Approximate Hypergraph Vertex Cover and Generalized Tuza’s Conjecture

Using known results regarding PCP, we present simple proofs of the inapproximability of vertex cover for hypergraphs. Specifically, we show that 1 Approximating the size of the minimum vertex cover in O(1)-regular hypergraphs to within a factor of 1.99999 is NP-hard. 2 Approximating the size of the minimum vertex cover in 4-regular hypergraphs to within a factor of 1.49999 is NP-hard. Both results are inferior to known results (by Trevisan (2001) and Holmerin (2001)), but they are derived using much simpler proofs. Furthermore, these proofs demonstrate the applicability of the FGLSS-reduction in the context of reductions among combinatorial optimization problems.

In this paper, we consider the minimal vertex cover and minimal dominating sets with capacity and/or connectivity constraint enumeration problems. We develop polynomial-delay enumeration algorithms for these problems on bounded-degree graphs. For the case of minimal connected vertex covers, our algorithms run in polynomial delay, even on the class of d-claw free graphs. This result is extended for bounded-degree graphs and outputs in quasi-polynomial time on general graphs. To complement these algorithmic results, we show that the minimal connected vertex cover, minimal connected dominating set, and minimal capacitated vertex cover enumeration problems in 2-degenerated bipartite graphs are at least as hard as enumerating minimal transversals in hypergraphs.

The minimum vertex cover problem is a well-known optimization problem; it has been used in a wide variety of applications. This paper focuses on rough set-based approach for the minimum vertex cover problem of the dynamic and static hypergraphs. First, we demonstrate the relationship between the attribute reduction of decision table and the minimum vertex cover of hypergraph, and the minimum vertex cover problem is converted to an attribute reduction problem based on this relationship. Then, we discuss the update mechanism of minimum vertex cover from the perspective of attribute reduction, and two types of incremental attribute reduction algorithms are proposed, one is the dynamic increase of single vertex and the other is the dynamic increase of multiple vertices. Our algorithms can quickly update the minimum vertex cover in a dynamic hypergraph and improve the rough sets-based method for the minimum vertex cover problem of a static hypergraph in terms of the computational time and the solution quality. The experimental results show the advantages and limitations of the proposed algorithms compared with the existing algorithms.

In a beautiful result, Raghavendra established optimal Unique Games Conjecture (UGC)-based inapproximability for a large class of constraint satisfaction problems (CSPs). In the class of CSPs he considers, of which Maximum Cut is a prominent example, the goal is to find an assignment which maximizes a weighted fraction of constraints satisfied. He gave a generic semi-definite program (SDP) for this class of problems and showed how the approximability of each problem is determined by the corresponding SDP (upto an arbitrarily small additive error) assuming the UGC. He noted that his techniques do no apply to CSPs with strict constraints (all of which must be satisfied) such as Vertex Cover.In this paper we address the approximability of these strict-CSPs. In the class of CSPs we consider, one is given a set of constraints over a set of variables, and a cost function over the assignments, the goal is to find an assignment to the variables of minimum cost which satisfies all the constraints. We present a generic linear program (LP) for a large class of strict-CSPs and give a systematic way to convert integrality gaps for this LP into UGC-based inapproximability results. Some important problems whose approximability our framework captures are Vertex Cover, Hypergraph Vertex Cover, k-partite-Hypergraph Vertex Cover, Independent Set and other covering and packing problems over q-ary alphabets, and a scheduling problem. For the covering and packing problems, which occur quite commonly in practice as well, we provide a matching rounding algorithm, thus settling their approximability upto an arbitrarily small additive error.

We consider the (precedence constrained) Minimum Feedback Arc Set problem with triangle inequalities on the weights, which finds important applications in problems of ranking with inconsistent information. We present a surprising structural insight showing that the problem is a special case of the minimum vertex cover in hypergraphs with edges of size at most 3. This result leads to combinatorial approximation algorithms for the problem and opens the road to studying the problem as a vertex cover problem.KeywordsApproximation AlgorithmTriangle InequalityVertex CoverValid InequalitySingle Machine ScheduleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Exact algorithms for finding minimum transversals in rank-3 hypergraphs

