Volume
LIPIcs, Volume 317
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)
Publication Details
- published at: 2024-09-16
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-348-5
- DBLP: db/conf/approx/approx2024
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Complete Volume
LIPIcs, Volume 317, APPROX/RANDOM 2024, Complete Volume
Abstract
LIPIcs, Volume 317, APPROX/RANDOM 2024, Complete Volume
Cite as
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 1-1496, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@Proceedings{kumar_et_al:LIPIcs.APPROX/RANDOM.2024,
title = {{LIPIcs, Volume 317, APPROX/RANDOM 2024, Complete Volume}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {1--1496},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024},
URN = {urn:nbn:de:0030-drops-209921},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024},
annote = {Keywords: LIPIcs, Volume 317, APPROX/RANDOM 2024, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 0:i-0:xxvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{kumar_et_al:LIPIcs.APPROX/RANDOM.2024.0,
author = {Kumar, Amit and Ron-Zewi, Noga},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {0:i--0:xxvi},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.0},
URN = {urn:nbn:de:0030-drops-209933},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
APPROX
A (3/2 + 1/e)-Approximation Algorithm for Ordered TSP
Abstract
We present a new (3/2 + 1/e)-approximation algorithm for the Ordered Traveling Salesperson Problem (Ordered TSP). Ordered TSP is a variant of the classic metric Traveling Salesperson Problem (TSP) where a specified subset of vertices needs to appear on the output Hamiltonian cycle in a given order, and the task is to compute a cheapest such cycle. Our approximation guarantee of approximately 1.868 holds with respect to the value of a natural new linear programming (LP) relaxation for Ordered TSP. Our result significantly improves upon the previously best known guarantee of 5/2 for this problem and thereby considerably reduces the gap between approximability of Ordered TSP and metric TSP. Our algorithm is based on a decomposition of the LP solution into weighted trees that serve as building blocks in our tour construction.
Cite as
Susanne Armbruster, Matthias Mnich, and Martin Nägele. A (3/2 + 1/e)-Approximation Algorithm for Ordered TSP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{armbruster_et_al:LIPIcs.APPROX/RANDOM.2024.1,
author = {Armbruster, Susanne and Mnich, Matthias and N\"{a}gele, Martin},
title = {{A (3/2 + 1/e)-Approximation Algorithm for Ordered TSP}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {1:1--1:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.1},
URN = {urn:nbn:de:0030-drops-209943},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.1},
annote = {Keywords: Travelling Salesperson Problem, precedence constraints, linear programming, approximation algorithms}
}
Document
APPROX
Online Time-Windows TSP with Predictions
Abstract
In the Time-Windows TSP (TW-TSP) we are given requests at different locations on a network; each request is endowed with a reward and an interval of time; the goal is to find a tour that visits as much reward as possible during the corresponding time window. For the online version of this problem, where each request is revealed at the start of its time window, no finite competitive ratio can be obtained. We consider a version of the problem where the algorithm is presented with predictions of where and when the online requests will appear, without any knowledge of the quality of this side information.
Vehicle routing problems such as the TW-TSP can be very sensitive to errors or changes in the input due to the hard time-window constraints, and it is unclear whether imperfect predictions can be used to obtain a finite competitive ratio. We show that good performance can be achieved by explicitly building slack into the solution. Our main result is an online algorithm that achieves a competitive ratio logarithmic in the diameter of the underlying network, matching the performance of the best offline algorithm to within factors that depend on the quality of the provided predictions. The competitive ratio degrades smoothly as a function of the quality and we show that this dependence is tight within constant factors.
Cite as
Shuchi Chawla and Dimitris Christou. Online Time-Windows TSP with Predictions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 2:1-2:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chawla_et_al:LIPIcs.APPROX/RANDOM.2024.2,
author = {Chawla, Shuchi and Christou, Dimitris},
title = {{Online Time-Windows TSP with Predictions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {2:1--2:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.2},
URN = {urn:nbn:de:0030-drops-209954},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.2},
annote = {Keywords: Travelling Salesman Problem, Predictions, Learning-Augmented Algorithms, Approximation}
}
Document
APPROX
Degrees and Network Design: New Problems and Approximations
Abstract
While much of network design focuses mostly on cost (number or weight of edges), node degrees have also played an important role. They have traditionally either appeared as an objective, to minimize the maximum degree (e.g., the Minimum Degree Spanning Tree problem), or as constraints that might be violated to give bicriteria approximations (e.g., the Minimum Cost Degree Bounded Spanning Tree problem). We extend the study of degrees in network design in two ways. First, we introduce and study a new variant of the Survivable Network Design Problem where in addition to the traditional objective of minimizing the cost of the chosen edges, we add a constraint that the 𝓁_p-norm of the node degree vector is bounded by an input parameter. This interpolates between the classical settings of maximum degree (the 𝓁_∞-norm) and the number of edges (the 𝓁₁-degree), and has natural applications in distributed systems and VLSI design. We give a constant bicriteria approximation in both measures using convex programming. Second, we provide a polylogarithmic bicriteria approximation for the Degree Bounded Group Steiner problem on bounded treewidth graphs, solving an open problem from [Guy Kortsarz and Zeev Nutov, 2022] and [X. Guo et al., 2022].
Cite as
Michael Dinitz, Guy Kortsarz, and Shi Li. Degrees and Network Design: New Problems and Approximations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 3:1-3:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{dinitz_et_al:LIPIcs.APPROX/RANDOM.2024.3,
author = {Dinitz, Michael and Kortsarz, Guy and Li, Shi},
title = {{Degrees and Network Design: New Problems and Approximations}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {3:1--3:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.3},
URN = {urn:nbn:de:0030-drops-209969},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.3},
annote = {Keywords: Network Design, Degrees}
}
Document
APPROX
Hybrid k-Clustering: Blending k-Median and k-Center
Abstract
We propose a novel clustering model encompassing two well-known clustering models: k-center clustering and k-median clustering. In the Hybrid k-Clustering problem, given a set P of points in ℝ^d, an integer k, and a non-negative real r, our objective is to position k closed balls of radius r to minimize the sum of distances from points not covered by the balls to their closest balls. Equivalently, we seek an optimal L₁-fitting of a union of k balls of radius r to a set of points in the Euclidean space. When r = 0, this corresponds to k-median; when the minimum sum is zero, indicating complete coverage of all points, it is k-center.
Our primary result is a bicriteria approximation algorithm that, for a given ε > 0, produces a hybrid k-clustering with balls of radius (1+ε)r. This algorithm achieves a cost at most 1+ε of the optimum, and it operates in time 2^{(kd/ε)^𝒪(1)} ⋅ n^𝒪(1). Notably, considering the established lower bounds on k-center and k-median, our bicriteria approximation stands as the best possible result for Hybrid k-Clustering.
Cite as
Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi. Hybrid k-Clustering: Blending k-Median and k-Center. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{fomin_et_al:LIPIcs.APPROX/RANDOM.2024.4,
author = {Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Saurabh, Saket and Zehavi, Meirav},
title = {{Hybrid k-Clustering: Blending k-Median and k-Center}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {4:1--4:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.4},
URN = {urn:nbn:de:0030-drops-209975},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.4},
annote = {Keywords: clustering, k-center, k-median, Euclidean space, fpt approximation}
}
Document
APPROX
Asynchronous Majority Dynamics on Binomial Random Graphs
Abstract
We study information aggregation in networks when agents interact to learn a binary state of the world. Initially each agent privately observes an independent signal which is correct with probability 1/2+δ for some δ > 0. At each round, a node is selected uniformly at random to update their public opinion to match the majority of their neighbours (breaking ties in favour of their initial private signal). Our main result shows that for sparse and connected binomial random graphs G(n,p) the process stabilizes in a correct consensus in 𝒪(nlog² n/log log n) steps with high probability. In fact, when log n/n ≪ p = o(1) the process terminates at time T^ = (1+o(1))nlog n, where T^ is the first time when all nodes have been selected at least once. However, in dense binomial random graphs with p = Ω(1), there is an information cascade where the process terminates in the incorrect consensus with probability bounded away from zero.
Cite as
Divyarthi Mohan and Paweł Prałat. Asynchronous Majority Dynamics on Binomial Random Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{mohan_et_al:LIPIcs.APPROX/RANDOM.2024.5,
author = {Mohan, Divyarthi and Pra{\l}at, Pawe{\l}},
title = {{Asynchronous Majority Dynamics on Binomial Random Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {5:1--5:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.5},
URN = {urn:nbn:de:0030-drops-209985},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.5},
annote = {Keywords: Opinion dynamics, Social learning, Stochastic processes, Random Graphs, Consensus}
}
Document
APPROX
Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3
Abstract
In a disk graph, every vertex corresponds to a disk in ℝ² and two vertices are connected by an edge whenever the two corresponding disks intersect. Disk graphs form an important class of geometric intersection graphs, which generalizes both planar graphs and unit-disk graphs. We study a fundamental optimization problem in algorithmic graph theory, Bipartization (also known as Odd Cycle Transversal), on the class of disk graphs. The goal of Bipartization is to delete a minimum number of vertices from the input graph such that the resulting graph is bipartite. A folklore (polynomial-time) 3-approximation algorithm for Bipartization on disk graphs follows from the classical framework of Goemans and Williamson [Combinatorica'98] for cycle-hitting problems. For over two decades, this result has remained the best known approximation for the problem (in fact, even for Bipartization on unit-disk graphs). In this paper, we achieve the first improvement upon this result, by giving a (3-α)-approximation algorithm for Bipartization on disk graphs, for some constant α > 0. Our algorithm directly generalizes to the broader class of pseudo-disk graphs. Furthermore, our algorithm is robust in the sense that it does not require a geometric realization of the input graph to be given.
Cite as
Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, and Meirav Zehavi. Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{lokshtanov_et_al:LIPIcs.APPROX/RANDOM.2024.6,
author = {Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Xue, Jie and Zehavi, Meirav},
title = {{Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {6:1--6:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.6},
URN = {urn:nbn:de:0030-drops-209990},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.6},
annote = {Keywords: bipartization, geometric intersection graphs, approximation algorithms}
}
Document
APPROX
A Logarithmic Approximation of Linearly-Ordered Colourings
Abstract
A linearly ordered (LO) k-colouring of a hypergraph assigns to each vertex a colour from the set {0,1,…,k-1} in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO k-colouring of an LO 2-colourable 3-uniform hypergraph for any constant k ≥ 2 [STACS'21] but even the case k = 3 is still open. Nakajima and Živný gave polynomial-time algorithms for finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with O^*(√n) colours [ICALP'22] and an LO colouring with O^*(n^(1/3)) colours [ACM ToCT'23]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with O^*(n^(1/5)) colours. We present two simple polynomial-time algorithms that find an LO colouring with O(log₂(n)) colours, which is an exponential improvement.
Cite as
Johan Håstad, Björn Martinsson, Tamio-Vesa Nakajima, and Stanislav Živný. A Logarithmic Approximation of Linearly-Ordered Colourings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 7:1-7:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{hastad_et_al:LIPIcs.APPROX/RANDOM.2024.7,
author = {H\r{a}stad, Johan and Martinsson, Bj\"{o}rn and Nakajima, Tamio-Vesa and \v{Z}ivn\'{y}, Stanislav},
title = {{A Logarithmic Approximation of Linearly-Ordered Colourings}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {7:1--7:6},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.7},
URN = {urn:nbn:de:0030-drops-210006},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.7},
annote = {Keywords: Linear ordered colouring, Hypergraph, Approximation, Promise Constraint Satisfaction Problems}
}
Document
APPROX
Speed-Robust Scheduling Revisited
Abstract
Speed-robust scheduling is the following two-stage problem of scheduling n jobs on m uniformly related machines. In the first stage, the algorithm receives the value of m and the processing times of n jobs; it has to partition the jobs into b groups called bags. In the second stage, the machine speeds are revealed and the bags are assigned to the machines, i.e., the algorithm produces a schedule where all the jobs in the same bag are assigned to the same machine. The objective is to minimize the makespan (the length of the schedule). The algorithm is compared to the optimal schedule and it is called ρ-robust, if its makespan is always at most ρ times the optimal one.
Our main result is an improved bound for equal-size jobs for b = m. We give an upper bound of 1.6. This improves previous bound of 1.8 and it is almost tight in the light of previous lower bound of 1.58. Second, for infinitesimally small jobs, we give tight upper and lower bounds for the case when b ≥ m. This generalizes and simplifies the previous bounds for b = m. Finally, we introduce a new special case with relatively small jobs for which we give an algorithm whose robustness is close to that of infinitesimal jobs and thus gives better than 2-robust for a large class of inputs.
Cite as
Josef Minařík and Jiří Sgall. Speed-Robust Scheduling Revisited. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{minarik_et_al:LIPIcs.APPROX/RANDOM.2024.8,
author = {Mina\v{r}{\'\i}k, Josef and Sgall, Ji\v{r}{\'\i}},
title = {{Speed-Robust Scheduling Revisited}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {8:1--8:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.8},
URN = {urn:nbn:de:0030-drops-210010},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.8},
annote = {Keywords: scheduling, approximation algorithms, makespan, uniform speeds}
}
Document
APPROX
On the Generalized Mean Densest Subgraph Problem: Complexity and Algorithms
Abstract
Dense subgraph discovery is an important problem in graph mining and network analysis with several applications. Two canonical polynomial-time solvable problems here are to find a maxcore (subgraph of maximum min degree) and to find a densest subgraph (subgraph of maximum average degree). Both of these problems can be solved in polynomial time. Veldt, Benson, and Kleinberg [Veldt et al., 2021] introduced the generalized p-mean densest subgraph problem which captures the maxcore problem when p = -∞ and the densest subgraph problem when p = 1. They observed that for p ≥ 1, the objective function is supermodular and hence the problem can be solved in polynomial time. In this work, we focus on the p-mean densest subgraph problem for p ∈ (-∞, 1). We prove that for every p ∈ (-∞,1), the problem is NP-hard, thus resolving an open question from [Veldt et al., 2021]. We also show that for every p ∈ (0,1), the weighted version of the problem is APX-hard. On the algorithmic front, we describe two simple 1/2-approximation algorithms for every p ∈ (-∞, 1). We complement the approximation algorithms by exhibiting non-trivial instances on which the algorithms simultaneously achieve an approximation factor of at most 1/2.
Cite as
Karthekeyan Chandrasekaran, Chandra Chekuri, Manuel R. Torres, and Weihao Zhu. On the Generalized Mean Densest Subgraph Problem: Complexity and Algorithms. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chandrasekaran_et_al:LIPIcs.APPROX/RANDOM.2024.9,
author = {Chandrasekaran, Karthekeyan and Chekuri, Chandra and Torres, Manuel R. and Zhu, Weihao},
title = {{On the Generalized Mean Densest Subgraph Problem: Complexity and Algorithms}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {9:1--9:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.9},
URN = {urn:nbn:de:0030-drops-210025},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.9},
annote = {Keywords: Densest subgraph problem, Hardness of approximation, Approximation algorithms}
}
Document
APPROX
Improved Online Load Balancing with Known Makespan
Abstract
We break the barrier of 3/2 for the problem of online load balancing with known makespan, also known as bin stretching. In this problem, m identical machines and the optimal makespan are given. The load of a machine is the total size of all the jobs assigned to it and the makespan is the maximum load of all the machines. Jobs arrive online and the goal is to assign each job to a machine while staying within a small factor (the competitive ratio) of the optimal makespan.
We present an algorithm that maintains a competitive ratio of 139/93 < 1.495 for sufficiently large values of m, improving the previous bound of 3/2. The value 3/2 represents a natural bound for this problem: as long as the online bins are of size at least 3/2 of the offline bin, all items that fit at least two times in an offline bin have two nice properties. They fit three times in an online bin and a single such item can be packed together with an item of any size in an online bin. These properties are now both lost, which means that putting even one job on a wrong machine can leave some job unassigned at the end. It also makes it harder to determine good thresholds for the item types. This was one of the main technical issues in getting below 3/2.
The analysis consists of an intricate mixture of size and weight arguments.
