Algorithms to Approximate Column-Sparse Packing Problems

Brubach, Brian; Sankararaman, Karthik Abinav; Srinivasan, Aravind; Xu, Pan

doi:10.1137/1.9781611975031.22

Cited by 6 publications

(11 citation statements)

References 52 publications

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“…Our contribution: We improve upon the recent bound of Brubach et al [6]. Our algorithm also works in the case where we replace counting constraints on rows with arbitrary matroids.…”

Section: Problems Overview Known Results and Our Contributionsmentioning

confidence: 99%

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Random Order Contention Resolution Schemes

Adamczyk

Włodarczyk

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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Contention resolution schemes have proven to be an incredibly powerful concept which allows to tackle a broad class of problems. The framework has been initially designed to handle submodular optimization under various types of constraints, that is, intersections of exchange systems (including matroids), knapsacks, and unsplittable flows on trees. Later on, it turned out that this framework perfectly extends to optimization under uncertainty, like stochastic probing and online selection problems, which further can be applied to mechanism design. We add to this line of work by showing how to create contention resolution schemes for intersection of matroids and knapsacks when we work in the random order setting. More precisely, we do know the whole universe of elements in advance, but they appear in an order given by a random permutation. Upon arrival we need to irrevocably decide whether to take an element or not. We bring a novel technique for analyzing procedures in the random order setting that is based on the martingale theory. This unified approach makes it easier to combine constraints, and we do not need to rely on the monotonicity of contention resolution schemes.Our paper fills the gaps, extends, and creates connections between many previous results and techniques. The main application of our framework is a k + 4 + ε approximation ratio for the Bayesian multi-parameter unit-demand mechanism design under the constraint of k matroids intersection, which improves upon the previous bounds of 4k − 2 and e(k + 1). Other results include improved approximation ratios for stochastic k-set packing and submodular stochastic probing over arbitrary non-negative submodular objective function, whereas previous results required the objective to be monotone. * This is an extended version of a paper whose preliminary version appeared in

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“…Our contribution: We improve upon the recent bound of Brubach et al [6]. Our algorithm also works in the case where we replace counting constraints on rows with arbitrary matroids.…”

Section: Problems Overview Known Results and Our Contributionsmentioning

confidence: 99%

“…They have presented a 2k-approximation algorithm for it, and a (k + 1)-approximation algorithm with an assumption that the outcomes of size vectors L e are monotone. Recently Brubach et al [6] improved the approximation ratio to k + o(k) in the general case.…”

Section: Problems Overview Known Results and Our Contributionsmentioning

confidence: 99%

Random Order Contention Resolution Schemes

Adamczyk

Włodarczyk

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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“…Although not stated explicitly in [2], their algorithm implies (1) for g(e) = |e| + 1 + o(|e|). (A formal proof of this is given in [4].) More precisely, they present a randomized procedure returning a matching that contains each edge e with probability at least…”

Section: Introductionmentioning

confidence: 99%

“…x(e) |e|+1+o(|e|) , where x is a fractional matching. Motivated by this, Brubach, Sankararaman, Srinivasan, and Xu [4] presented a strengthening of the algorithm in [2] to sample a matching containing each edge e with probability at least…”

Section: Introductionmentioning

confidence: 99%

Simpler and Stronger Approaches for Non-Uniform Hypergraph Matching and the Füredi, Kahn, and Seymour Conjecture

Anegg¹,

Angelidakis²,

Zenklusen³

2020

Preprint

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A well-known conjecture of Füredi, Kahn, and Seymour (1993) on non-uniform hypergraph matching states that for any hypergraph with edge weights w, there exists a matching M such that the inequality e∈M g(e)w(e) ≥ OPT LP holds with g(e) = |e| − 1 + 1 /|e|, where OPT LP denotes the optimal value of the canonical LP relaxation. While the conjecture remains open, the strongest result towards it was very recently obtained by Brubach, Sankararaman, Srinivasan, and Xu (2020)-building on and strengthening prior work by Bansal, Gupta, Li, Mestre, Nagarajan, and Rudra (2012)-showing that the aforementioned inequality holds with g(e) = |e| + O(|e| exp(−|e|)). Actually, their method works in a more general sampling setting, where, given a point x of the canonical LP relaxation, the task is to efficiently sample a matching M containing each edge e with probability at least x(e) /g(e).We present simpler and easy-to-analyze procedures leading to improved results. More precisely, for any solution x to the canonical LP, we introduce a simple algorithm based on exponential clocks for Brubach et al.'s sampling setting achieving g(e) = |e| − (|e| − 1)x(e). Apart from the slight improvement in g, our technique may open up new ways to attack the original conjecture. Moreover, we provide a short and arguably elegant analysis showing that a natural greedy approach for the original setting of the conjecture shows the inequality for the same g(e) = |e| − (|e| − 1)x(e) even for the more general hypergraph b-matching problem.

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“…Thus, the FKS conjecture is closely related to approximation algorithms for matchings and there has been considerable interest in the FKS conjecture from the computer science perspective. A number of recent papers [3,1] have shown weakened versions of Conjecture 1.1; most recently, [1] showed the following:…”

Section: Introductionmentioning

confidence: 99%

Some remarks on hypergraph matching and the Füredi-Kahn-Seymour conjecture

Bansal¹,

Harris²

2020

Preprint

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A classic conjecture of Füredi, Kahn and Seymour (1993) states that given any hypergraph with non-negative edge weights w(e), there exists a matching M such that e∈M (|e| − 1 + 1/|e|) w(e) ≥ w * , where w * is the value of an optimum fractional matching. We show the conjecture is true for rank-3 hypergraphs, and is achieved by a natural iterated rounding algorithm. While the general conjecture remains open, we give several new improved bounds. In particular, we show that the iterated rounding algorithm gives e∈M (|e| − δ(e)) w(e) ≥ w * , where δ(e) = |e|/(|e| 2 + |e| − 1), improving upon the baseline guarantee of e∈M |e| w(e) ≥ w * .

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Algorithms to Approximate Column-Sparse Packing Problems

Cited by 6 publications

References 52 publications

Random Order Contention Resolution Schemes

Random Order Contention Resolution Schemes

Simpler and Stronger Approaches for Non-Uniform Hypergraph Matching and the Füredi, Kahn, and Seymour Conjecture

Some remarks on hypergraph matching and the Füredi-Kahn-Seymour conjecture

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