Summer has come again. And what better way is there to spend a summer than to relax on a sandy beach, on a mountain top, or at a park's picnic tables, and... think theory ! Summer is a particularly good time to attack the big questions whose openness just plain annoys you. In light of Reingold's L = SL result, does L-vs.-RL tempt you? If so, take it on! (But perhaps peek first at Reingold--Trevisan--Vadhan's ECCC TR05-022.) Are you convinced that UP = NP should imply the collapse of the polynomial hierarchy? Make it so! (But if you hope to do so via showing that UP is contained in the low hierarchy, peek first at Long--Sheu's 1996 MST article.) You know that S 2 ⊆ S NP∩coNP 2 ⊆ ZPP NP and S NP 2 ⊆ ZPP NP NP , but believe that some of those "⊆"s are "=" s or are (gasp!) provably strict containments (thus giving some insight into whether recent improvements (Cai in FOCS 2001; Cai et al. in Information and Computation 2005) in the collapses of the polynomial hierarchy from assumptions that NP is in P/poly, (NP ∩ coNP)/poly, or coNP/poly are strict improvements)? Well, prove an "=" or find evidence for a strict containment... or to really make this a summer for us all to remember, prove a strict containment! Wishing you happy theorems.This article gives an overview of recent PCP constructions based on the Long Code and the inapproximability results implied by these constructions. We cover the inapproximability results for (1) MAX-3SAT and CLIQUE (2) Results for SPARSEST CUT, VERTEX COVER and MAX-CUT implied by the Unique Games Conjecture and (3) HYPERGRAPH VERTEX COVER and coloring problems on hypergraphs. The article aims at explaining the general framework that, loosely speaking, incorporates all the above results.

We consider the (precedence constrained) Minimum Feedback Arc Set problem with triangle inequalities on the weights, which finds important applications in problems of ranking with inconsistent information. We present a surprising structural insight showing that the problem is a special case of the minimum vertex cover in hypergraphs with edges of size at most 3.

Random $(d_{v},d_{c})$ - regular low-density parity-check (LDPC) codes, where each variable is involved in $d_{v}$ parity checks and each parity check involves $d_{c}$ variables are well-known to achieve the Shannon capacity of the binary symmetric channel, for sufficiently large $d_{v}$ and $d_{c}$ , under exponential time decoding. However, polynomial time algorithms are only known to correct a much smaller fraction of errors. One of the most powerful polynomial-time algorithms with a formal analysis is the linear programming (LP) decoding algorithm of Feldman et al., which is known to correct an $\Omega (1/d_{c})$ fraction of errors. In this paper, we show that fairly powerful extensions of LP decoding, based on the Sherali–Adams and Lasserre hierarchies, fail to correct much more errors than the basic LP-decoder. In particular, we show that: 1) for any values of $d_{v}$ and $d_{c}$ , a linear number of rounds of the Sherali–Adams LP hierarchy cannot correct more than an $O(1/d_{c})$ fraction of errors on a random $(d_{v},d_{c})$ -regular LDPC code; and 2) for any value of $d_{v}$ and infinitely many values of $d_{c}$ , a linear number of rounds of the Lasserre SDP hierarchy cannot correct more than an $O(1/d_{c})$ fraction of errors on a random $(d_{v},d_{c})$ -regular LDPC code. Our proofs use a new stretching and collapsing technique that allows us to leverage recent progress in the study of the limitations of LP/SDP hierarchies for Maximum Constraint Satisfaction Problems (Max-CSPs). The problem then reduces to the construction of special balanced pairwise independent distributions for Sherali–Adams and special cosets of balanced pairwise independent subgroups for Lasserre. Our (algebraic) construction for the Lasserre hierarchy is based on designing sets of points in ${\mathbb F}_{q}^{d}$ (for $q$ any power of 2 and $d = 2,3$ ) with special hyperplane-incidence properties—constructions that may be of independent interest. An intriguing consequence of our work is that expansion seems to be both the strength and the weakness of random regular LDPC codes. Some of our techniques are more generally applicable to a large class of Boolean CSPs called Min-Ones. In particular, for $k$ -Hypergraph Vertex Cover, we obtain an improved integrality gap of $k-1-\epsilon $ that holds after a linear number of rounds of the Lasserre hierarchy, for any $k = q+1$ with $q$ an arbitrary prime power. The best previous gap for a linear number of rounds was equal to $2-\epsilon $ and due to Schoenebeck.