Cite as
Martin Böhm, Matej Lieskovský, Sören Schmitt, Jiří Sgall, and Rob van Stee. Improved Online Load Balancing with Known Makespan. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bohm_et_al:LIPIcs.APPROX/RANDOM.2024.10,
author = {B\"{o}hm, Martin and Lieskovsk\'{y}, Matej and Schmitt, S\"{o}ren and Sgall, Ji\v{r}{\'\i} and van Stee, Rob},
title = {{Improved Online Load Balancing with Known Makespan}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {10:1--10:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.10},
URN = {urn:nbn:de:0030-drops-210032},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.10},
annote = {Keywords: Online algorithms, bin stretching, bin packing}
}
Document
APPROX
On the NP-Hardness Approximation Curve for Max-2Lin(2)
Abstract
In the Max-2Lin(2) problem you are given a system of equations on the form x_i + x_j ≡ b mod 2, and your objective is to find an assignment that satisfies as many equations as possible. Let c ∈ [0.5, 1] denote the maximum fraction of satisfiable equations. In this paper we construct a curve s (c) such that it is NP-hard to find a solution satisfying at least a fraction s of equations. This curve either matches or improves all of the previously known inapproximability NP-hardness results for Max-2Lin(2). In particular, we show that if c ⩾ 0.9232 then (1 - s(c))/(1 - c) > 1.48969, which improves the NP-hardness inapproximability constant for the min deletion version of Max-2Lin(2). Our work complements the work of O'Donnell and Wu that studied the same question assuming the Unique Games Conjecture.
Similar to earlier inapproximability results for Max-2Lin(2), we use a gadget reduction from the (2^k - 1)-ary Hadamard predicate. Previous works used k ranging from 2 to 4. Our main result is a procedure for taking a gadget for some fixed k, and use it as a building block to construct better and better gadgets as k tends to infinity. Our method can be used to boost the result of both smaller gadgets created by hand (k = 3) or larger gadgets constructed using a computer (k = 4).
Cite as
Björn Martinsson. On the NP-Hardness Approximation Curve for Max-2Lin(2). In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 11:1-11:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{martinsson:LIPIcs.APPROX/RANDOM.2024.11,
author = {Martinsson, Bj\"{o}rn},
title = {{On the NP-Hardness Approximation Curve for Max-2Lin(2)}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {11:1--11:38},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.11},
URN = {urn:nbn:de:0030-drops-210049},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.11},
annote = {Keywords: Inapproximability, NP-hardness, 2Lin(2), Max-Cut, Gadget}
}
Document
APPROX
Universal Optimization for Non-Clairvoyant Subadditive Joint Replenishment
Abstract
The online joint replenishment problem (JRP) is a fundamental problem in the area of online problems with delay. Over the last decade, several works have studied generalizations of JRP with different cost functions for servicing requests. Most prior works on JRP and its generalizations have focused on the clairvoyant setting. Recently, Touitou [Noam Touitou, 2023] developed a non-clairvoyant framework that provided an O(√{n log n}) upper bound for a wide class of generalized JRP, where n is the number of request types.
We advance the study of non-clairvoyant algorithms by providing a simpler, modular framework that matches the competitive ratio established by Touitou for the same class of generalized JRP. Our key insight is to leverage universal algorithms for Set Cover to approximate arbitrary monotone subadditive functions using a simple class of functions termed disjoint. This allows us to reduce the problem to several independent instances of the TCP Acknowledgement problem, for which a simple 2-competitive non-clairvoyant algorithm is known. The modularity of our framework is a major advantage as it allows us to tailor the reduction to specific problems and obtain better competitive ratios. In particular, we obtain tight O(√n)-competitive algorithms for two significant problems: Multi-Level Aggregation and Weighted Symmetric Subadditive Joint Replenishment. We also show that, in contrast, Touitou’s algorithm is Ω(√{n log n})-competitive for both of these problems.
Cite as
Tomer Ezra, Stefano Leonardi, Michał Pawłowski, Matteo Russo, and Seeun William Umboh. Universal Optimization for Non-Clairvoyant Subadditive Joint Replenishment. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 12:1-12:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{ezra_et_al:LIPIcs.APPROX/RANDOM.2024.12,
author = {Ezra, Tomer and Leonardi, Stefano and Paw{\l}owski, Micha{\l} and Russo, Matteo and Umboh, Seeun William},
title = {{Universal Optimization for Non-Clairvoyant Subadditive Joint Replenishment}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {12:1--12:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.12},
URN = {urn:nbn:de:0030-drops-210050},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.12},
annote = {Keywords: Set Cover, Joint Replenishment, TCP-Acknowledgment, Subadditive Function Approximation, Multi-Level Aggregation}
}
Document
APPROX
The Average-Value Allocation Problem
Abstract
We initiate the study of centralized algorithms for welfare-maximizing allocation of goods to buyers subject to average-value constraints. We show that this problem is NP-hard to approximate beyond a factor of e/(e-1), and provide a 4e/(e-1)-approximate offline algorithm. For the online setting, we show that no non-trivial approximations are achievable under adversarial arrivals. Under i.i.d. arrivals, we present a polytime online algorithm that provides a constant approximation of the optimal (computationally-unbounded) online algorithm. In contrast, we show that no constant approximation of the ex-post optimum is achievable by an online algorithm.
Cite as
Kshipra Bhawalkar, Zhe Feng, Anupam Gupta, Aranyak Mehta, David Wajc, and Di Wang. The Average-Value Allocation Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 13:1-13:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bhawalkar_et_al:LIPIcs.APPROX/RANDOM.2024.13,
author = {Bhawalkar, Kshipra and Feng, Zhe and Gupta, Anupam and Mehta, Aranyak and Wajc, David and Wang, Di},
title = {{The Average-Value Allocation Problem}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {13:1--13:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.13},
URN = {urn:nbn:de:0030-drops-210062},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.13},
annote = {Keywords: Resource allocation, return-on-spend constraint, approximation algorithm, online algorithm}
}
Document
APPROX
Scheduling on a Stochastic Number of Machines
Abstract
We consider a new scheduling problem on parallel identical machines in which the number of machines is initially not known, but it follows a given probability distribution. Only after all jobs are assigned to a given number of bags, the actual number of machines is revealed. Subsequently, the jobs need to be assigned to the machines without splitting the bags. This is the stochastic version of a related problem introduced by Stein and Zhong [SODA 2018, TALG 2020] and it is, for example, motivated by bundling jobs that need to be scheduled by data centers. We present two PTASs for the stochastic setting, computing job-to-bag assignments that (i) minimize the expected maximum machine load and (ii) maximize the expected minimum machine load (like in the Santa Claus problem), respectively. The former result follows by careful enumeration combined with known PTASs. For the latter result, we introduce an intricate dynamic program that we apply to a suitably rounded instance.
Cite as
Moritz Buchem, Franziska Eberle, Hugo Kooki Kasuya Rosado, Kevin Schewior, and Andreas Wiese. Scheduling on a Stochastic Number of Machines. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{buchem_et_al:LIPIcs.APPROX/RANDOM.2024.14,
author = {Buchem, Moritz and Eberle, Franziska and Kasuya Rosado, Hugo Kooki and Schewior, Kevin and Wiese, Andreas},
title = {{Scheduling on a Stochastic Number of Machines}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {14:1--14:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.14},
URN = {urn:nbn:de:0030-drops-210073},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.14},
annote = {Keywords: scheduling, approximation algorithms, stochastic machines, makespan, max-min fair allocation, dynamic programming}
}
Document
APPROX
Distributional Online Weighted Paging with Limited Horizon
Abstract
In this work we study the classic problem of online weighted paging with a probabilistic prediction model, in which we are given additional information about the input in the form of distributions over page requests, known as distributional online paging (DOP). This work continues a recent line of research on learning-augmented algorithms that incorporates machine-learning predictions in online algorithms, so as to go beyond traditional worst-case competitive analysis, thus circumventing known lower bounds for online paging. We first provide an efficient online algorithm that achieves a constant factor competitive ratio with respect to the best online algorithm (policy) for weighted DOP that follows from earlier work on the stochastic k-server problem.
Our main contribution concerns the question of whether distributional information over a limited horizon suffices for obtaining a constant competitive factor. To this end, we define in a natural way a new predictive model with limited horizon, which we call Per-Request Stochastic Prediction (PRSP). We show that we can obtain a constant factor competitive algorithm with respect to the optimal online algorithm for this model.
Cite as
Yaron Fairstein, Joseph (Seffi) Naor, and Tomer Tsachor. Distributional Online Weighted Paging with Limited Horizon. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{fairstein_et_al:LIPIcs.APPROX/RANDOM.2024.15,
author = {Fairstein, Yaron and Naor, Joseph (Seffi) and Tsachor, Tomer},
title = {{Distributional Online Weighted Paging with Limited Horizon}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {15:1--15:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.15},
URN = {urn:nbn:de:0030-drops-210088},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.15},
annote = {Keywords: Online algorithms, Caching, Stochastic analysis, Predictions}
}
Document
APPROX
Weighted Matching in the Random-Order Streaming and Robust Communication Models
Abstract
We study the maximum weight matching problem in the random-order semi-streaming model and in the robust communication model. Unlike many other sublinear models, in these two frameworks, there is a large gap between the guarantees of the best known algorithms for the unweighted and weighted versions of the problem.
In the random-order semi-streaming setting, the edges of an n-vertex graph arrive in a stream in a random order. The goal is to compute an approximate maximum weight matching with a single pass over the stream using O(npolylog n) space. Our main result is a (2/3-ε)-approximation algorithm for maximum weight matching in random-order streams, using space O(n log n log R), where R is the ratio between the heaviest and the lightest edge in the graph. Our result nearly matches the best known unweighted (2/3+ε₀)-approximation (where ε₀ ∼ 10^{-14} is a small constant) achieved by Assadi and Behnezhad [Assadi and Behnezhad, 2021], and significantly improves upon previous weighted results. Our techniques also extend to the related robust communication model, in which the edges of a graph are partitioned randomly between Alice and Bob. Alice sends a single message of size O(npolylog n) to Bob, who must compute an approximate maximum weight matching. We achieve a (5/6-ε)-approximation using O(n log n log R) words of communication, matching the results of Azarmehr and Behnezhad [Azarmehr and Behnezhad, 2023] for unweighted graphs.
Cite as
Diba Hashemi and Weronika Wrzos-Kaminska. Weighted Matching in the Random-Order Streaming and Robust Communication Models. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 16:1-16:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{hashemi_et_al:LIPIcs.APPROX/RANDOM.2024.16,
author = {Hashemi, Diba and Wrzos-Kaminska, Weronika},
title = {{Weighted Matching in the Random-Order Streaming and Robust Communication Models}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {16:1--16:26},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.16},
URN = {urn:nbn:de:0030-drops-210097},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.16},
annote = {Keywords: Maximum Weight Matching, Streaming, Random-Order Streaming, Robust Communication Complexity}
}
Document
APPROX
Competitive Query Minimization for Stable Matching with One-Sided Uncertainty
Abstract
We study the two-sided stable matching problem with one-sided uncertainty for two sets of agents A and B, with equal cardinality. Initially, the preference lists of the agents in A are given but the preferences of the agents in B are unknown. An algorithm can make queries to reveal information about the preferences of the agents in B. We examine three query models: comparison queries, interviews, and set queries. Using competitive analysis, our aim is to design algorithms that minimize the number of queries required to solve the problem of finding a stable matching or verifying that a given matching is stable (or stable and optimal for the agents of one side). We present various upper and lower bounds on the best possible competitive ratio as well as results regarding the complexity of the offline problem of determining the optimal query set given full information.
Cite as
Evripidis Bampis, Konstantinos Dogeas, Thomas Erlebach, Nicole Megow, Jens Schlöter, and Amitabh Trehan. Competitive Query Minimization for Stable Matching with One-Sided Uncertainty. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bampis_et_al:LIPIcs.APPROX/RANDOM.2024.17,
author = {Bampis, Evripidis and Dogeas, Konstantinos and Erlebach, Thomas and Megow, Nicole and Schl\"{o}ter, Jens and Trehan, Amitabh},
title = {{Competitive Query Minimization for Stable Matching with One-Sided Uncertainty}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {17:1--17:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.17},
URN = {urn:nbn:de:0030-drops-210100},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.17},
annote = {Keywords: Matching under Preferences, Stable Marriage, Query-Competitive Algorithms, Uncertainty}
}
Document
APPROX
A Constant Factor Approximation for Directed Feedback Vertex Set in Graphs of Bounded Genus
Abstract
The minimum directed feedback vertex set problem consists in finding the minimum set of vertices that should be removed in order to make a directed graph acyclic. This is a well-known NP-hard optimization problem with applications in various fields, such as VLSI chip design, bioinformatics and transaction processing deadlock prevention and node-weighted network design. We show a constant factor approximation for the directed feedback vertex set problem in graphs of bounded genus.
Cite as
Hao Sun. A Constant Factor Approximation for Directed Feedback Vertex Set in Graphs of Bounded Genus. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{sun:LIPIcs.APPROX/RANDOM.2024.18,
author = {Sun, Hao},
title = {{A Constant Factor Approximation for Directed Feedback Vertex Set in Graphs of Bounded Genus}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {18:1--18:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.18},
URN = {urn:nbn:de:0030-drops-210112},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.18},
annote = {Keywords: Feedback Vertex Set, Combinatorial Optimization, Approximation Algorithms, min-max relation, linear programming}
}
Document
APPROX
More Basis Reduction for Linear Codes: Backward Reduction, BKZ, Slide Reduction, and More
Abstract
We expand on recent exciting work of Debris-Alazard, Ducas, and van Woerden [Transactions on Information Theory, 2022], which introduced the notion of basis reduction for codes, in analogy with the extremely successful paradigm of basis reduction for lattices. We generalize DDvW’s LLL algorithm and size-reduction algorithm from codes over 𝔽₂ to codes over 𝔽_q, and we further develop the theory of proper bases. We then show how to instantiate for codes the BKZ and slide-reduction algorithms, which are the two most important generalizations of the LLL algorithm for lattices.
Perhaps most importantly, we show a new and very efficient basis-reduction algorithm for codes, called full backward reduction. This algorithm is quite specific to codes and seems to have no analogue in the lattice setting. We prove that this algorithm finds vectors as short as LLL does in the worst case (i.e., within the Griesmer bound) and does so in less time. We also provide both heuristic and empirical evidence that it outperforms LLL in practice, and we give a variant of the algorithm that provably outperforms LLL (in some sense) for random codes.
Finally, we explore the promise and limitations of basis reduction for codes. In particular, we show upper and lower bounds on how "good" of a basis a code can have, and we show two additional illustrative algorithms that demonstrate some of the promise and the limitations of basis reduction for codes.
Cite as
Surendra Ghentiyala and Noah Stephens-Davidowitz. More Basis Reduction for Linear Codes: Backward Reduction, BKZ, Slide Reduction, and More. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 19:1-19:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{ghentiyala_et_al:LIPIcs.APPROX/RANDOM.2024.19,
author = {Ghentiyala, Surendra and Stephens-Davidowitz, Noah},
title = {{More Basis Reduction for Linear Codes: Backward Reduction, BKZ, Slide Reduction, and More}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {19:1--19:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.19},
URN = {urn:nbn:de:0030-drops-210120},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.19},
annote = {Keywords: Linear Codes, Basis Reduction}
}
Document
APPROX
Online k-Median with Consistent Clusters
Abstract
We consider the problem in which n points arrive online over time, and upon arrival must be irrevocably assigned to one of k clusters where the objective is the standard k-median objective. Lower-bound instances show that for this problem no online algorithm can achieve a competitive ratio bounded by any function of n. Thus we turn to a beyond worst-case analysis approach, namely we assume that the online algorithm is a priori provided with a predicted budget B that is an upper bound to the optimal objective value (e.g., obtained from past instances). Our main result is an online algorithm whose competitive ratio (measured against B) is solely a function of k. We also give a lower bound showing that the competitive ratio of every algorithm must depend on k.
Cite as
Benjamin Moseley, Heather Newman, and Kirk Pruhs. Online k-Median with Consistent Clusters. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 20:1-20:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{moseley_et_al:LIPIcs.APPROX/RANDOM.2024.20,
author = {Moseley, Benjamin and Newman, Heather and Pruhs, Kirk},
title = {{Online k-Median with Consistent Clusters}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {20:1--20:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.20},
URN = {urn:nbn:de:0030-drops-210133},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.20},
annote = {Keywords: k-median, online algorithms, learning-augmented algorithms, beyond worst-case analysis}
}
Document
APPROX
The Telephone k-Multicast Problem
Abstract
We consider minimum time multicasting problems in directed and undirected graphs: given a root node and a subset of t terminal nodes, multicasting seeks to find the minimum number of rounds within which all terminals can be informed with a message originating at the root. In each round, the telephone model we study allows the information to move via a matching from the informed nodes to the uninformed nodes.
