397-3},
ISSN = {1868-8969},
year = {2025},
volume = {353},
editor = {Ene, Alina and Chattopadhyay, Eshan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.68},
URN = {urn:nbn:de:0030-drops-244345},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2025.68},
annote = {Keywords: coupling independence, Glauber dynamics, mixing times, relaxation times, spin systems}
}