Given a graph G = (V, E) and an integer k ∈ ℕ, we study k-Vertex Separator (resp. k-Edge Separator), where the goal is to remove the minimum number of vertices (resp. edges) such that each connected component in the resulting graph has at most k vertices. Our primary focus is on the case where k is either a constant or a slowly growing function of n (e.g. O(log n) or no(1)). Our problems can be interpreted as a special case of three general classes of problems that have been studied separately (balanced graph partitioning, Hypergraph Vertex Cover (HVC), and fixed parameter tractability (FPT)).Our main result is an O(log k)-approximation algorithm for k-Vertex Separator that runs in time 2O(k)nO(1), and an O(log k)-approximation algorithm for k-Edge Separator that runs in time nO(1). Our result on k-Edge Separator improves the best previous graph partitioning algorithm [24] for small k. Our result on k-Vertex Separator improves the simple (k + 1)-approximation from HVC [3]. When OPT > k, the running time 2O(k)nO(1) is faster than the lower bound kΩ(OPT)nΩ(1) for exact algorithms assuming the Exponential Time Hypothesis [12]. While the running time of 2O(k)nO(1) for k-Vertex Separator seems unsatisfactory, we show that the superpolynomial dependence on k may be needed to achieve a polylogarithmic approximation ratio, based on hardness of Densest k-Subgraph.We also study k-Path Transversal, where the goal is to remove the minimum number of vertices such that there is no simple path of length k. With additional ideas from FPT algorithms and graph theory, we present an O(log k)-approximation algorithm for k-Path Transversal that runs in time 2O(k3 log k)nO(1). Previously, the existence of even (1 − δ)k-approximation algorithm for fixed δ > 0 was open [9].

Given a graph G = (V, E) and an integer k ∊ ℕ, we study k-Vertex Separator (resp. k-Edge Separator), where the goal is to remove the minimum number of vertices (resp. edges) such that each connected component in the resulting graph has at most k vertices. Our primary focus is on the case where k is either a constant or a slowly growing function of n (e.g. O(logn) or no(1)). Our problems can be interpreted as a special case of three general classes of problems that have been studied separately (balanced graph partitioning, Hypergraph Vertex Cover (HVC), and fixed parameter tractability (FPT)).Our main result is an O(log k)-approximation algorithm for k-Vertex Separator that runs in time 2O(k)nO(1), and an O(log k)-approximation algorithm for k-Edge Separator that runs in time nO(1). Our result on k-Edge Separator improves the best previous graph partitioning algorithm [24] for small k. Our result on k-Vertex Separator improves the simple (k + 1)- approximation from HVC [3]. When OPT > k, the running time 2O(k)nO(1) is faster than the lower bound kΩ(OPT)n°(1) for exact algorithms assuming the Exponential Time Hypothesis [12]. While the running time of 2O(k)nO(1) for k-Vertex Separator seems unsatisfactory, we show that the superpolynomial dependence on k may be needed to achieve a polylogarithmic approximation ratio, based on hardness of Densest k-Subgraph.We also study k-Path Transversal, where the goal is to remove the minimum number of vertices such that there is no simple path of length k. With additional ideas from FPT algorithms and graph theory, we present an O(log k)-approximation algorithm for k-Path Transversal that runs in time 2O(k log k)nO(1). Previously, the existence of even (1 - δ)k-approximation algorithm for fixed δ > 0 was open [9].

We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f. The approximation factor of our algorithm is (f+varepsilon ). Let varDelta denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires O(log {varDelta } / log log varDelta ) rounds, for constants varepsilon in (0,1] and fin {mathbb {N}}^+. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms. For constant values of f and varepsilon , our algorithm improves over the (f+varepsilon )-approximation algorithm of Kuhn et al. (SODA, 2006)whose running time is O(log varDelta + log W), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f-approximation for the problem in O(flog n) rounds, improving over the classical result of Khuller et al. (J Algorithms, 1994) that achieves a running time of O(flog ^2 n). Finally, for weighted vertex cover (f=2) our algorithm achieves a deterministic running time of O(log n), matching the randomized previously best result of Koufogiannakis and Young (Distrib Comput, 2011). We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (flceil log _2(M)+1 rceil +varepsilon )-approximate integral solution in O(1+f/logn)·logΔloglogΔ+(f·logM)1.01·logε-1·(logΔ)0.01\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} O\\left( (1+f/\\log n)\\cdot \\left( {\\frac{\\log \\varDelta }{ \\log \\log \\varDelta } + ({f\\cdot \\log M})^{1.01}\\cdot \\log \\varepsilon ^{-1}\\cdot (\\log \\varDelta )^{0.01}}\\right) \\right) \\end{aligned}$$\\end{document}rounds, where f bounds the number of variables in a constraint, varDelta bounds the number of constraints a variable appears in, and M=max left{ 1, lceil 1/a_{min } rceil right} , where a_{min } is the smallest normalized constraint coefficient.