Since minimum time multicasting in digraphs is poorly understood compared to the undirected variant, we study an intermediate problem in undirected graphs that specifies a target k < t, and requires the only k of the terminals be informed in the minimum number of rounds. For this problem, we improve implications of prior results and obtain an Õ(t^{1/3}) multiplicative approximation. For the directed version, we obtain an additive Õ(k^{1/2}) approximation algorithm (with a poly-logarithmic multiplicative factor). Our algorithms are based on reductions to the related problems of finding k-trees of minimum poise (sum of maximum degree and diameter) and applying a combination of greedy network decomposition techniques and set covering under partition matroid constraints.
Cite as
Daniel Hathcock, Guy Kortsarz, and R. Ravi. The Telephone k-Multicast Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{hathcock_et_al:LIPIcs.APPROX/RANDOM.2024.21,
author = {Hathcock, Daniel and Kortsarz, Guy and Ravi, R.},
title = {{The Telephone k-Multicast Problem}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {21:1--21:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.21},
URN = {urn:nbn:de:0030-drops-210148},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.21},
annote = {Keywords: Network Design, Multicast, Steiner Poise}
}
Document
APPROX
Scheduling Splittable Jobs on Configurable Machines
Abstract
Motivated by modern architectures allowing for the partitioning of a GPU into hardware separated instances, we initiate the study of scheduling splittable jobs on configurable machines. We consider machines that can be configured into smaller instances, which we call blocks, in multiple ways, each of which is referred to as a configuration. We introduce the Configurable Machine Scheduling (cms) problem, where we are given n jobs and a set C of configurations. A schedule consists of a set of machines, each assigned some configuration in C with each block in the configuration assigned to process one job. The amount of a job’s demand that is satisfied by a block is given by an arbitrary function of the job and block. The objective is to construct a schedule using as few machines as possible. We provide a tight logarithmic factor approximation algorithm for this problem in the general setting, a factor (3 + ε) approximation algorithm for arbitrary ε > 0 when there are O(1) input configurations, and a polynomial time approximation scheme when both the number and size of configurations are O(1). Finally, we utilize a technique for finding conic integer combinations in fixed dimension to develop an optimal polynomial time algorithm in the case with O(1) jobs, O(1) blocks, and every configuration up to a given size.
Cite as
Matthew Casey, Rajmohan Rajaraman, David Stalfa, and Cheng Tan. Scheduling Splittable Jobs on Configurable Machines. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{casey_et_al:LIPIcs.APPROX/RANDOM.2024.22,
author = {Casey, Matthew and Rajaraman, Rajmohan and Stalfa, David and Tan, Cheng},
title = {{Scheduling Splittable Jobs on Configurable Machines}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {22:1--22:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.22},
URN = {urn:nbn:de:0030-drops-210157},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.22},
annote = {Keywords: Scheduling algorithms, Approximation algorithms, Configurable machines, Splittable jobs, Linear programming}
}
Document
APPROX
On Instance-Optimal Algorithms for a Generalization of Nuts and Bolts and Generalized Sorting
Abstract
We generalize the classical nuts and bolts problem to a setting where the input is a collection of n nuts and m bolts, and there is no promise of any matching pairs. It is not allowed to compare a nut directly with a nut or a bolt directly with a bolt, and the goal is to perform the fewest nut-bolt comparisons to discover the partial order between the nuts and bolts. We term this problem bipartite sorting.
We show that instances of bipartite sorting of the same size exhibit a wide range of complexity, and propose to perform a fine-grained analysis for this problem. We rule out straightforward notions of instance-optimality as being too stringent, and adopt a neighborhood-based definition. Our definition may be of independent interest as a unifying lens for instance-optimal algorithms for other static problems existing in literature. This includes problems like sorting (Estivill-Castro and Woods, ACM Comput. Surv. 1992), convex hull (Afshani, Barbay and Chan, JACM 2017), adaptive joins (Demaine, López-Ortiz and Munro, SODA 2000), and the recent concept of universal optimality for graphs (Haeupler, Hladík, Rozhoň, Tarjan and Tětek, 2023).
As our main result on bipartite sorting, we give a randomized algorithm that is within a factor of O(log³(n+m)) of being instance-optimal w.h.p., with respect to the neighborhood-based definition.
As our second contribution, we generalize bipartite sorting to DAG sorting, when the underlying DAG is not necessarily bipartite. As an unexpected consequence of a simple algorithm for DAG sorting, we rule out a potential lower bound on the widely-studied problem of sorting with priced information, posed by (Charikar, Fagin, Guruswami, Kleinberg, Raghavan and Sahai, STOC 2000). In this problem, comparing keys i and j has a known cost c_{ij} ∈ ℝ^+ ∪ {∞}, and the goal is to sort the keys in an instance-optimal way, by keeping the total cost of an algorithm as close as possible to ∑_{i=1}^{n-1} c_{x(i)x(i+1)}. Here x(1) < ⋯ < x(n) is the sorted order. While several special cases of cost functions have received a lot of attention in the community, no progress on the general version with arbitrary costs has been reported so far. One reason for this lack of progress seems to be a widely-cited Ω(n) lower bound on the competitive ratio for finding the maximum. This Ω(n) lower bound by (Gupta and Kumar, FOCS 2000) uses costs in {0,1,n, ∞}, and although not extended to sorting, this barrier seems to have stalled any progress on the general cost case.
We rule out such a potential lower bound by showing the existence of an algorithm with a Õ(n^{3/4}) competitive ratio for the {0,1,n,∞} cost version. This generalizes the setting of generalized sorting proposed by (Huang, Kannan and Khanna, FOCS 2011), where the costs are either 1 or infinity, and the cost of the cheapest proof is always n-1.
Cite as
Mayank Goswami and Riko Jacob. On Instance-Optimal Algorithms for a Generalization of Nuts and Bolts and Generalized Sorting. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 23:1-23:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{goswami_et_al:LIPIcs.APPROX/RANDOM.2024.23,
author = {Goswami, Mayank and Jacob, Riko},
title = {{On Instance-Optimal Algorithms for a Generalization of Nuts and Bolts and Generalized Sorting}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {23:1--23:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.23},
URN = {urn:nbn:de:0030-drops-210168},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.23},
annote = {Keywords: Sorting, Priced Information, Instance Optimality, Nuts and Bolts}
}
Document
APPROX
Learning-Augmented Maximum Independent Set
Abstract
We study the Maximum Independent Set (MIS) problem on general graphs within the framework of learning-augmented algorithms. The MIS problem is known to be NP-hard and is also NP-hard to approximate to within a factor of n^(1-δ) for any δ > 0. We show that we can break this barrier in the presence of an oracle obtained through predictions from a machine learning model that answers vertex membership queries for a fixed MIS with probability 1/2+ε. In the first setting we consider, the oracle can be queried once per vertex to know if a vertex belongs to a fixed MIS, and the oracle returns the correct answer with probability 1/2 + ε. Under this setting, we show an algorithm that obtains an Õ((√Δ)/ε)-approximation in O(m) time where Δ is the maximum degree of the graph. In the second setting, we allow multiple queries to the oracle for a vertex, each of which is correct with probability 1/2 + ε. For this setting, we show an O(1)-approximation algorithm using O(n/ε²) total queries and Õ(m) runtime.
Cite as
Vladimir Braverman, Prathamesh Dharangutte, Vihan Shah, and Chen Wang. Learning-Augmented Maximum Independent Set. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{braverman_et_al:LIPIcs.APPROX/RANDOM.2024.24,
author = {Braverman, Vladimir and Dharangutte, Prathamesh and Shah, Vihan and Wang, Chen},
title = {{Learning-Augmented Maximum Independent Set}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {24:1--24:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.24},
URN = {urn:nbn:de:0030-drops-210179},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.24},
annote = {Keywords: Learning-augmented algorithms, maximum independent set, graph algorithms}
}
Document
APPROX
Maximum Unique Coverage on Streams: Improved FPT Approximation Scheme and Tighter Space Lower Bound
Abstract
We consider the Max Unique Coverage problem, including applications to the data stream model. The input is a universe of n elements, a collection of m subsets of this universe, and a cardinality constraint, k. The goal is to select a subcollection of at most k sets that maximizes unique coverage, i.e, the number of elements contained in exactly one of the selected sets. The Max Unique Coverage problem has applications in wireless networks, radio broadcast, and envy-free pricing.
Our first main result is a fixed-parameter tractable approximation scheme (FPT-AS) for Max Unique Coverage, parameterized by k and the maximum element frequency, r, which can be implemented on a data stream. Our FPT-AS finds a (1-ε)-approximation while maintaining a kernel of size Õ(k r/ε), which can be combined with subsampling to use Õ(k² r / ε³) space overall. This significantly improves on the previous-best FPT-AS with the same approximation, but a kernel of size Õ(k² r / ε²). In order to achieve our first result, we show upper bounds on the ratio of a collection’s coverage to the unique coverage of a maximizing subcollection; this is by constructing explicit algorithms that find a subcollection with unique coverage at least a logarithmic ratio of the collection’s coverage. We complement our algorithms with our second main result, showing that Ω(m / k²) space is necessary to achieve a (1.5 + o(1))/(ln k - 1)-approximation in the data stream. This dramatically improves the previous-best lower bound showing that Ω(m / k²) is necessary to achieve better than a e^{-1+1/k}-approximation.
Cite as
Philip Cervenjak, Junhao Gan, Seeun William Umboh, and Anthony Wirth. Maximum Unique Coverage on Streams: Improved FPT Approximation Scheme and Tighter Space Lower Bound. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 25:1-25:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{cervenjak_et_al:LIPIcs.APPROX/RANDOM.2024.25,
author = {Cervenjak, Philip and Gan, Junhao and Umboh, Seeun William and Wirth, Anthony},
title = {{Maximum Unique Coverage on Streams: Improved FPT Approximation Scheme and Tighter Space Lower Bound}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {25:1--25:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.25},
URN = {urn:nbn:de:0030-drops-210183},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.25},
annote = {Keywords: Maximum unique coverage, maximum coverage, approximate kernel, data streams}
}
Document
APPROX
Improved Streaming Algorithm for the Klee’s Measure Problem and Generalizations
Abstract
Estimating the size of the union of a stream of sets S₁, S₂, …, S_M where each set is a subset of a known universe Ω is a fundamental problem in data streaming. This problem naturally generalizes the well-studied 𝖥₀ estimation problem in the streaming literature, where each set contains a single element from the universe. We consider the general case when the sets S_i can be succinctly represented and allow efficient membership, cardinality, and sampling queries (called a Delphic family of sets). A notable example in this framework is the Klee’s Measure Problem (KMP), where every set S_i is an axis-parallel rectangle in d-dimensional spaces (Ω = [Δ]^d where [Δ] := {1, … ,Δ} and Δ ∈ ℕ). Recently, Meel, Chakraborty, and Vinodchandran (PODS-21, PODS-22) designed a streaming algorithm for (ε,δ)-estimation of the size of the union of set streams over Delphic family with space and update time complexity O((log³|Ω|)/ε² ⋅ log 1/δ) and Õ((log⁴|Ω|)/ε² ⋅ log 1/(δ)), respectively.
This work presents a new, sampling-based algorithm for estimating the size of the union of Delphic sets that has space and update time complexity Õ((log²|Ω|)/ε² ⋅ log 1/(δ)). This improves the space complexity bound by a log|Ω| factor and update time complexity bound by a log² |Ω| factor.
A critical question is whether quadratic dependence of log|Ω| on space and update time complexities is necessary. Specifically, can we design a streaming algorithm for estimating the size of the union of sets over Delphic family with space and complexity linear in log|Ω| and update time poly(log|Ω|)? While this appears technically challenging, we show that establishing a lower bound of ω(log|Ω|) with poly(log|Ω|) update time is beyond the reach of current techniques. Specifically, we show that under certain hard-to-prove computational complexity hypothesis, there is a streaming algorithm for the problem with optimal space complexity O(log|Ω|) and update time poly(log(|Ω|)). Thus, establishing a space lower bound of ω(log|Ω|) will lead to break-through complexity class separation results.
Cite as
Mridul Nandi, N. V. Vinodchandran, Arijit Ghosh, Kuldeep S. Meel, Soumit Pal, and Sourav Chakraborty. Improved Streaming Algorithm for the Klee’s Measure Problem and Generalizations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 26:1-26:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{nandi_et_al:LIPIcs.APPROX/RANDOM.2024.26,
author = {Nandi, Mridul and Vinodchandran, N. V. and Ghosh, Arijit and Meel, Kuldeep S. and Pal, Soumit and Chakraborty, Sourav},
title = {{Improved Streaming Algorithm for the Klee’s Measure Problem and Generalizations}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {26:1--26:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.26},
URN = {urn:nbn:de:0030-drops-210191},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.26},
annote = {Keywords: Sampling, Streaming, Klee’s Measure Problem}
}
Document
APPROX
An EPTAS for Cardinality Constrained Multiple Knapsack via Iterative Randomized Rounding
Abstract
In [Math. Oper. Res., 2011], Fleischer et al. introduced a powerful technique for solving the generic class of separable assignment problems (SAP), in which a set of items of given values and weights needs to be packed into a set of bins subject to separable assignment constraints, so as to maximize the total value. The approach of Fleischer at al. relies on solving a configuration LP and sampling a configuration for each bin independently based on the LP solution. While there is a SAP variant for which this approach yields the best possible approximation ratio, for various special cases, there are discrepancies between the approximation ratios obtained using the above approach and the state-of-the-art approximations. This raises the following natural question: Can we do better by iteratively solving the configuration LP and sampling a few bins at a time?
To assess the potential of the iterative approach we consider a specific SAP variant as a case-study, Uniform Cardinality Constrained Multiple Knapsack, for which we answer this question affirmatively. The input is a set of items, each has a value and a weight, and a set of uniform capacity bins. The goal is to assign a subset of the items of maximum total value to the bins such that (i) the capacity of any bin is not exceeded, and (ii) the number of items assigned to each bin satisfies a given cardinality constraint. While the technique of Fleischer et al. yields a (1-1/e)-approximation for the problem, we show that iterative randomized rounding leads to efficient polynomial time approximation scheme (EPTAS), thus essentially resolving the complexity status of the problem. Our analysis of iterative randomized rounding may be useful for solving other SAP variants.
Cite as
Ilan Doron-Arad, Ariel Kulik, and Hadas Shachnai. An EPTAS for Cardinality Constrained Multiple Knapsack via Iterative Randomized Rounding. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{doronarad_et_al:LIPIcs.APPROX/RANDOM.2024.27,
author = {Doron-Arad, Ilan and Kulik, Ariel and Shachnai, Hadas},
title = {{An EPTAS for Cardinality Constrained Multiple Knapsack via Iterative Randomized Rounding}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {27:1--27:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.27},
URN = {urn:nbn:de:0030-drops-210204},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.27},
annote = {Keywords: multiple knapsack, cardinality constraint, EPTAS, iterative randomized rounding}
}
Document
APPROX
Rectangle Tiling Binary Arrays
Abstract
The problem of rectangle tiling binary arrays is defined as follows. Given an n × n array A of zeros and ones and a natural number p, our task is to partition A into at most p rectangular tiles, so that the maximal weight of a tile is minimized. A tile is any rectangular subarray of A. The weight of a tile is the sum of elements that fall within it. We present a linear (O(n²)) time (3/2 + p²/w(A))-approximation algorithm for this problem, where w(A) denotes the weight of the whole array A. This improves on the previously known approximation with the ratio 2 when p²/w(A) < 1/2.
The result is best possible in the following sense. The algorithm employs the lower bound of L = ⌈w(A)/p⌉, which is the only known and used bound on the optimum in all algorithms for rectangle tiling. We prove that a better approximation factor for the binary RTile cannot be achieved using L, because there exist arrays, whose every partition contains a tile with weight at least (3/2 + p/w(A))L. We also consider the dual problem of rectangle tiling for binary arrays, where we are given an upper bound on the weight of the tiles, and we have to cover the array A with the minimum number of non-overlapping tiles. Both problems have natural extensions to d-dimensional versions, for which we provide analogous results.
