549 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On Optimization and Counting of Non-Broken Bases of Matroids

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    Given a matroid M = (E,I), and a total ordering over the elements E, a broken circuit is a circuit where the smallest element is removed and an NBC independent set is an independent set in I with no broken circuit. The set of NBC independent sets of any matroid M define a simplicial complex called the broken circuit complex which has been the subject of intense study in combinatorics. Recently, Adiprasito, Huh and Katz showed that the face of numbers of any broken circuit complex form a log-concave sequence, proving a long-standing conjecture of Rota. We study counting and optimization problems on NBC bases of a generic matroid. We find several fundamental differences with the independent set complex: for example, we show that it is NP-hard to find the max-weight NBC base of a matroid or that the convex hull of NBC bases of a matroid has edges of arbitrary large length. We also give evidence that the natural down-up walk on the space of NBC bases of a matroid may not mix rapidly by showing that for some family of matroids it is NP-hard to count the number of NBC bases after certain conditionings

    Advancing Scalability in Decentralized Storage: A Novel Approach to Proof-of-Replication via Polynomial Evaluation

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    Proof-of-Replication (PoRep) plays a pivotal role in decentralized storage networks, serving as a mechanism to verify that provers consistently store retrievable copies of specific data. While PoRep’s utility is unquestionable, its implementation in large-scale systems, such as Filecoin, has been hindered by scalability challenges. Most existing PoRep schemes, such as Fisch’s (Eurocrypt 2019), face an escalating number of challenges and growing computational overhead as the number of stored files increases. This paper introduces a novel PoRep scheme distinctively tailored for expansive decentralized storage networks. At its core, our approach hinges on polynomial evaluation, diverging from the probabilistic checking prevalent in prior works. Remarkably, our design requires only a single challenge, irrespective of the number of files, ensuring both prover’s and verifier’s run-times remain manageable even as file counts soar. Our approach introduces a paradigm shift in PoRep designs, offering a blueprint for highly scalable and efficient decentralized storage solutions

    A proof of the Ryser-Brualdi-Stein conjecture for large even nn

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    A Latin square of order nn is an nn by nn grid filled using nn symbols so that each symbol appears exactly once in each row and column. A transversal in a Latin square is a collection of cells which share no symbol, row or column. The Ryser-Brualdi-Stein conjecture, with origins from 1967, states that every Latin square of order nn contains a transversal with n1n-1 cells, and a transversal with nn cells if nn is odd. Keevash, Pokrovskiy, Sudakov and Yepremyan recently improved the long-standing best known bounds towards this conjecture by showing that every Latin square of order nn has a transversal with nO(logn/loglogn)n-O(\log n/\log\log n) cells. Here, we show, for sufficiently large nn, that every Latin square of order nn has a transversal with n1n-1 cells. We also apply our methods to show that, for sufficiently large nn, every Steiner triple system of order nn has a matching containing at least (n4)/3(n-4)/3 edges. This improves a recent result of Keevash, Pokrovskiy, Sudakov and Yepremyan, who found such matchings with n/3O(logn/loglogn)n/3-O(\log n/\log\log n) edges, and proves a conjecture of Brouwer from 1981 for large nn.Comment: 71 pages, 13 figure

    Quadratic Speedups in Parallel Sampling from Determinantal Distributions

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    We study the problem of parallelizing sampling from distributions related to determinants: symmetric, nonsymmetric, and partition-constrained determinantal point processes, as well as planar perfect matchings. For these distributions, the partition function, a.k.a. the count, can be obtained via matrix determinants, a highly parallelizable computation; Csanky proved it is in NC. However, parallel counting does not automatically translate to parallel sampling, as classic reductions between the two are inherently sequential. We show that a nearly quadratic parallel speedup over sequential sampling can be achieved for all the aforementioned distributions. If the distribution is supported on subsets of size kk of a ground set, we show how to approximately produce a sample in O~(k12+c)\widetilde{O}(k^{\frac{1}{2} + c}) time with polynomially many processors for any constant c>0c>0. In the two special cases of symmetric determinantal point processes and planar perfect matchings, our bound improves to O~(k)\widetilde{O}(\sqrt k) and we show how to sample exactly in these cases. As our main technical contribution, we fully characterize the limits of batching for the steps of sampling-to-counting reductions. We observe that only O(1)O(1) steps can be batched together if we strive for exact sampling, even in the case of nonsymmetric determinantal point processes. However, we show that for approximate sampling, Ω~(k12c)\widetilde{\Omega}(k^{\frac{1}{2}-c}) steps can be batched together, for any entropically independent distribution, which includes all mentioned classes of determinantal point processes. Entropic independence and related notions have been the source of breakthroughs in Markov chain analysis in recent years, so we expect our framework to prove useful for distributions beyond those studied in this work.Comment: 33 pages, SPAA 202

