We consider approximation algorithms for packing integer programs (PIPs) of the form @math where @math , @math , and @math are nonnegative. We let @math denote the width of @math which is at least @math . Previous work by bansal-sparse obtained an @math -approximation ratio where @math is the maximum number of nonzeroes in any column of @math (in other words the @math -column sparsity of @math ). They raised the question of obtaining approximation ratios based on the @math -column sparsity of @math (denoted by @math ) which can be much smaller than @math . Motivated by recent work on covering integer programs (CIPs) cq,chs-16 we show that simple algorithms based on randomized rounding followed by alteration, similar to those of bansal-sparse (but with a twist), yield approximation ratios for PIPs based on @math . First, following an integrality gap example from bansal-sparse , we observe that the case of @math is as hard as maximum independent set even when @math . In sharp contrast to this negative result, as soon as width is strictly larger than one, we obtain positive results via the natural LP relaxation. For PIPs with width @math where @math , we obtain an @math -approximation. In the large width regime, when @math , we obtain an @math -approximation. We also obtain a @math -approximation when @math .
In terms of hardness of approximation, PIPs generalize MIS and hence one cannot obtain a ratio better than @math unless @math @cite_10 @cite_9 . Building on MIS, @cite_4 shows that PIPs are hard to approximate within a @math factor for any constant width @math . Hardness of MIS in bounded degree graphs @cite_14 and hardness for @math -set-packing @cite_7 imply that PIPs are hard to approximate to within @math and to within @math when @math is a sufficiently large constant. These hardness results are based on @math matrices for which @math and @math coincide.
@cite_14 @cite_4 @cite_7 @cite_9 @cite_10
2001663593 782508147 2952428427 2008724760
par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . We give new approximation algorithms for packing integer programs (PIPs) by employing the method of randomized rounding combined with alterations. Our first result is a simpler approximation algorithm for general PIPs which matches the best known bounds, and which admits an efficient parallel implementation. We also extend these results to a multi-criteria version of PIPs. Our second result is for the class of packing integer programs (PIPs) that are column sparse, i. e., where there is a specified upper bound k on the number of constraints that each variable appears in. We give an (ek+ o(k))-approximation algorithm for k-column sparse PIPs, improving over previously known O(k 2 )-approximation ratios. We also generalize our result to the case of maximizing non-negative monotone submodular functions over k-column sparse packing constraints, and obtain an e 2k We present a simple distributed @math -approximation algorithm for maximum weight independent set (MaxIS) in the @math model which completes in @math rounds, where @math is the maximum degree, @math is the number of rounds needed to compute a maximal independent set (MIS) on @math , and @math is the maximum weight of a node. Whether our algorithm is randomized or deterministic depends on the MIS algorithm used as a black-box. Plugging in the best known algorithm for MIS gives a randomized solution in @math rounds, where @math is the number of nodes. We also present a deterministic @math -round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a @math -approximation for maximum weight matching without incurring any additional round penalty in the @math model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of for which we describe a mechanism that allows the simulation to run in the @math model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to ( @math ) allows us to devise a distributed algorithm requiring @math rounds for any constant @math . For the unweighted case, we can even obtain a @math -approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on @math . In this paper we study a fundamental open problem in the area of probabilistic checkable proofs: What is the smallest s such that NP ⊆ naPCP1,s[O(log n),3]? In the language of hardness of approximation, this problem is equivalent to determining the smallest s such that getting an s-approximation for satisfiable 3-bit constraint satisfaction problems ("3-CSPs") is NP-hard. The previous best upper bound and lower bound for s are 20 27+µ by Khot and Saket [KS06], and 5 8 (assuming NP subseteq BPP) by Zwick [Zwi98]. In this paper we close the gap assuming Khot's d-to-1 Conjecture. Formally, we prove that if Khot's d-to-1 Conjecture holds for any finite constant integer d, then NP naPCP1,5 8+ µ[O(log n),3] for any constant µ > 0. Our conditional result also solves Hastad's open question [Has01] on determining the inapproximability of satisfiable Max-NTW ("Not Two") instances and confirms Zwick's conjecture [Zwi98] that the 5 8-approximation algorithm for satisfiable 3-CSPs is optimal.