Cite as
Pratik Ghosal, Syed Mohammad Meesum, and Katarzyna Paluch. Rectangle Tiling Binary Arrays. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{ghosal_et_al:LIPIcs.APPROX/RANDOM.2024.28,
author = {Ghosal, Pratik and Meesum, Syed Mohammad and Paluch, Katarzyna},
title = {{Rectangle Tiling Binary Arrays}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {28:1--28:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.28},
URN = {urn:nbn:de:0030-drops-210214},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.28},
annote = {Keywords: Rectangle Tiling, RTILE, DRTILE}
}
Document
APPROX
Approximation Algorithms for Correlated Knapsack Orienteering
Abstract
We consider the correlated knapsack orienteering (CorrKO) problem: we are given a travel budget B, processing-time budget W, finite metric space (V,d) with root ρ ∈ V, where each vertex is associated with a job with possibly correlated random size and random reward that become known only when the job completes. Random variables are independent across different vertices. The goal is to compute a ρ-rooted path of length at most B, in a possibly adaptive fashion, that maximizes the reward collected from jobs that processed by time W. To our knowledge, CorrKO has not been considered before, though prior work has considered the uncorrelated problem, stochastic knapsack orienteering, and correlated orienteering, which features only one budget constraint on the sum of travel-time and processing-times.
Gupta et al. [Gupta et al., 2015] showed that the uncorrelated version of this problem has a constant-factor adaptivity gap. We show that, perhaps surprisingly and in stark contrast to the uncorrelated problem, the adaptivity gap of CorrKO is is at least Ω(max{√log(B),√(log log(W))}). Complementing this result, we devise non-adaptive algorithms that obtain: (a) O(log log W)-approximation in quasi-polytime; and (b) O(log W)-approximation in polytime. This also establishes that the adaptivity gap for CorrKO is at most O(log log W). We obtain similar guarantees for CorrKO with cancellations, wherein a job can be cancelled before its completion time, foregoing its reward. We show that an α-approximation for CorrKO implies an O(α)-approximation for CorrKO with cancellations.
We also consider the special case of CorrKO where job sizes are weighted Bernoulli distributions, and more generally where the distributions are supported on at most two points (2CorrKO). Although weighted Bernoulli distributions suffice to yield an Ω(√{log log B}) adaptivity-gap lower bound for (uncorrelated) stochastic orienteering, we show that they are easy instances for CorrKO. We develop non-adaptive algorithms that achieve O(1)-approximation, in polytime for weighted Bernoulli distributions, and in (n+log B)^O(log W)-time for 2CorrKO. (Thus, our adaptivity-gap lower-bound example, which uses distributions of support-size 3, is tight in terms of support-size of the distributions.)
Finally, we leverage our techniques to provide a quasi-polynomial time O(log log B) approximation algorithm for correlated orienteering improving upon the approximation guarantee in [Bansal and Nagarajan, 2015].
Cite as
David Alemán Espinosa and Chaitanya Swamy. Approximation Algorithms for Correlated Knapsack Orienteering. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 29:1-29:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{alemanespinosa_et_al:LIPIcs.APPROX/RANDOM.2024.29,
author = {Alem\'{a}n Espinosa, David and Swamy, Chaitanya},
title = {{Approximation Algorithms for Correlated Knapsack Orienteering}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {29:1--29:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.29},
URN = {urn:nbn:de:0030-drops-210224},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.29},
annote = {Keywords: Approximation algorithms, Stochastic orienteering, Adaptivity gap, Vehicle routing problems, LP rounding algorithms}
}
Document
APPROX
Greedy Heuristics and Linear Relaxations for the Random Hitting Set Problem
Abstract
Consider the Hitting Set problem where, for a given universe 𝒳 = {1, ..., n} and a collection of subsets 𝒮₁, ..., 𝒮_m, one seeks to identify the smallest subset of 𝒳 which has a nonempty intersection with every element in the collection. We study a probabilistic formulation of this problem, where the underlying subsets are formed by including each element of the universe independently with probability p. We rigorously analyze integrality gaps between linear programming and integer programming solutions to the problem. In particular, we prove the absence of an integrality gap in the sparse regime mp ≲ log(n) and the presence of a non-vanishing integrality gap in the dense regime mp ≫ log{n}. Moreover, for large enough values of n, we look at the performance of Lovász’s celebrated Greedy algorithm [Lovász, 1975] with respect to the chosen input distribution, and prove that it finds optimal solutions up to multiplicative constants. This highlights separation of Greedy performance between average-case and worst-case settings.
Cite as
Gabriel Arpino, Daniil Dmitriev, and Nicolo Grometto. Greedy Heuristics and Linear Relaxations for the Random Hitting Set Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 30:1-30:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{arpino_et_al:LIPIcs.APPROX/RANDOM.2024.30,
author = {Arpino, Gabriel and Dmitriev, Daniil and Grometto, Nicolo},
title = {{Greedy Heuristics and Linear Relaxations for the Random Hitting Set Problem}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {30:1--30:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.30},
URN = {urn:nbn:de:0030-drops-210234},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.30},
annote = {Keywords: Hitting Set, Random Hypergraph, Integrality Gap, Greedy Algorithm}
}
Document
RANDOM
The Expander Hitting Property When the Sets Are Arbitrarily Unbalanced
Abstract
Numerous works have studied the probability that a length t-1 random walk on an expander is confined to a given rectangle S_1 × … × S_t, providing both upper and lower bounds for this probability. However, when the densities of the sets S_i may depend on the walk length (e.g., when all set are equal and the density is 1-1/t), the currently best known upper and lower bounds are very far from each other. We give an improved confinement lower bound that almost matches the upper bound.
We also study the more general question, of how well random walks fool various classes of test functions. Recently, Golowich and Vadhan proved that random walks on λ-expanders fool Boolean, symmetric functions up to a O(λ) error in total variation distance, with no dependence on the labeling bias. Our techniques extend this result to cases not covered by it, e.g., to functions testing confinement to S_1 × … × S_t, where each set S_i either has density ρ or 1-ρ, for arbitrary ρ.
Technique-wise, we extend Beck’s framework for analyzing what is often referred to as the "flow" of linear operators, reducing it to bounding the entries of a product of 2×2 matrices.
Cite as
Amnon Ta-Shma and Ron Zadicario. The Expander Hitting Property When the Sets Are Arbitrarily Unbalanced. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 31:1-31:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{tashma_et_al:LIPIcs.APPROX/RANDOM.2024.31,
author = {Ta-Shma, Amnon and Zadicario, Ron},
title = {{The Expander Hitting Property When the Sets Are Arbitrarily Unbalanced}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {31:1--31:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.31},
URN = {urn:nbn:de:0030-drops-210246},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.31},
annote = {Keywords: Expander random walks, Expander hitting property}
}
Document
RANDOM
Near-Linear Time Samplers for Matroid Independent Sets with Applications
Abstract
We give a Õ(n) time almost uniform sampler for independent sets of a matroid, whose ground set has n elements and is given by an independence oracle. As a consequence, one can sample connected spanning subgraphs of a given graph G = (V,E) in Õ(|E|) time, whereas the previous best algorithm takes O(|E||V|) time. This improvement, in turn, leads to a faster running time on estimating all-terminal network reliability. Furthermore, we generalise this near-linear time sampler to the random cluster model with q ≤ 1.
Cite as
Xiaoyu Chen, Heng Guo, Xinyuan Zhang, and Zongrui Zou. Near-Linear Time Samplers for Matroid Independent Sets with Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 32:1-32:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2024.32,
author = {Chen, Xiaoyu and Guo, Heng and Zhang, Xinyuan and Zou, Zongrui},
title = {{Near-Linear Time Samplers for Matroid Independent Sets with Applications}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {32:1--32:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.32},
URN = {urn:nbn:de:0030-drops-210254},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.32},
annote = {Keywords: Network reliability, Random cluster modek, Matroid, Bases-exchange walk}
}
Document
RANDOM
On the Amortized Complexity of Approximate Counting
Abstract
Naively storing a counter up to value n would require Ω(log n) bits of memory. Nelson and Yu [Jelani Nelson and Huacheng Yu, 2022], following work of Morris [Robert H. Morris, 1978], showed that if the query answers need only be (1+ε)-approximate with probability at least 1 - δ, then O(log log n + log log(1/δ) + log(1/ε)) bits suffice, and in fact this bound is tight. Morris' original motivation for studying this problem though, as well as modern applications, require not only maintaining one counter, but rather k counters for k large. This motivates the following question: for k large, can k counters be simultaneously maintained using asymptotically less memory than k times the cost of an individual counter? That is to say, does this problem benefit from an improved amortized space complexity bound?
We answer this question in the negative. Specifically, we prove a lower bound for nearly the full range of parameters showing that, in terms of memory usage, there is no asymptotic benefit possible via amortization when storing multiple counters. Our main proof utilizes a certain notion of "information cost" recently introduced by Braverman, Garg and Woodruff [Mark Braverman et al., 2020] to prove lower bounds for streaming algorithms.
Cite as
Ishaq Aden-Ali, Yanjun Han, Jelani Nelson, and Huacheng Yu. On the Amortized Complexity of Approximate Counting. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{adenali_et_al:LIPIcs.APPROX/RANDOM.2024.33,
author = {Aden-Ali, Ishaq and Han, Yanjun and Nelson, Jelani and Yu, Huacheng},
title = {{On the Amortized Complexity of Approximate Counting}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {33:1--33:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.33},
URN = {urn:nbn:de:0030-drops-210264},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.33},
annote = {Keywords: streaming, approximate counting, information complexity, lower bounds}
}
Document
RANDOM
Matrix Multiplication Reductions
Abstract
In this paper we study a worst case to average case reduction for the problem of matrix multiplication over finite fields. Suppose we have an efficient average case algorithm, that given two random matrices A,B outputs a matrix that has a non-trivial correlation with their product A ⋅ B. Can we transform it into a worst case algorithm, that outputs the correct answer for all inputs without incurring a significant overhead in the running time? We present two results in this direction.
- Two-sided error in the high agreement regime. We begin with a brief remark about a reduction for high agreement algorithms, i.e., an algorithm which agrees with the correct output on a large (say > 0.9) fraction of entries, and show that the standard self-correction of linearity allows us to transform such algorithms into algorithms that work in worst case.
- One-sided error in the low agreement regime. Focusing on average case algorithms with one-sided error, we show that over 𝔽₂ there is a reduction that gets an O(T) time average case algorithm that given a random input A,B outputs a matrix that agrees with A ⋅ B on at least 51% of the entries (i.e., has only a slight advantage over the trivial algorithm), and transforms it into an Õ(T) time worst case algorithm, that outputs the correct answer for all inputs with high probability.
Cite as
Ashish Gola, Igor Shinkar, and Harsimran Singh. Matrix Multiplication Reductions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{gola_et_al:LIPIcs.APPROX/RANDOM.2024.34,
author = {Gola, Ashish and Shinkar, Igor and Singh, Harsimran},
title = {{Matrix Multiplication Reductions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {34:1--34:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.34},
URN = {urn:nbn:de:0030-drops-210274},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.34},
annote = {Keywords: Matrix Multiplication, Reductions, Worst case to average case reductions}
}
Document
RANDOM
Testing Intersectingness of Uniform Families
Abstract
A set family F is called intersecting if every two members of F intersect, and it is called uniform if all members of F share a common size. A uniform family F ⊆ binom([n],k) of k-subsets of [n] is ε-far from intersecting if one has to remove more than ε ⋅ binom(n,k) of the sets of F to make it intersecting. We study the property testing problem that given query access to a uniform family F ⊆ binom([n],k), asks to distinguish between the case that F is intersecting and the case that it is ε-far from intersecting. We prove that for every fixed integer r, the problem admits a non-adaptive two-sided error tester with query complexity O((ln n)/ε) for ε ≥ Ω((k/n)^r) and a non-adaptive one-sided error tester with query complexity O((ln k)/ε) for ε ≥ Ω((k²/n)^r). The query complexities are optimal up to the logarithmic terms. For ε ≥ Ω((k²/n)²), we further provide a non-adaptive one-sided error tester with optimal query complexity of O(1/ε). Our findings show that the query complexity of the problem behaves differently from that of testing intersectingness of non-uniform families, studied recently by Chen, De, Li, Nadimpalli, and Servedio (ITCS, 2024).
Cite as
Ishay Haviv and Michal Parnas. Testing Intersectingness of Uniform Families. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{haviv_et_al:LIPIcs.APPROX/RANDOM.2024.35,
author = {Haviv, Ishay and Parnas, Michal},
title = {{Testing Intersectingness of Uniform Families}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {35:1--35:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.35},
URN = {urn:nbn:de:0030-drops-210288},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.35},
annote = {Keywords: Intersecting family, Uniform family, Property testing}
}
Document
RANDOM
On the Houdré-Tetali Conjecture About an Isoperimetric Constant of Graphs
Abstract
Houdré and Tetali defined a class of isoperimetric constants φ_p of graphs for 0 ≤ p ≤ 1, and conjectured a Cheeger-type inequality for φ_(1/2) of the form λ₂ ≲ φ_(1/2) ≲ √λ₂, where λ₂ is the second smallest eigenvalue of the normalized Laplacian matrix. If true, the conjecture would be a strengthening of the hard direction of the classical Cheeger’s inequality. Morris and Peres proved Houdré and Tetali’s conjecture up to an additional log factor, using techniques from evolving sets. We present the following related results on this conjecture.
1) We provide a family of counterexamples to the conjecture of Houdré and Tetali, showing that the logarithmic factor is needed.
2) We match Morris and Peres’s bound using standard spectral arguments.
3) We prove that Houdré and Tetali’s conjecture is true for any constant p strictly bigger than 1/2, which is also a strengthening of the hard direction of Cheeger’s inequality. Furthermore, our results can be extended to directed graphs using Chung’s definition of eigenvalues for directed graphs.
Cite as
Lap Chi Lau and Dante Tjowasi. On the Houdré-Tetali Conjecture About an Isoperimetric Constant of Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{lau_et_al:LIPIcs.APPROX/RANDOM.2024.36,
author = {Lau, Lap Chi and Tjowasi, Dante},
title = {{On the Houdr\'{e}-Tetali Conjecture About an Isoperimetric Constant of Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {36:1--36:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.36},
URN = {urn:nbn:de:0030-drops-210295},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.36},
annote = {Keywords: Isoperimetric constant, Markov chains, Cheeger’s inequality}
}
Document
RANDOM
Nearly Optimal Bounds for Sample-Based Testing and Learning of k-Monotone Functions
Abstract
We study monotonicity testing of functions f : {0,1}^d → {0,1} using sample-based algorithms, which are only allowed to observe the value of f on points drawn independently from the uniform distribution. A classic result by Bshouty-Tamon (J. ACM 1996) proved that monotone functions can be learned with exp(Õ(min{(1/ε)√d,d})) samples and it is not hard to show that this bound extends to testing. Prior to our work the only lower bound for this problem was Ω(√{exp(d)/ε}) in the small ε parameter regime, when ε = O(d^{-3/2}), due to Goldreich-Goldwasser-Lehman-Ron-Samorodnitsky (Combinatorica 2000). Thus, the sample complexity of monotonicity testing was wide open for ε ≫ d^{-3/2}. We resolve this question, obtaining a nearly tight lower bound of exp(Ω(min{(1/ε)√d,d})) for all ε at most a sufficiently small constant. In fact, we prove a much more general result, showing that the sample complexity of k-monotonicity testing and learning for functions f : {0,1}^d → [r] is exp(Ω(min{(rk/ε)√d,d})). For testing with one-sided error we show that the sample complexity is exp(Ω(d)).
Beyond the hypercube, we prove nearly tight bounds (up to polylog factors of d,k,r,1/ε in the exponent) of exp(Θ̃(min{(rk/ε)√d,d})) on the sample complexity of testing and learning measurable k-monotone functions f : ℝ^d → [r] under product distributions. Our upper bound improves upon the previous bound of exp(Õ(min{(k/ε²)√d,d})) by Harms-Yoshida (ICALP 2022) for Boolean functions (r = 2).
Cite as
Hadley Black. Nearly Optimal Bounds for Sample-Based Testing and Learning of k-Monotone Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 37:1-37:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{black:LIPIcs.APPROX/RANDOM.2024.37,
author = {Black, Hadley},
title = {{Nearly Optimal Bounds for Sample-Based Testing and Learning of k-Monotone Functions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {37:1--37:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.37},
URN = {urn:nbn:de:0030-drops-210308},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.37},
annote = {Keywords: Property testing, learning, Boolean functions, monotonicity, k-monotonicity}
}
Document
RANDOM
Approximating the Number of Relevant Variables in a Parity Implies Proper Learning
Abstract
Consider the model where we can access a parity function through random uniform labeled examples in the presence of random classification noise. In this paper, we show that approximating the number of relevant variables in the parity function is as hard as properly learning parities.