We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank ƒ. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by ƒ. The approximation factor of our algorithm is (ƒ + e). Let Δ denote the maximum degree in the hypergraph. Our algorithm runs in the CONGEST model and requires O(log Δ/log log Δ) rounds, for constants e ∈ (0,1] and ƒ ∈ N+. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of emphprovably optimal distributed algorithms.For constant values of ƒ and e, our algorithm improves over the (&3402; + e)-approximation algorithm of [16] whose running time is O(log Δ + log W), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an ƒ-approximation for the problem in O(ƒ log n) rounds, improving over the classical result of [13] that achieves a running time of O(ƒ log 2 n). Finally, for weighted vertex cover (ƒ=2) our algorithm achieves a deterministic running time of O(log n), matching the randomized previously best result of [14].We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (ƒ + e)-approximate integral solution in O(1 + ƒ /log n)⋅ log Δ over log log Δ+(ƒ ⋅ log M)1.01⋅ log e-1 ⋅(log Δ)0.01)) rounds, where ƒ bounds the number of variables in a constraint, Δ bounds the number of constraints a variable appears in, and M=max{1,1/a min},, amin, where amin is the smallest normalized constraint coefficient. This significantly improves over the results of [16] for the integral case, which achieves the same guarantees in O(e-4 ⋅ ƒ4 ⋅ log ƒ ⋅ log(M ⋅ Δ)) rounds.

A famous conjecture of Tuza states that the minimum number of edges needed to cover all the triangles in a graph is at most twice the maximum number of edge-disjoint triangles. This conjecture was couched in a broader setting by Aharoni and Zerbib who proposed a hypergraph version of this conjecture, and also studied its implied fractional versions. We establish the fractional version of the Aharoni-Zerbib conjecture up to lower order terms. Specifically, we give a factor approximation based on LP rounding for an algorithmic version of the hypergraph Turán problem (AHTP). The objective in AHTP is to pick the smallest collection of (t–1)-sized subsets of vertices of an input t-uniform hypergraph such that every hyperedge contains one of these subsets. Aharoni and Zerbib also posed whether Tuza's conjecture and its hypergraph versions could follow from non-trivial duality gaps between vertex covers and matchings on hypergraphs that exclude certain sub-hypergraphs, for instance, a “tent” structure that cannot occur in the incidence of triangles and edges. We give a strong negative answer to this question, by exhibiting tent-free hypergraphs, and indeed ℱ-free hypergraphs for any finite family ℱ of excluded sub-hypergraphs, whose vertex covers must include almost all the vertices. The algorithmic questions arising in the above study can be phrased as instances of vertex cover on simple hypergraphs, whose hyperedges can pairwise share at most one vertex. We prove that the trivial factor t approximation for vertex cover is hard to improve for simple t-uniform hypergraphs. However, for set cover on simple n-vertex hypergraphs, the greedy algorithm achieves a factor (ln n)/2, better than the optimal ln n factor for general hypergraphs.

This work studies the inapproximability of two well-known scheduling problems: Concurrent Open Shop and the Assembly Line problem. For both these problems, Bansal and Khot [1] obtained tight (2 - ε)-factor inapproximability, assuming the Unique Games Conjecture (UGC). In this paper, we prove optimal (2 - ε)-factor NP-hardness of approximation for both these problems unconditionally, i.e., without assuming UGC. Our results for the scheduling problems follow from a structural hardness for Minimum Vertex Cover on hypergraphs - an unconditional but weaker analog of a similar result of Bansal and Khot [1] which, however, is based on UGC. Formally, we prove that for any ε > 0, and positive integers q, T > 1, the following is NP-hard: Given a qT-uniform hypergraph G(V, E), decide between the following cases, YES Case: There is a partition of V into V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , ⋯, V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> , X with |V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> | = ⋯ = |V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> | ≥ (1 - ε/q) |V| such that the vertices of any hyperedge e can be partitioned into T blocks of q vertices each so that at least T - 1 of the blocks each contain at most one vertex from V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> , for any j ∈ [q]. Since T > 1, this implies that for any j ∈ [q], V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> ∪ X is a vertex cover of size at most (1/q + ε) |V|. NO Case: Every vertex cover in G has size at least (1 - ε)|V |. The above hardness result suffices to prove optimal inapproximability for the scheduling problems mentioned above, via black-box hardness reductions. Using this result, we also prove a super constant hardness factor for Two Stage Stochastic Vehicle Routing, for which a similar inapproximability was shown by Gørtz, Nagarajan, and Saket [2] assuming the UGC, and based on the result of [1].