An optimal monotone contention resolution scheme for bipartite matchings via a polyhedral viewpoint
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An optimal monotone contention resolution scheme for bipartite matchings via a polyhedral viewpoint Simon Bruggmann1
· Rico Zenklusen1
Received: 22 May 2019 / Accepted: 19 September 2020 © The Author(s) 2020
Abstract Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes round a fractional point by first rounding each coordinate independently, and then dropping some elements to reach a feasible set. Also the second step, where elements are dropped, is typically randomized. This leads to an additional source of randomization within the procedure, which can complicate the analysis. We suggest a different, polyhedral viewpoint to design contention resolution schemes, which avoids to deal explicitly with the randomization in the second step. This is achieved by focusing on the marginals of a dropping procedure. Apart from avoiding one source of randomization, our viewpoint allows for employing polyhedral techniques. Both can significantly simplify the construction and analysis of contention resolution schemes. We show how, through our framework, one can obtain an optimal monotone contention resolution scheme for bipartite matchings, which has a balancedness of 0.4762. So far, only very few results are known about optimality of monotone contention resolution schemes. Our contention resolution scheme for the bipartite case also improves the lower bound on the correlation gap for bipartite matchings. Furthermore, we derive a monotone contention resolution scheme for matchings that significantly improves over the previously best one. More precisely, we obtain a balancedness of 0.4326, improving on a prior 0.1997-balanced scheme. At the same time, our scheme implies that the currently best lower bound on the correlation gap for matchings is not tight. Our results lead to improved approximation factors for various constrained submodular function maximization problems over a combination of matching constraints with further constraints. Keywords Submodular function maximization · Approximation algorithms · Contention resolution schemes
Supported by Swiss National Science Foundation grant 200021_165866. Extended author information available on the last page of the article
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S. Bruggmann, R. Zenklusen
Mathematics Subject Classification 90C27 · 68W20 · 68W25
1 Introduction Submodular function maximization problems enjoyed a surge of interest recently, both within the theory community and in application-focused areas. This is due to a wide set of applications in fields as diverse as combinatorial optimization, economics, algorithmic game theory, or machine learning (see, e.g., [5,9,29,37,39,50] and references therein). The breadth of settings where submodular functions are found is not surprising in view of the fact that submodularity formalizes a very natural property, namely the property of diminishing marginal returns. More fo
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