More specifically, let γ:ℝ^+ → ℝ^+, where γ(x) ≥ x, be any strictly increasing function. In our first result, we show that from any polynomial-time algorithm that returns a γ-approximation, D (i.e., γ^{-1}(d(f)) ≤ D ≤ γ(d(f))), of the number of relevant variables d(f) for any parity f, we can, in polynomial time, construct a solution to the long-standing open problem of polynomial-time learning k(n)-sparse parities (parities with k(n) ≤ n relevant variables), where k(n) = ω_n(1).
In our second result, we show that from any T(n)-time algorithm that, for any parity f, returns a γ-approximation of the number of relevant variables d(f) of f, we can, in polynomial time, construct a poly(Γ(n))T(Γ(n)²)-time algorithm that properly learns parities, where Γ(x) = γ(γ(x)).
If T(Γ(n)²) = exp({o(n/log n)}), this would resolve another long-standing open problem of properly learning parities in the presence of random classification noise in time exp(o(n/log n)).
Cite as
Nader H. Bshouty and George Haddad. Approximating the Number of Relevant Variables in a Parity Implies Proper Learning. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bshouty_et_al:LIPIcs.APPROX/RANDOM.2024.38,
author = {Bshouty, Nader H. and Haddad, George},
title = {{Approximating the Number of Relevant Variables in a Parity Implies Proper Learning}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {38:1--38:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.38},
URN = {urn:nbn:de:0030-drops-210316},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.38},
annote = {Keywords: PAC Learning, Random Classification Noise, Uniform Distribution, Parity, Sparcity Approximation}
}
Document
RANDOM
The Number of Random 2-SAT Solutions Is Asymptotically Log-Normal
Abstract
We prove that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem. This implies that the log of the number of satisfying assignments exhibits fluctuations of order √n, with n the number of variables. The formula for the variance can be evaluated effectively. By contrast, for numerous other random constraint satisfaction problems the typical fluctuations of the logarithm of the number of solutions are bounded throughout all or most of the satisfiable regime.
Cite as
Arnab Chatterjee, Amin Coja-Oghlan, Noela Müller, Connor Riddlesden, Maurice Rolvien, Pavel Zakharov, and Haodong Zhu. The Number of Random 2-SAT Solutions Is Asymptotically Log-Normal. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chatterjee_et_al:LIPIcs.APPROX/RANDOM.2024.39,
author = {Chatterjee, Arnab and Coja-Oghlan, Amin and M\"{u}ller, Noela and Riddlesden, Connor and Rolvien, Maurice and Zakharov, Pavel and Zhu, Haodong},
title = {{The Number of Random 2-SAT Solutions Is Asymptotically Log-Normal}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {39:1--39:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.39},
URN = {urn:nbn:de:0030-drops-210329},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.39},
annote = {Keywords: satisfiability problem, 2-SAT, random satisfiability, central limit theorem}
}
Document
RANDOM
Private Counting of Distinct Elements in the Turnstile Model and Extensions
Abstract
Privately counting distinct elements in a stream is a fundamental data analysis problem with many applications in machine learning. In the turnstile model, Jain et al. [NeurIPS2023] initiated the study of this problem parameterized by the maximum flippancy of any element, i.e., the number of times that the count of an element changes from 0 to above 0 or vice versa. They give an item-level (ε,δ)-differentially private algorithm whose additive error is tight with respect to that parameterization. In this work, we show that a very simple algorithm based on the sparse vector technique achieves a tight additive error for item-level (ε,δ)-differential privacy and item-level ε-differential privacy with regards to a different parameterization, namely the sum of all flippancies. Our second result is a bound which shows that for a large class of algorithms, including all existing differentially private algorithms for this problem, the lower bound from item-level differential privacy extends to event-level differential privacy. This partially answers an open question by Jain et al. [NeurIPS2023].
Cite as
Monika Henzinger, A. R. Sricharan, and Teresa Anna Steiner. Private Counting of Distinct Elements in the Turnstile Model and Extensions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 40:1-40:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{henzinger_et_al:LIPIcs.APPROX/RANDOM.2024.40,
author = {Henzinger, Monika and Sricharan, A. R. and Steiner, Teresa Anna},
title = {{Private Counting of Distinct Elements in the Turnstile Model and Extensions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {40:1--40:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.40},
URN = {urn:nbn:de:0030-drops-210335},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.40},
annote = {Keywords: differential privacy, turnstile model, counting distinct elements}
}
Document
RANDOM
Hilbert Functions and Low-Degree Randomness Extractors
Abstract
For S ⊆ 𝔽ⁿ, consider the linear space of restrictions of degree-d polynomials to S. The Hilbert function of S, denoted h_S(d,𝔽), is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets S of arbitrary finite grids in 𝔽ⁿ with a fixed size |S|. We achieve this by proving that this value coincides with a combinatorial quantity, namely the smallest number of low Hamming weight points in a down-closed set of size |S|.
Understanding the smallest values of Hilbert functions is closely related to the study of degree-d closure of sets, a notion introduced by Nie and Wang (Journal of Combinatorial Theory, Series A, 2015). We use bounds on the Hilbert function to obtain a tight bound on the size of degree-d closures of subsets of 𝔽_qⁿ, which answers a question posed by Doron, Ta-Shma, and Tell (Computational Complexity, 2022).
We use the bounds on the Hilbert function and degree-d closure of sets to prove that a random low-degree polynomial is an extractor for samplable randomness sources. Most notably, we prove the existence of low-degree extractors and dispersers for sources generated by constant-degree polynomials and polynomial-size circuits. Until recently, even the existence of arbitrary deterministic extractors for such sources was not known.
Cite as
Alexander Golovnev, Zeyu Guo, Pooya Hatami, Satyajeet Nagargoje, and Chao Yan. Hilbert Functions and Low-Degree Randomness Extractors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 41:1-41:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{golovnev_et_al:LIPIcs.APPROX/RANDOM.2024.41,
author = {Golovnev, Alexander and Guo, Zeyu and Hatami, Pooya and Nagargoje, Satyajeet and Yan, Chao},
title = {{Hilbert Functions and Low-Degree Randomness Extractors}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {41:1--41:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.41},
URN = {urn:nbn:de:0030-drops-210345},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.41},
annote = {Keywords: Extractors, Dispersers, Circuits, Hilbert Function, Randomness, Low Degree Polynomials}
}
Document
RANDOM
Matrix Multiplication Verification Using Coding Theory
Abstract
We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three n × n matrices A, B, and C as input, to decide whether AB = C. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in Õ(n²) time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in o(n^ω) time).
To that end, we give two algorithms for MMV in the case where AB - C is sparse. Specifically, when AB - C has at most O(n^δ) non-zero entries for a constant 0 ≤ δ < 2, we give (1) a deterministic O(n^(ω-ε))-time algorithm for constant ε = ε(δ) > 0, and (2) a randomized Õ(n²)-time algorithm using δ/2 ⋅ log₂ n + O(1) random bits. The former algorithm is faster than the deterministic algorithm of Künnemann (ESA, 2018) when δ ≥ 1.056, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses log₂ n + O(1) random bits (in turn fewer than Freivalds’s algorithm).
Our algorithms are simple and use techniques from coding theory. Let H be a parity-check matrix of a Maximum Distance Separable (MDS) code, and let G = (I | G') be a generator matrix of a (possibly different) MDS code in systematic form. Our deterministic algorithm uses fast rectangular matrix multiplication to check whether HAB = HC and H(AB)^T = H(C^T), and our randomized algorithm samples a uniformly random row g' from G' and checks whether g'AB = g'C and g'(AB)^T = g'C^T.
We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require Ω(n^ω) time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic Õ(n²)-time reductions).
Cite as
Huck Bennett, Karthik Gajulapalli, Alexander Golovnev, and Evelyn Warton. Matrix Multiplication Verification Using Coding Theory. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bennett_et_al:LIPIcs.APPROX/RANDOM.2024.42,
author = {Bennett, Huck and Gajulapalli, Karthik and Golovnev, Alexander and Warton, Evelyn},
title = {{Matrix Multiplication Verification Using Coding Theory}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {42:1--42:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.42},
URN = {urn:nbn:de:0030-drops-210352},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.42},
annote = {Keywords: Matrix Multiplication Verification, Derandomization, Sparse Matrices, Error-Correcting Codes, Hardness Barriers, Reductions}
}
Document
RANDOM
Interactive Coding with Unbounded Noise
Abstract
Interactive coding allows two parties to conduct a distributed computation despite noise corrupting a certain fraction of their communication. Dani et al. (Inf. and Comp., 2018) suggested a novel setting in which the amount of noise is unbounded and can significantly exceed the length of the (noise-free) computation. While no solution is possible in the worst case, under the restriction of oblivious noise, Dani et al. designed a coding scheme that succeeds with a polynomially small failure probability.
We revisit the question of conducting computations under this harsh type of noise and devise a computationally-efficient coding scheme that guarantees the success of the computation, except with an exponentially small probability. This higher degree of correctness matches the case of coding schemes with a bounded fraction of noise.
Our simulation of an N-bit noise-free computation in the presence of T corruptions, communicates an optimal number of O(N+T) bits and succeeds with probability 1-2^(-Ω(N)). We design this coding scheme by introducing an intermediary noise model, where an oblivious adversary can choose the locations of corruptions in a worst-case manner, but the effect of each corruption is random: the noise either flips the transmission with some probability or otherwise erases it. This randomized abstraction turns out to be instrumental in achieving an optimal coding scheme.
Cite as
Eden Fargion, Ran Gelles, and Meghal Gupta. Interactive Coding with Unbounded Noise. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{fargion_et_al:LIPIcs.APPROX/RANDOM.2024.43,
author = {Fargion, Eden and Gelles, Ran and Gupta, Meghal},
title = {{Interactive Coding with Unbounded Noise}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {43:1--43:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.43},
URN = {urn:nbn:de:0030-drops-210361},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.43},
annote = {Keywords: Distributed Computation with Noisy Links, Interactive Coding, Noise Resilience, Unbounded Noise, Random Erasure-Flip Noise}
}
Document
RANDOM
Optimal Pseudorandom Generators for Low-Degree Polynomials over Moderately Large Fields
Abstract
We construct explicit pseudorandom generators that fool n-variate polynomials of degree at most d over a finite field 𝔽_q. The seed length of our generators is O(d log n + log q), over fields of size exponential in d and characteristic at least d(d-1)+1. Previous constructions such as Bogdanov’s (STOC 2005) and Derksen and Viola’s (FOCS 2022) had either suboptimal seed length or required the field size to depend on n.
Our approach follows Bogdanov’s paradigm while incorporating techniques from Lecerf’s factorization algorithm (J. Symb. Comput. 2007) and insights from the construction of Derksen and Viola regarding the role of indecomposability of polynomials.
Cite as
Ashish Dwivedi, Zeyu Guo, and Ben Lee Volk. Optimal Pseudorandom Generators for Low-Degree Polynomials over Moderately Large Fields. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{dwivedi_et_al:LIPIcs.APPROX/RANDOM.2024.44,
author = {Dwivedi, Ashish and Guo, Zeyu and Volk, Ben Lee},
title = {{Optimal Pseudorandom Generators for Low-Degree Polynomials over Moderately Large Fields}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {44:1--44:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.44},
URN = {urn:nbn:de:0030-drops-210370},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.44},
annote = {Keywords: Pseudorandom Generators, Low Degree Polynomials}
}
Document
RANDOM
Refining the Adaptivity Notion in the Huge Object Model
Abstract
The Huge Object model for distribution testing, first defined by Goldreich and Ron in 2022, combines the features of classical string testing and distribution testing. In this model we are given access to independent samples from an unknown distribution P over the set of strings {0,1}ⁿ, but are only allowed to query a few bits from the samples. The distinction between adaptive and non-adaptive algorithms, which occurs naturally in the realm of string testing (while being irrelevant for classical distribution testing), plays a substantial role also in the Huge Object model.
In this work we show that the full picture in the Huge Object model is much richer than just that of the adaptive vs. non-adaptive dichotomy. We define and investigate several models of adaptivity that lie between the fully-adaptive and the completely non-adaptive extremes. These models are naturally grounded by observing the querying process from each sample independently, and considering the "algorithmic flow" between them. For example, if we allow no information at all to cross over between samples (up to the final decision), then we obtain the locally bounded adaptive model, arguably the "least adaptive" one apart from being completely non-adaptive. A slightly stronger model allows only a "one-way" information flow. Even stronger (but still far from being fully adaptive) models follow by taking inspiration from the setting of streaming algorithms. To show that we indeed have a hierarchy, we prove a chain of exponential separations encompassing most of the models that we define.
Cite as
Tomer Adar and Eldar Fischer. Refining the Adaptivity Notion in the Huge Object Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 45:1-45:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{adar_et_al:LIPIcs.APPROX/RANDOM.2024.45,
author = {Adar, Tomer and Fischer, Eldar},
title = {{Refining the Adaptivity Notion in the Huge Object Model}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {45:1--45:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.45},
URN = {urn:nbn:de:0030-drops-210383},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.45},
annote = {Keywords: Huge-Object model, Property Testing}
}
Document
RANDOM
Support Testing in the Huge Object Model
Abstract
The Huge Object model is a distribution testing model in which we are given access to independent samples from an unknown distribution over the set of strings {0,1}ⁿ, but are only allowed to query a few bits from the samples. We investigate the problem of testing whether a distribution is supported on m elements in this model. It turns out that the behavior of this property is surprisingly intricate, especially when also considering the question of adaptivity.
We prove lower and upper bounds for both adaptive and non-adaptive algorithms in the one-sided and two-sided error regime. Our bounds are tight when m is fixed to a constant (and the distance parameter ε is the only variable). For the general case, our bounds are at most O(log m) apart. In particular, our results show a surprising O(log ε^{-1}) gap between the number of queries required for non-adaptive testing as compared to adaptive testing. For one-sided error testing, we also show that an O(log m) gap between the number of samples and the number of queries is necessary. Our results utilize a wide variety of combinatorial and probabilistic methods.
Cite as
Tomer Adar, Eldar Fischer, and Amit Levi. Support Testing in the Huge Object Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{adar_et_al:LIPIcs.APPROX/RANDOM.2024.46,
author = {Adar, Tomer and Fischer, Eldar and Levi, Amit},
title = {{Support Testing in the Huge Object Model}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {46:1--46:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.46},
URN = {urn:nbn:de:0030-drops-210399},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.46},
annote = {Keywords: Huge-Object model, Property Testing}
}
Document
RANDOM
Upper Bounds on the 2-Colorability Threshold of Random d-Regular k-Uniform Hypergraphs for k ≥ 3
Abstract
For a large class of random constraint satisfaction problems (csp), deep but non-rigorous theory from statistical physics predict the location of the sharp satisfiability transition. The works of Ding, Sly, Sun (2014, 2016) and Coja-Oghlan, Panagiotou (2014) established the satisfiability threshold for random regular k-nae-sat, random k-sat, and random regular k-sat for large enough k ≥ k₀ where k₀ is a large non-explicit constant. Establishing the same for small values of k ≥ 3 remains an important open problem in the study of random csps.
In this work, we study two closely related models of random csps, namely the 2-coloring on random d-regular k-uniform hypergraphs and the random d-regular k-nae-sat model. For every k ≥ 3, we prove that there is an explicit d_⋆(k) which gives a satisfiability upper bound for both of the models. Our upper bound d_⋆(k) for k ≥ 3 matches the prediction from statistical physics for the hypergraph 2-coloring by Dall’Asta, Ramezanpour, Zecchina (2008), thus conjectured to be sharp. Moreover, d_⋆(k) coincides with the satisfiability threshold of random regular k-nae-sat for large enough k ≥ k₀ by Ding, Sly, Sun (2014).