    Fault-Tolerant Spanners against Bounded-Degree Edge Failures: Linearly More Faults, Almost For Free

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    We study a new and stronger notion of fault-tolerant graph structures whose size bounds depend on the degree of the failing edge set, rather than the total number of faults. For a subset of faulty edges FGF \subseteq G, the faulty-degree deg(F)\deg(F) is the largest number of faults in FF incident to any given vertex. We design new fault-tolerant structures with size comparable to previous constructions, but which tolerate every fault set of small faulty-degree deg(F)\deg(F), rather than only fault sets of small size F|F|. Our main results are: - New FT-Certificates: For every nn-vertex graph GG and degree threshold ff, one can compute a connectivity certificate HGH \subseteq G with E(H)=O~(fn)|E(H)| = \widetilde{O}(fn) edges that has the following guarantee: for any edge set FF with faulty-degree deg(F)f\deg(F)\leq f and every vertex pair u,vu,v, it holds that uu and vv are connected in HFH \setminus F iff they are connected in GFG \setminus F. This bound on E(H)|E(H)| is nearly tight. Since our certificates handle some fault sets of size up to F=O(fn)|F|=O(fn), prior work did not imply any nontrivial upper bound for this problem, even when f=1f=1. - New FT-Spanners: We show that every nn-vertex graph GG admits a (2k1)(2k-1)-spanner HH with E(H)=Ok(f11/kn1+1/k)|E(H)| = O_k(f^{1-1/k} n^{1+1/k}) edges, which tolerates any fault set FF of faulty-degree at most ff. This bound on E(H)|E(H)| optimal up to its hidden dependence on kk, and it is close to the bound of Ok(F1/2n1+1/k+Fn)O_k(|F|^{1/2} n^{1+1/k} + |F|n) that is known for the case where the total number of faults is F|F| [Bodwin, Dinitz, Robelle SODA '22]. Our proof of this theorem is non-constructive, but by following a proof strategy of Dinitz and Robelle [PODC '20], we show that the runtime can be made polynomial by paying an additional polylog n\text{polylog } n factor in spanner size

    Effective Resistances in Non-Expander Graphs

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    Stability of Homomorphisms, Coverings and Cocycles I: Equivalence

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    This paper is motivated by recent developments in group stability, high dimensional expansion, local testability of error correcting codes and topological property testing. In Part I, we formulate and motivate three stability problems: 1. Homomorphism stability: Are almost homomorphisms close to homomorphisms? 2. Covering stability: Are almost coverings of a cell complex close to genuine coverings of it? 3. Cocycle stability: Are 1-cochains whose coboundary is small close to 1-cocycles? We then prove that these three problems are equivalent.Comment: 32 page

    Gap Preserving Reductions Between Reconfiguration Problems

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    Combinatorial reconfiguration is a growing research field studying problems on the transformability between a pair of solutions for a search problem. For example, in SAT Reconfiguration, for a Boolean formula ? and two satisfying truth assignments ?_? and ?_? for ?, we are asked to determine whether there is a sequence of satisfying truth assignments for ? starting from ?_? and ending with ?_?, each resulting from the previous one by flipping a single variable assignment. We consider the approximability of optimization variants of reconfiguration problems; e.g., Maxmin SAT Reconfiguration requires to maximize the minimum fraction of satisfied clauses of ? during transformation from ?_? to ?_?. Solving such optimization variants approximately, we may be able to obtain a reasonable reconfiguration sequence comprising almost-satisfying truth assignments. In this study, we prove a series of gap-preserving reductions to give evidence that a host of reconfiguration problems are PSPACE-hard to approximate, under some plausible assumption. Our starting point is a new working hypothesis called the Reconfiguration Inapproximability Hypothesis (RIH), which asserts that a gap version of Maxmin CSP Reconfiguration is PSPACE-hard. This hypothesis may be thought of as a reconfiguration analogue of the PCP theorem. Our main result is PSPACE-hardness of approximating Maxmin 3-SAT Reconfiguration of bounded occurrence under RIH. The crux of its proof is a gap-preserving reduction from Maxmin Binary CSP Reconfiguration to itself of bounded degree. Because a simple application of the degree reduction technique using expander graphs due to Papadimitriou and Yannakakis (J. Comput. Syst. Sci., 1991) does not preserve the perfect completeness, we modify the alphabet as if each vertex could take a pair of values simultaneously. To accomplish the soundness requirement, we further apply an explicit family of near-Ramanujan graphs and the expander mixing lemma. As an application of the main result, we demonstrate that under RIH, optimization variants of popular reconfiguration problems are PSPACE-hard to approximate, including Nondeterministic Constraint Logic due to Hearn and Demaine (Theor. Comput. Sci., 2005), Independent Set Reconfiguration, Clique Reconfiguration, and Vertex Cover Reconfiguration
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