Cite as
Evan Chang, Neel Kolhe, and Youngtak Sohn. Upper Bounds on the 2-Colorability Threshold of Random d-Regular k-Uniform Hypergraphs for k ≥ 3. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 47:1-47:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chang_et_al:LIPIcs.APPROX/RANDOM.2024.47,
author = {Chang, Evan and Kolhe, Neel and Sohn, Youngtak},
title = {{Upper Bounds on the 2-Colorability Threshold of Random d-Regular k-Uniform Hypergraphs for k ≥ 3}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {47:1--47:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.47},
URN = {urn:nbn:de:0030-drops-210402},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.47},
annote = {Keywords: Random constraint satisfaction problem, replica symmetry breaking, interpolation bound}
}
Document
RANDOM
Improved Bounds for High-Dimensional Equivalence and Product Testing Using Subcube Queries
Abstract
We study property testing in the subcube conditional model introduced by Bhattacharyya and Chakraborty (2017). We obtain the first equivalence test for n-dimensional distributions that is quasi-linear in n, improving the previously known Õ(n²/ε²) query complexity bound to Õ(n/ε²). We extend this result to general finite alphabets with logarithmic cost in the alphabet size.
By exploiting the specific structure of the queries that we use (which are more restrictive than general subcube queries), we obtain a cubic improvement over the best known test for distributions over {1,…,N} under the interval querying model of Canonne, Ron and Servedio (2015), attaining a query complexity of Õ((log N)/ε²), which for fixed ε almost matches the known lower bound of Ω((log N)/log log N). We also derive a product test for n-dimensional distributions with Õ(n/ε²) queries, and provide an Ω(√n/ε²) lower bound for this property.
Cite as
Tomer Adar, Eldar Fischer, and Amit Levi. Improved Bounds for High-Dimensional Equivalence and Product Testing Using Subcube Queries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 48:1-48:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{adar_et_al:LIPIcs.APPROX/RANDOM.2024.48,
author = {Adar, Tomer and Fischer, Eldar and Levi, Amit},
title = {{Improved Bounds for High-Dimensional Equivalence and Product Testing Using Subcube Queries}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {48:1--48:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.48},
URN = {urn:nbn:de:0030-drops-210418},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.48},
annote = {Keywords: Distribution testing, conditional sampling, sub-cube sampling}
}
Document
RANDOM
Parallelising Glauber Dynamics
Abstract
For distributions over discrete product spaces ∏_{i=1}^n Ω_i', Glauber dynamics is a Markov chain that at each step, resamples a random coordinate conditioned on the other coordinates. We show that k-Glauber dynamics, which resamples a random subset of k coordinates, mixes k times faster in χ²-divergence, and assuming approximate tensorization of entropy, mixes k times faster in KL-divergence. We apply this to obtain parallel algorithms in two settings: (1) For the Ising model μ_{J,h}(x) ∝ exp(1/2 ⟨x,Jx⟩ + ⟨h,x⟩) with ‖J‖ < 1-c (the regime where fast mixing is known), we show that we can implement each step of Θ(n/‖J‖_F)-Glauber dynamics efficiently with a parallel algorithm, resulting in a parallel algorithm with running time Õ(‖J‖_F) = Õ(√n). (2) For the mixed p-spin model at high enough temperature, we show that with high probability we can implement each step of Θ(√n)-Glauber dynamics efficiently and obtain running time Õ(√n).
Cite as
Holden Lee. Parallelising Glauber Dynamics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 49:1-49:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{lee:LIPIcs.APPROX/RANDOM.2024.49,
author = {Lee, Holden},
title = {{Parallelising Glauber Dynamics}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {49:1--49:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.49},
URN = {urn:nbn:de:0030-drops-210424},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.49},
annote = {Keywords: sampling, Ising model, parallel algorithm, Markov chain, Glauber dynamics}
}
Document
RANDOM
Towards Simpler Sorting Networks and Monotone Circuits for Majority
Abstract
In this paper, we study the problem of computing the majority function by low-depth monotone circuits and a related problem of constructing low-depth sorting networks. We consider both the classical setting with elementary operations of arity 2 and the generalized setting with operations of arity k, where k is a parameter. For both problems and both settings, there are various constructions known, the minimal known depth being logarithmic. However, there is currently no known efficient deterministic construction that simultaneously achieves sub-log-squared depth, simplicity, and has a potential to be used in practice. In this paper we make progress towards resolution of this problem.
For computing majority by standard monotone circuits (gates of arity 2) we provide an explicit monotone circuit of depth O(log₂^{5/3} n). The construction is a combination of several known and not too complicated ideas. Essentially, for this result we gradually derandomize the construction of Valiant (1984).
As one of the intermediate steps in our result we need an efficient construction of a sorting network with gates of arity k for arbitrary fixed k. For this we provide a new sorting network architecture inspired by representation of inputs as a high-dimensional cube. As a result we obtain a simple construction that improves previous upper bound of 4 log_k² n to 2 log_k² n. We prove the similar bound for the depth of the circuit computing majority of n bits consisting of gates computing majority of k bits. Note, that for both problems there is an explicit construction of depth O(log_k n) known, but the construction is complicated and the constant hidden in O-notation is huge.
Cite as
Natalia Dobrokhotova-Maikova, Alexander Kozachinskiy, and Vladimir Podolskii. Towards Simpler Sorting Networks and Monotone Circuits for Majority. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 50:1-50:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{dobrokhotovamaikova_et_al:LIPIcs.APPROX/RANDOM.2024.50,
author = {Dobrokhotova-Maikova, Natalia and Kozachinskiy, Alexander and Podolskii, Vladimir},
title = {{Towards Simpler Sorting Networks and Monotone Circuits for Majority}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {50:1--50:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.50},
URN = {urn:nbn:de:0030-drops-210436},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.50},
annote = {Keywords: Sorting networks, constant depth, lower bounds, threshold circuits}
}
Document
RANDOM
Consequences of Randomized Reductions from SAT to Time-Bounded Kolmogorov Complexity
Abstract
A central open question within meta-complexity is that of NP-hardness of problems such as MCSP and MK^{t}P. Despite a large body of work giving consequences of and barriers for NP-hardness of these problems under (restricted) deterministic reductions, very little is known in the setting of randomized reductions. In this work, we give consequences of randomized NP-hardness reductions for both approximating and exactly computing time-bounded and time-unbounded Kolmogorov complexity.
In the setting of approximate K^{poly} complexity, our results are as follows.
1) Under a derandomization assumption, for any constant δ > 0, if approximating K^t complexity within n^{δ} additive error is hard for SAT under an honest randomized non-adaptive Turing reduction running in time polynomially less than t, then NP = coNP.
2) Under the same assumptions, the worst-case hardness of NP is equivalent to the existence of one-way functions. Item 1 above may be compared with a recent work of Saks and Santhanam [Michael E. Saks and Rahul Santhanam, 2022], which makes the same assumptions except with ω(log n) additive error, obtaining the conclusion NE = coNE.
In the setting of exact K^{poly} complexity, where the barriers of Item 1 and [Michael E. Saks and Rahul Santhanam, 2022] do not apply, we show:
3) If computing K^t complexity is hard for SAT under reductions as in Item 1, then the average-case hardness of NP is equivalent to the existence of one-way functions. That is, "Pessiland" is excluded.
Finally, we give consequences of NP-hardness of exact time-unbounded Kolmogorov complexity under randomized reductions.
4) If computing Kolmogorov complexity is hard for SAT under a randomized many-one reduction running in time t_R and with failure probability at most 1/(t_R)^16, then coNP is contained in non-interactive statistical zero-knowledge; thus NP ⊆ coAM. Also, the worst-case hardness of NP is equivalent to the existence of one-way functions. We further exploit the connection to NISZK along with a previous work of Allender et al. [Eric Allender et al., 2023] to show that hardness of K complexity under randomized many-one reductions is highly robust with respect to failure probability, approximation error, output length, and threshold parameter.
Cite as
Halley Goldberg and Valentine Kabanets. Consequences of Randomized Reductions from SAT to Time-Bounded Kolmogorov Complexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 51:1-51:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{goldberg_et_al:LIPIcs.APPROX/RANDOM.2024.51,
author = {Goldberg, Halley and Kabanets, Valentine},
title = {{Consequences of Randomized Reductions from SAT to Time-Bounded Kolmogorov Complexity}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {51:1--51:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.51},
URN = {urn:nbn:de:0030-drops-210444},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.51},
annote = {Keywords: Meta-complexity, Randomized reductions, NP-hardness, Worst-case complexity, Time-bounded Kolmogorov complexity}
}
Document
RANDOM
Trace Reconstruction from Local Statistical Queries
Abstract
The goal of trace reconstruction is to reconstruct an unknown n-bit string x given only independent random traces of x, where a random trace of x is obtained by passing x through a deletion channel. A Statistical Query (SQ) algorithm for trace reconstruction is an algorithm which can only access statistical information about the distribution of random traces of x rather than individual traces themselves. Such an algorithm is said to be 𝓁-local if each of its statistical queries corresponds to an 𝓁-junta function over some block of 𝓁 consecutive bits in the trace. Since several - but not all - known algorithms for trace reconstruction fall under the local statistical query paradigm, it is interesting to understand the abilities and limitations of local SQ algorithms for trace reconstruction.
In this paper we establish nearly-matching upper and lower bounds on local Statistical Query algorithms for both worst-case and average-case trace reconstruction. For the worst-case problem, we show that there is an Õ(n^{1/5})-local SQ algorithm that makes all its queries with tolerance τ ≥ 2^{-Õ(n^{1/5})}, and also that any Õ(n^{1/5})-local SQ algorithm must make some query with tolerance τ ≤ 2^{-Ω̃(n^{1/5})}. For the average-case problem, we show that there is an O(log n)-local SQ algorithm that makes all its queries with tolerance τ ≥ 1/poly(n), and also that any O(log n)-local SQ algorithm must make some query with tolerance τ ≤ 1/poly(n).
Cite as
Xi Chen, Anindya De, Chin Ho Lee, and Rocco A. Servedio. Trace Reconstruction from Local Statistical Queries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 52:1-52:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2024.52,
author = {Chen, Xi and De, Anindya and Lee, Chin Ho and Servedio, Rocco A.},
title = {{Trace Reconstruction from Local Statistical Queries}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {52:1--52:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.52},
URN = {urn:nbn:de:0030-drops-210459},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.52},
annote = {Keywords: trace reconstruction, statistical queries, algorithmic statistics}
}
Document
RANDOM
When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?
Abstract
The Gilbert-Varshamov (GV) bound is a classical existential result in coding theory. It implies that a random linear binary code of rate ε² has relative distance at least 1/2 - O(ε) with high probability. However, it is a major challenge to construct explicit codes with similar parameters.
One hope to derandomize the Gilbert-Varshamov construction is with code concatenation: We begin with a (hopefully explicit) outer code 𝒞_out over a large alphabet, and concatenate that with a small binary random linear code 𝒞_in. It is known that when we use independent small codes for each coordinate, then the result lies on the GV bound with high probability, but this still uses a lot of randomness. In this paper, we consider the question of whether code concatenation with a single random linear inner code 𝒞_in can lie on the GV bound; and if so what conditions on 𝒞_out are sufficient for this.
We show that first, there do exist linear outer codes 𝒞_out that are "good" for concatenation in this sense (in fact, most linear codes codes are good). We also provide two sufficient conditions for 𝒞_out, so that if 𝒞_out satisfies these, 𝒞_out∘𝒞_in will likely lie on the GV bound. We hope that these conditions may inspire future work towards constructing explicit codes 𝒞_out.
Cite as
Dean Doron, Jonathan Mosheiff, and Mary Wootters. When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 53:1-53:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{doron_et_al:LIPIcs.APPROX/RANDOM.2024.53,
author = {Doron, Dean and Mosheiff, Jonathan and Wootters, Mary},
title = {{When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound?}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {53:1--53:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.53},
URN = {urn:nbn:de:0030-drops-210467},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.53},
annote = {Keywords: Error-correcting codes, Concatenated codes, Derandomization, Gilbert-Varshamov bound}
}
Document
RANDOM
Parallel Repetition of k-Player Projection Games
Abstract
We study parallel repetition of k-player games where the constraints satisfy the projection property. We prove exponential decay in the value of a parallel repetition of projection games with a value less than 1.
Cite as
Amey Bhangale, Mark Braverman, Subhash Khot, Yang P. Liu, and Dor Minzer. Parallel Repetition of k-Player Projection Games. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{bhangale_et_al:LIPIcs.APPROX/RANDOM.2024.54,
author = {Bhangale, Amey and Braverman, Mark and Khot, Subhash and Liu, Yang P. and Minzer, Dor},
title = {{Parallel Repetition of k-Player Projection Games}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {54:1--54:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.54},
URN = {urn:nbn:de:0030-drops-210475},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.54},
annote = {Keywords: Parallel Repetition, Multiplayer games, Projection games}
}
Document
RANDOM
Faster Algorithms for Schatten-p Low Rank Approximation
Abstract
We study algorithms for the Schatten-p Low Rank Approximation (LRA) problem. First, we show that by using fast rectangular matrix multiplication algorithms and different block sizes, we can improve the running time of the algorithms in the recent work of Bakshi, Clarkson and Woodruff (STOC 2022). We then show that by carefully combining our new algorithm with the algorithm of Li and Woodruff (ICML 2020), we can obtain even faster algorithms for Schatten-p LRA.
While the block-based algorithms are fast in the real number model, we do not have a stability analysis which shows that the algorithms work when implemented on a machine with polylogarithmic bits of precision. We show that the LazySVD algorithm of Allen-Zhu and Li (NeurIPS 2016) can be implemented on a floating point machine with only logarithmic, in the input parameters, bits of precision. As far as we are aware, this is the first stability analysis of any algorithm using O((k/√ε)poly(log n)) matrix-vector products with the matrix A to output a 1+ε approximate solution for the rank-k Schatten-p LRA problem.
Cite as
Praneeth Kacham and David P. Woodruff. Faster Algorithms for Schatten-p Low Rank Approximation. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 55:1-55:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{kacham_et_al:LIPIcs.APPROX/RANDOM.2024.55,
author = {Kacham, Praneeth and Woodruff, David P.},
title = {{Faster Algorithms for Schatten-p Low Rank Approximation}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {55:1--55:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.55},
URN = {urn:nbn:de:0030-drops-210488},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.55},
annote = {Keywords: Low Rank Approximation, Schatten Norm, Rectangular Matrix Multiplication, Stability Analysis}
}
Document
RANDOM
Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs
Abstract
We study the worst-case mixing time of the global Kawasaki dynamics for the fixed-magnetization Ising model on the class of graphs of maximum degree Δ. Proving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that below the tree uniqueness threshold, the Kawasaki dynamics mix rapidly for all magnetizations. Disproving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that the regime of fast mixing does not extend throughout the regime of tractability for this model: there is a range of parameters for which there exist efficient sampling algorithms for the fixed-magnetization Ising model on max-degree Δ graphs, but the Kawasaki dynamics can take exponential time to mix. Our techniques involve showing spectral independence in the fixed-magnetization Ising model and proving a sharp threshold for the existence of multiple metastable states in the Ising model with external field on random regular graphs.
Cite as
Aiya Kuchukova, Marcus Pappik, Will Perkins, and Corrine Yap. Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 56:1-56:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{kuchukova_et_al:LIPIcs.APPROX/RANDOM.2024.56,
author = {Kuchukova, Aiya and Pappik, Marcus and Perkins, Will and Yap, Corrine},
title = {{Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {56:1--56:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.56},
URN = {urn:nbn:de:0030-drops-210493},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.56},
annote = {Keywords: ferromagnetic Ising model, fixed-magnetization Ising model, Kawasaki dynamics, Glauber dynamics, mixing time}
}
Document
RANDOM
Stochastic Distance in Property Testing
Abstract
We introduce a novel concept termed "stochastic distance" for property testing. Diverging from the traditional definition of distance, where a distance t implies that there exist t edges that can be added to ensure a graph possesses a certain property (such as k-edge-connectivity), our new notion implies that there is a high probability that adding t random edges will endow the graph with the desired property. While formulating testers based on this new distance proves challenging in a sequential environment, it is much easier in a distributed setting. Taking k-edge-connectivity as a case study, we design ultra-fast testing algorithms in the CONGEST model. Our introduction of stochastic distance offers a more natural fit for the distributed setting, providing a promising avenue for future research in emerging models of computation.
Cite as
Uri Meir, Gregory Schwartzman, and Yuichi Yoshida. Stochastic Distance in Property Testing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 57:1-57:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{meir_et_al:LIPIcs.APPROX/RANDOM.2024.57,
author = {Meir, Uri and Schwartzman, Gregory and Yoshida, Yuichi},
title = {{Stochastic Distance in Property Testing}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {57:1--57:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.57},
URN = {urn:nbn:de:0030-drops-210506},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.57},
annote = {Keywords: Connectivity, k-edge connectivity}
}
Document
RANDOM
Expanderizing Higher Order Random Walks
Abstract
We study a variant of the down-up (also known as the Glauber dynamics) and up-down walks over an n-partite simplicial complex, which we call expanderized higher order random walks - where the sequence of updated coordinates correspond to the sequence of vertices visited by a random walk over an auxiliary expander graph H. When H is the clique with self loops on [n], this random walk reduces to the usual down-up walk and when H is the directed cycle on [n], this random walk reduces to the well-known systematic scan Glauber dynamics. We show that whenever the usual higher order random walks satisfy a log-Sobolev inequality or a Poincaré inequality, the expanderized walks satisfy the same inequalities with a loss of quality related to the two-sided expansion of the auxillary graph H. Our construction can be thought as a higher order random walk generalization of the derandomized squaring algorithm of Rozenman and Vadhan (RANDOM 2005).
We study the mixing times of our expanderized walks in two example cases: We show that when initiated with an expander graph our expanderized random walks have mixing time (i) O(n log n) for sampling a uniformly random list colorings of a graph G of maximum degree Δ = O(1) where each vertex has at least (11/6 - ε) Δ and at most O(Δ) colors, (ii) O_h((n log n)/(1 - ‖J‖_op)²) for sampling the Ising model with a PSD interaction matrix J ∈ ℝ^{n×n} satisfying ‖J‖_op ≤ 1 and the external field h ∈ ℝⁿ- here the O(•) notation hides a constant that depends linearly on the largest entry of h. As expander graphs can be very sparse, this decreases the amount of randomness required to simulate the down-up walks by a logarithmic factor.
We also prove some simple results which enable us to argue about log-Sobolev constants of higher order random walks and provide a simple and self-contained analysis of local-to-global Φ-entropy contraction in simplicial complexes - giving simpler proofs for many pre-existing results.
Cite as
Vedat Levi Alev and Shravas Rao. Expanderizing Higher Order Random Walks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 58:1-58:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{alev_et_al:LIPIcs.APPROX/RANDOM.2024.58,
author = {Alev, Vedat Levi and Rao, Shravas},
title = {{Expanderizing Higher Order Random Walks}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {58:1--58:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.58},
URN = {urn:nbn:de:0030-drops-210510},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.58},
annote = {Keywords: Higher Order Random Walks, Expander Graphs, Glauber Dynamics, Derandomized Squaring, High Dimensional Expansion, Spectral Independence, Entropic Independence}
}
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RANDOM
Ramsey Properties of Randomly Perturbed Hypergraphs
Abstract
We study Ramsey properties of randomly perturbed 3-uniform hypergraphs. For t ≥ 2, write K^(3)_t to denote the 3-uniform expanded clique hypergraph obtained from the complete graph K_t by expanding each of the edges of the latter with a new additional vertex. For an even integer t ≥ 4, let M denote the asymmetric maximal density of the pair (K^(3)_t, K^(3)_{t/2}). We prove that adding a set F of random hyperedges satisfying |F| ≫ n^{3-1/M} to a given n-vertex 3-uniform hypergraph H with non-vanishing edge density asymptotically almost surely results in a perturbed hypergraph enjoying the Ramsey property for K^(3)_t and two colours. We conjecture that this result is asymptotically best possible with respect to the size of F whenever t ≥ 6 is even. The key tools of our proof are a new variant of the hypergraph regularity lemma accompanied with a tuple lemma providing appropriate control over joint link graphs. Our variant combines the so called strong and the weak hypergraph regularity lemmata.
Cite as
Elad Aigner-Horev, Dan Hefetz, and Mathias Schacht. Ramsey Properties of Randomly Perturbed Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 59:1-59:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{aignerhorev_et_al:LIPIcs.APPROX/RANDOM.2024.59,
author = {Aigner-Horev, Elad and Hefetz, Dan and Schacht, Mathias},
title = {{Ramsey Properties of Randomly Perturbed Hypergraphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {59:1--59:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.59},
URN = {urn:nbn:de:0030-drops-210528},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.59},
annote = {Keywords: Ramsey Theory, Smoothed Analysis, Random Hypergraphs}
}
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RANDOM
Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs
Abstract
In this paper, we study the problem of locally constructing a sparse spanning subgraph (LSSG), introduced by Levi, Ron, and Rubinfeld (ALGO'20). In this problem, the goal is to locally decide for each e ∈ E if it is in G' where G' is a connected subgraph of G (determined only by G and the randomness of the algorithm). We provide an LSSG that receives as a parameter a lower bound, ϕ, on the conductance of G whose query complexity is Õ(√n/ϕ²). This is almost optimal when ϕ is a constant since Ω(√n) queries are necessary even when G is an expander. Furthermore, this improves the state of the art of Õ(n^{2/3}) queries for ϕ = Ω(1/n^{1/12}).
We then extend our result for (k, ϕ_in, ϕ_out)-clusterable graphs and provide an algorithm whose query complexity is Õ(√n + ϕ_out n) for constant k and ϕ_in. This bound is almost optimal when ϕ_out = O(1/√n).
Cite as
Reut Levi, Moti Medina, and Omer Tubul. Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 60:1-60:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{levi_et_al:LIPIcs.APPROX/RANDOM.2024.60,
author = {Levi, Reut and Medina, Moti and Tubul, Omer},
title = {{Nearly Optimal Local Algorithms for Constructing Sparse Spanners of Clusterable Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {60:1--60:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.60},
URN = {urn:nbn:de:0030-drops-210537},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.60},
annote = {Keywords: Locally Computable Algorithms, Sublinear algorithms, Spanning Subgraphs, Clusterbale Graphs}
}
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RANDOM
When Can an Expander Code Correct Ω(n) Errors in O(n) Time?
Abstract
Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph G together with a linear inner code C₀. Expander codes are Tanner codes whose defining bipartite graph G has good expansion property. This paper is motivated by the following natural and fundamental problem in decoding expander codes:
What are the sufficient and necessary conditions that δ and d₀ must satisfy, so that every bipartite expander G with vertex expansion ratio δ and every linear inner code C₀ with minimum distance d₀ together define an expander code that corrects Ω(n) errors in O(n) time?
For C₀ being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT'96) showed that δ > 3/4 is sufficient; later Viderman (ACM-TOCT'13) improved this to δ > 2/3-Ω(1) and he also showed that δ > 1/2 is necessary. For general linear code C₀, the previously best-known result of Dowling and Gao (IEEE-TIT'18) showed that d₀ = Ω(cδ^{-2}) is sufficient, where c is the left-degree of G.
In this paper, we give a near-optimal solution to the above question for general C₀ by showing that δ d₀ > 3 is sufficient and δ d₀ > 1 is necessary, thereby also significantly improving Dowling-Gao’s result. We present two novel algorithms for decoding expander codes, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius.
Cite as
Kuan Cheng, Minghui Ouyang, Chong Shangguan, and Yuanting Shen. When Can an Expander Code Correct Ω(n) Errors in O(n) Time?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 61:1-61:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{cheng_et_al:LIPIcs.APPROX/RANDOM.2024.61,
author = {Cheng, Kuan and Ouyang, Minghui and Shangguan, Chong and Shen, Yuanting},
title = {{When Can an Expander Code Correct \Omega(n) Errors in O(n) Time?}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {61:1--61:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.61},
URN = {urn:nbn:de:0030-drops-210543},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.61},
annote = {Keywords: expander codes, expander graphs, linear-time decoding}
}
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RANDOM
Coboundary and Cosystolic Expansion Without Dependence on Dimension or Degree
Abstract
We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of SL_n(𝔽_q). The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and Oppenheim.
Our new expansion constants do not depend on the degree of the complex nor on its dimension, nor on the group of coefficients. This implies improved bounds on Gromov’s topological overlap constant, and on Dinur and Meshulam’s cover stability, which may have applications for agreement testing.
In comparison, existing bounds decay exponentially with the ambient dimension (for spherical buildings) and in addition decay linearly with the degree (for all known bounded-degree high dimensional expanders). Our results are based on several new techniques:
- We develop a new "color-restriction" technique which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes.
- We give a new "spectral" proof for Evra and Kaufman’s local-to-global theorem, deriving better bounds and getting rid of the dependence on the degree. This theorem bounds the cosystolic expansion of a complex using coboundary expansion and spectral expansion of the links.
- We derive absolute bounds on the coboundary expansion of the spherical building (and any order complex of a homogeneous geometric lattice) by constructing a novel family of very short cones.
Cite as
Yotam Dikstein and Irit Dinur. Coboundary and Cosystolic Expansion Without Dependence on Dimension or Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 62:1-62:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{dikstein_et_al:LIPIcs.APPROX/RANDOM.2024.62,
author = {Dikstein, Yotam and Dinur, Irit},
title = {{Coboundary and Cosystolic Expansion Without Dependence on Dimension or Degree}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {62:1--62:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.62},
URN = {urn:nbn:de:0030-drops-210556},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.62},
annote = {Keywords: High Dimensional Expanders, HDX, Spectral Expansion, Coboundary Expansion, Cocycle Expansion, Cosystolic Expansion}
}
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RANDOM
Rapid Mixing of the Down-Up Walk on Matchings of a Fixed Size
Abstract
Let G = (V,E) be a graph on n vertices and let m^*(G) denote the size of a maximum matching in G. We show that for any δ > 0 and for any 1 ≤ k ≤ (1-δ)m^*(G), the down-up walk on matchings of size k in G mixes in time polynomial in n. Previously, polynomial mixing was not known even for graphs with maximum degree Δ, and our result makes progress on a conjecture of Jain, Perkins, Sah, and Sawhney [STOC, 2022] that the down-up walk mixes in optimal time O_{Δ,δ}(nlog{n}).
In contrast with recent works analyzing mixing of down-up walks in various settings using the spectral independence framework, we bound the spectral gap by constructing and analyzing a suitable multi-commodity flow. In fact, we present constructions demonstrating the limitations of the spectral independence approach in our setting.
Cite as
Vishesh Jain and Clayton Mizgerd. Rapid Mixing of the Down-Up Walk on Matchings of a Fixed Size. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{jain_et_al:LIPIcs.APPROX/RANDOM.2024.63,
author = {Jain, Vishesh and Mizgerd, Clayton},
title = {{Rapid Mixing of the Down-Up Walk on Matchings of a Fixed Size}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {63:1--63:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.63},
URN = {urn:nbn:de:0030-drops-210563},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.63},
annote = {Keywords: Down-up walk, Matchings, MCMC}
}
Document
RANDOM
On the Communication Complexity of Finding a King in a Tournament
Abstract
A tournament is a complete directed graph. A source in a tournament is a vertex that has no in-neighbours (every other vertex is reachable from it via a path of length 1), and a king in a tournament is a vertex v such that every other vertex is reachable from v via a path of length at most 2. It is well known that every tournament has at least one king. In particular, a maximum out-degree vertex is a king. The tasks of finding a king and a maximum out-degree vertex in a tournament has been relatively well studied in the context of query complexity. We study the communication complexity of finding a king, of finding a maximum out-degree vertex, and of finding a source (if it exists) in a tournament, where the edges are partitioned between two players. The following are our main results for n-vertex tournaments:
- We show that the communication task of finding a source in a tournament is equivalent to the well-studied Clique vs. Independent Set (CIS) problem on undirected graphs. As a result, known bounds on the communication complexity of CIS [Yannakakis, JCSS'91, Göös, Pitassi, Watson, SICOMP'18] imply a bound of Θ̃(log² n) for finding a source (if it exists, or outputting that there is no source) in a tournament.
- The deterministic and randomized communication complexities of finding a king are Θ(n). The quantum communication complexity of finding a king is Θ̃(√n).
- The deterministic, randomized, and quantum communication complexities of finding a maximum out-degree vertex are Θ(n log n), Θ̃(n) and Θ̃(√n), respectively. Our upper bounds above hold for all partitions of edges, and the lower bounds for a specific partition of the edges.
One of our lower bounds uses a fooling-set based argument, and all our other lower bounds follow from carefully-constructed reductions from Set-Disjointness. An interesting point to note here is that while the deterministic query complexity of finding a king has been open for over two decades [Shen, Sheng, Wu, SICOMP'03], we are able to essentially resolve the complexity of this problem in a model (communication complexity) that is usually harder to analyze than query complexity.
Cite as
Nikhil S. Mande, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh. On the Communication Complexity of Finding a King in a Tournament. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 64:1-64:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{mande_et_al:LIPIcs.APPROX/RANDOM.2024.64,
author = {Mande, Nikhil S. and Paraashar, Manaswi and Sanyal, Swagato and Saurabh, Nitin},
title = {{On the Communication Complexity of Finding a King in a Tournament}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {64:1--64:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.64},
URN = {urn:nbn:de:0030-drops-210571},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.64},
annote = {Keywords: Communication complexity, tournaments, query complexity}
}
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RANDOM
Capacity-Achieving Gray Codes
Abstract
To ensure differential privacy, one can reveal an integer fuzzily in two ways: (a) add some Laplace noise to the integer, or (b) encode the integer as a binary string and add iid BSC noise. The former is simple and natural while the latter is flexible and affordable, especially when one wants to reveal a sparse vector of integers. In this paper, we propose an implementation of (b) that achieves the capacity of the BSC with positive error exponents. Our implementation adds error-correcting functionality to Gray codes by mimicking how software updates back up the files that are getting updated ("coded Gray code"). In contrast, the old implementation of (b) interpolates between codewords of a black-box error-correcting code ("Grayed code").
Cite as
Venkatesan Guruswami and Hsin-Po Wang. Capacity-Achieving Gray Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 65:1-65:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{guruswami_et_al:LIPIcs.APPROX/RANDOM.2024.65,
author = {Guruswami, Venkatesan and Wang, Hsin-Po},
title = {{Capacity-Achieving Gray Codes}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {65:1--65:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.65},
URN = {urn:nbn:de:0030-drops-210582},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.65},
annote = {Keywords: Gray codes, capacity-achieving codes, differential privacy}
}
Document
RANDOM
On Black-Box Meta Complexity and Function Inversion
Abstract
The relationships between various meta-complexity problems are not well understood in the worst-case regime, including whether the search version is harder than the decision version, whether the hardness scales with the "threshold", and how the hardness of different meta-complexity problems relate to one another, and to the task of function inversion.
In this work, we present resolutions to some of these questions with respect to the black-box analog of these problems. In more detail, let MK^t_M P[s] denote the language consisting of strings x with K_{M}^t(x) < s(|x|), where K_M^t(x) denotes the t-bounded Kolmogorov complexity of x with M as the underlying (Universal) Turing machine, and let search-MK^t_M P[s] denote the search version of the same problem.
We show that if for every Universal Turing machine U there exists a 2^{α n}poly(n)-size U-oracle aided circuit deciding MK^t_U P[n-O(1)], then for every function s, and every not necessarily universal Turing machine M, there exists a 2^{α s(n)}poly(n)-size M-oracle aided circuit solving search-MK^t_M P[s(n)]; this in turn yields circuits of roughly the same size for both the Minimum Circuit Size Problem (MCSP), and the function inversion problem, as they can be thought of as instantiating MK^t_M P with particular choices of (a non-universal) TMs M (the circuit emulator for the case of MCSP, and the function evaluation in the case of function inversion).
As a corollary of independent interest, we get that the complexity of black-box function inversion is (roughly) the same as the complexity of black-box deciding MK^t_U P[n-O(1)] for any universal TM U; that is, also in the worst-case regime, black-box function inversion is "equivalent" to black-box deciding MK^t_U P.
Cite as
Noam Mazor and Rafael Pass. On Black-Box Meta Complexity and Function Inversion. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 66:1-66:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{mazor_et_al:LIPIcs.APPROX/RANDOM.2024.66,
author = {Mazor, Noam and Pass, Rafael},
title = {{On Black-Box Meta Complexity and Function Inversion}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {66:1--66:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.66},
URN = {urn:nbn:de:0030-drops-210597},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.66},
annote = {Keywords: Meta Complexity, Kolmogorov complexity, function inversion}
}
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RANDOM
Explicit and Near-Optimal Construction of t-Rankwise Independent Permutations
Abstract
Letting t ≤ n, a family of permutations of [n] = {1,2,…, n} is called t-rankwise independent if for any t distinct entries in [n], when a permutation π is sampled uniformly at random from the family, the order of the t entries in π is uniform among the t! possibilities. Itoh et al. show a lower bound of (n/2)^⌊t/4⌋ for the number of members in such a family, and provide a construction of a t-rankwise independent permutation family of size n^O(t^2/ln(t)). We provide an explicit, deterministic construction of a t-rankwise independent family of size n^O(t) for arbitrary parameters t ≤ n. Our main ingredient is a way to make the elements of a t-independent family "more injective", which might be of independent interest.
Cite as
Nicholas Harvey and Arvin Sahami. Explicit and Near-Optimal Construction of t-Rankwise Independent Permutations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 67:1-67:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{harvey_et_al:LIPIcs.APPROX/RANDOM.2024.67,
author = {Harvey, Nicholas and Sahami, Arvin},
title = {{Explicit and Near-Optimal Construction of t-Rankwise Independent Permutations}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {67:1--67:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.67},
URN = {urn:nbn:de:0030-drops-210600},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.67},
annote = {Keywords: Rankwise independent permutations}
}
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RANDOM
Sparse High Dimensional Expanders via Local Lifts
Abstract
High dimensional expanders (HDXs) are a hypergraph generalization of expander graphs. They are extensively studied in the math and TCS communities due to their many applications. Like expander graphs, HDXs are especially interesting for applications when they are bounded degree, namely, if the number of edges adjacent to every vertex is bounded. However, only a handful of constructions are known to have this property, all of which rely on algebraic techniques. In particular, no random or combinatorial construction of bounded degree high dimensional expanders is known. As a result, our understanding of these objects is limited.
The degree of an i-face in an HDX is the number of (i+1)-faces that contain it. In this work we construct complexes whose higher dimensional faces have bounded degree. This is done by giving an elementary and deterministic algorithm that takes as input a regular k-dimensional HDX X and outputs another regular k-dimensional HDX X̂ with twice as many vertices. While the degree of vertices in X̂ grows, the degree of the (k-1)-faces in X̂ stays the same. As a result, we obtain a new "algebra-free" construction of HDXs whose (k-1)-face degree is bounded.
Our construction algorithm is based on a simple and natural generalization of the expander graph construction by Bilu and Linial [Yehonatan Bilu and Nathan Linial, 2006], which build expander graphs using lifts coming from edge signings. Our construction is based on local lifts of high dimensional expanders, where a local lift is a new complex whose top-level links are lifts of the links of the original complex. We demonstrate that a local lift of an HDX is also an HDX in many cases.
In addition, combining local lifts with existing bounded degree constructions creates new families of bounded degree HDXs with significantly different links than before. For every large enough D, we use this technique to construct families of bounded degree HDXs with links that have diameter ≥ D.
Cite as
Inbar Ben Yaacov, Yotam Dikstein, and Gal Maor. Sparse High Dimensional Expanders via Local Lifts. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 68:1-68:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{benyaacov_et_al:LIPIcs.APPROX/RANDOM.2024.68,
author = {Ben Yaacov, Inbar and Dikstein, Yotam and Maor, Gal},
title = {{Sparse High Dimensional Expanders via Local Lifts}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {68:1--68:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.68},
URN = {urn:nbn:de:0030-drops-210612},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.68},
annote = {Keywords: High Dimensional Expanders, HDX, Spectral Expansion, Lifts, Covers, Explicit Constructions, Randomized Constructions, Deterministic Constructions}
}
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RANDOM
Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors
Abstract
We study randomness extractors in AC⁰ and NC¹. For the AC⁰ setting, we give a logspace-uniform construction such that for every k ≥ n/poly log n, ε ≥ 2^{-poly log n}, it can extract from an arbitrary (n, k) source, with a small constant fraction entropy loss, and the seed length is O(log n/(ε)). The seed length and output length are optimal up to constant factors matching the parameters of the best polynomial time construction such as [Guruswami et al., 2009]. The range of k and ε almost meets the lower bound in [Goldreich et al., 2015] and [Cheng and Li, 2018]. We also generalize the main lower bound of [Goldreich et al., 2015] for extractors in AC⁰, showing that when k < n/poly log n, even strong dispersers do not exist in non-uniform AC⁰. For the NC¹ setting, we also give a logspace-uniform extractor construction with seed length O(log n/(ε)) and a small constant fraction entropy loss in the output. It works for every k ≥ O(log² n), ε ≥ 2^{-O(√k)}.
Our main techniques include a new error reduction process and a new output stretch process, based on low-depth circuit implementations for mergers, condensers, and somewhere extractors.
Cite as
Kuan Cheng and Ruiyang Wu. Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 69:1-69:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{cheng_et_al:LIPIcs.APPROX/RANDOM.2024.69,
author = {Cheng, Kuan and Wu, Ruiyang},
title = {{Randomness Extractors in AC⁰ and NC¹: Optimal up to Constant Factors}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {69:1--69:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.69},
URN = {urn:nbn:de:0030-drops-210623},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.69},
annote = {Keywords: randomness extractor, uniform AC⁰, error reduction, uniform NC¹, disperser}
}
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RANDOM
On Sampling from Ising Models with Spectral Constraints
Abstract
We consider the problem of sampling from the Ising model when the underlying interaction matrix has eigenvalues lying within an interval of length γ. Recent work in this setting has shown various algorithmic results that apply roughly when γ < 1, notably with nearly-linear running times based on the classical Glauber dynamics. However, the optimality of the range of γ was not clear since previous inapproximability results developed for the antiferromagnetic case (where the matrix has entries ≤ 0) apply only for γ > 2.
To this end, Kunisky (SODA'24) recently provided evidence that the problem becomes hard already when γ > 1 based on the low-degree hardness for an inference problem on random matrices. Based on this, he conjectured that sampling from the Ising model in the same range of γ is NP-hard.
Here we confirm this conjecture, complementing in particular the known algorithmic results by showing NP-hardness results for approximately counting and sampling when γ > 1, with strong inapproximability guarantees; we also obtain a more refined hardness result for matrices where only a constant number of entries per row are allowed to be non-zero. The main observation in our reductions is that, for γ > 1, Glauber dynamics mixes slowly when the interactions are all positive (ferromagnetic) for the complete and random regular graphs, due to a bimodality in the underlying distribution. While ferromagnetic interactions typically preclude NP-hardness results, here we work around this by introducing in an appropriate way mild antiferromagnetism, keeping the spectrum roughly within the same range. This allows us to exploit the bimodality of the aforementioned graphs and show the target NP-hardness by adapting suitably previous inapproximability techniques developed for antiferromagnetic systems.
Cite as
Andreas Galanis, Alkis Kalavasis, and Anthimos Vardis Kandiros. On Sampling from Ising Models with Spectral Constraints. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{galanis_et_al:LIPIcs.APPROX/RANDOM.2024.70,
author = {Galanis, Andreas and Kalavasis, Alkis and Kandiros, Anthimos Vardis},
title = {{On Sampling from Ising Models with Spectral Constraints}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {70:1--70:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.70},
URN = {urn:nbn:de:0030-drops-210638},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.70},
annote = {Keywords: Ising model, spectral constraints, Glauber dynamics, mean-field Ising, random regular graphs}
}
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RANDOM
Approximate Degree Composition for Recursive Functions
Abstract
Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e. functions obtained by composing a base function with itself a number of times. Let h^d denote the standard d-fold composition of the base function h. The main result of this work is to show that the approximate degree composes if either of the following conditions holds:
- The outer function f:{0,1}ⁿ → {0,1} is a recursive function of the form h^d, with h being any base function and d = Ω(log log n).
- The inner function is a recursive function of the form h^d, with h being any constant arity base function (other than AND and OR) and d = Ω(log log n), where n is the arity of the outer function.
In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be efficiently eliminated if the inner or outer function is a recursive function.
Cite as
Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, and Nitin Saurabh. Approximate Degree Composition for Recursive Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 71:1-71:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{chakraborty_et_al:LIPIcs.APPROX/RANDOM.2024.71,
author = {Chakraborty, Sourav and Kayal, Chandrima and Mittal, Rajat and Paraashar, Manaswi and Saurabh, Nitin},
title = {{Approximate Degree Composition for Recursive Functions}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {71:1--71:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.71},
URN = {urn:nbn:de:0030-drops-210642},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.71},
annote = {Keywords: Approximate degree, Boolean function, Composition theorem}
}
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RANDOM
Public Coin Interactive Proofs for Label-Invariant Distribution Properties
Abstract
Assume we are given sample access to an unknown distribution D over a large domain [N]. An emerging line of work has demonstrated that many basic quantities relating to the distribution, such as its distance from uniform and its Shannon entropy, despite being hard to approximate through the samples only, can be efficiently and verifiably approximated through interaction with an untrusted powerful prover, that knows the entire distribution [Herman and Rothblum, STOC 2022, FOCS 2023]. Concretely, these works provide an efficient proof system for approximation of any label-invariant distribution quantity (i.e. any function over the distribution that’s invariant to a re-labeling of the domain [N]).
In our main result, we present the first efficient public coin AM protocol, for any label-invariant property. Our protocol achieves sample complexity and communication complexity of magnitude Õ(N^{2/3}), while the proof can be generated in quasi-linear Õ(N) time.
On top of that, we also give a public-coin protocol for efficiently verifying the distance a between a samplable distribution D, and some explicitly given distribution Q.
Cite as
Tal Herman. Public Coin Interactive Proofs for Label-Invariant Distribution Properties. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 72:1-72:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{herman:LIPIcs.APPROX/RANDOM.2024.72,
author = {Herman, Tal},
title = {{Public Coin Interactive Proofs for Label-Invariant Distribution Properties}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {72:1--72:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.72},
URN = {urn:nbn:de:0030-drops-210654},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.72},
annote = {Keywords: Interactive Proof Systems, Distribution Testing, Public-Coin Protocols}
}
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RANDOM
Additive Noise Mechanisms for Making Randomized Approximation Algorithms Differentially Private
Abstract
The exponential increase in the amount of available data makes taking advantage of them without violating users' privacy one of the fundamental problems of computer science. This question has been investigated thoroughly under the framework of differential privacy. However, most of the literature has not focused on settings where the amount of data is so large that we are not even able to compute the exact answer in the non-private setting (such as in the streaming setting, sublinear-time setting, etc.). This can often make the use of differential privacy unfeasible in practice.
In this paper, we show a general approach for making Monte-Carlo randomized approximation algorithms differentially private. We only need to assume the error R of the approximation algorithm is sufficiently concentrated around 0 (e.g. 𝔼[|R|] is bounded) and that the function being approximated has a small global sensitivity Δ. Specifically, if we have a randomized approximation algorithm with sufficiently concentrated error which has time/space/query complexity T(n,ρ) with ρ being an accuracy parameter, we can generally speaking get an algorithm with the same accuracy and complexity T(n,Θ(ε ρ)) that is ε-differentially private.
Our technical results are as follows. First, we show that if the error is subexponential, then the Laplace mechanism with error magnitude proportional to the sum of the global sensitivity Δ and the subexponential diameter of the error of the algorithm makes the algorithm differentially private. This is true even if the worst-case global sensitivity of the algorithm is large or infinite. We then introduce a new additive noise mechanism, which we call the zero-symmetric Pareto mechanism. We show that using this mechanism, we can make an algorithm differentially private even if we only assume a bound on the first absolute moment of the error 𝔼[|R|].
Finally, we use our results to give either the first known or improved sublinear-complexity differentially private algorithms for various problems. This includes results for frequency moments, estimating the average degree of a graph in subliinear time, rank queries, or estimating the size of the maximum matching. Our results raise many new questions and we state multiple open problems.
Cite as
Jakub Tětek. Additive Noise Mechanisms for Making Randomized Approximation Algorithms Differentially Private. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{tetek:LIPIcs.APPROX/RANDOM.2024.73,
author = {T\v{e}tek, Jakub},
title = {{Additive Noise Mechanisms for Making Randomized Approximation Algorithms Differentially Private}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {73:1--73:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.73},
URN = {urn:nbn:de:0030-drops-210660},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.73},
annote = {Keywords: Differential privacy, Randomized approximation algorithms}
}
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RANDOM
Improved Bounds for Graph Distances in Scale Free Percolation and Related Models
Abstract
In this paper, we study graph distances in the geometric random graph models scale-free percolation SFP, geometric inhomogeneous random graphs GIRG, and hyperbolic random graphs HRG. Despite the wide success of the models, the parameter regime in which graph distances are polylogarithmic is poorly understood. We provide new and improved lower bounds. In a certain portion of the parameter regime, those match the known upper bounds.
Compared to the best previous lower bounds by Hao and Heydenreich [Hao and Heydenreich, 2023], our result has several advantages: it gives matching bounds for a larger range of parameters, thus settling the question for a larger portion of the parameter space. It strictly improves the lower bounds of [Hao and Heydenreich, 2023] for all parameters settings in which those bounds were not tight. It gives tail bounds on the probability of having short paths, which imply shape theorems for the k-neighbourhood of a vertex whenever our lower bounds are tight, and tight bounds for the size of this k-neighbourhood. And last but not least, our proof is much simpler and not much longer than two pages, and we demonstrate that it generalizes well by showing that the same technique also works for first passage percolation.
Cite as
Kostas Lakis, Johannes Lengler, Kalina Petrova, and Leon Schiller. Improved Bounds for Graph Distances in Scale Free Percolation and Related Models. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 74:1-74:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{lakis_et_al:LIPIcs.APPROX/RANDOM.2024.74,
author = {Lakis, Kostas and Lengler, Johannes and Petrova, Kalina and Schiller, Leon},
title = {{Improved Bounds for Graph Distances in Scale Free Percolation and Related Models}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {74:1--74:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.74},
URN = {urn:nbn:de:0030-drops-210676},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.74},
annote = {Keywords: Mathematics, Probability Theory, Combinatorics, Random Graphs, Random Metric Spaces}
}
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RANDOM
Derandomizing Multivariate Polynomial Factoring for Low Degree Factors
Abstract
Kaltofen [STOC 1986] gave a randomized algorithm to factor multivariate polynomials given by algebraic circuits. We derandomize the algorithm in some special cases.
For an n-variate polynomial f of degree d from a class 𝒞 of algebraic circuits, we design a deterministic algorithm to find all its irreducible factors of degree ≤ δ, for constant δ. The running time of this algorithm stems from a deterministic PIT algorithm for class 𝒞 and a deterministic algorithm that tests divisibility of f by a polynomial of degree ≤ δ.
By using the PIT algorithm for constant-depth circuits by Limaye, Srinivasan and Tavenas [FOCS 2021] and the divisibility results by Forbes [FOCS 2015], this generalizes and simplifies a recent result by Kumar, Ramanathan and Saptharishi [SODA 2024]. They designed a subexponential-time algorithm that, given a blackbox access to f computed by a constant-depth circuit, outputs its irreducible factors of degree ≤ δ. When the input f is sparse, the time complexity of our algorithm depends on a whitebox PIT algorithm for ∑_i m_i g_i^{d_i}, where m_i are monomials and deg(g_i) ≤ δ. All the previous algorithms required a blackbox PIT algorithm for the same class.
Our second main result considers polynomials f, where each irreducible factor has degree at most δ. We show that all the irreducible factors with their multiplicities can be computed in polynomial time with blackbox access to f.
Finally, we consider factorization of sparse polynomials. We show that in order to compute all the sparse irreducible factors efficiently, it suffices to derandomize irreducibility preserving bivariate projections for sparse polynomials.
Cite as
Pranjal Dutta, Amit Sinhababu, and Thomas Thierauf. Derandomizing Multivariate Polynomial Factoring for Low Degree Factors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 75:1-75:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
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@InProceedings{dutta_et_al:LIPIcs.APPROX/RANDOM.2024.75,
author = {Dutta, Pranjal and Sinhababu, Amit and Thierauf, Thomas},
title = {{Derandomizing Multivariate Polynomial Factoring for Low Degree Factors}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
pages = {75:1--75:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-348-5},
ISSN = {1868-8969},
year = {2024},
volume = {317},
editor = {Kumar, Amit and Ron-Zewi, Noga},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.75},
URN = {urn:nbn:de:0030-drops-210687},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.75},
annote = {Keywords: algebraic complexity, factoring, low degree, weight isolation, divisibility}
}