Abstract
Real-world problems often have parameters that are uncertain during the optimization phase; stochastic optimization or stochastic programming is a key approach introduced by Beale and by Dantzig in the 1950s to address such uncertainty. Matching is a classical problem in combinatorial optimization. Modern stochastic versions of this problem model problems in kidney exchange, for instance. We improve upon the current-best approximation bound of 3.709 for stochastic matching due to Adamczyk et al. (in: Algorithms-ESA 2015, Springer, Berlin, 2015) to 3.224; we also present improvements on Bansal et al. (Algorithmica 63(4):733–762, 2012) for hypergraph matching and for relaxed versions of the problem. These results are obtained by improved analyses and/or algorithms for rounding linear-programming relaxations of these problems.
Highlights
Stochastic optimization deals with problems where there is uncertainty inherent in the input [14]; this classical sub-area of optimization has received much attention in computer science over the last decade, especially from the viewpoints of approximation algorithms and of handling various models for the input
Matching is well-known to be a bedrock of combinatorial optimization – a problem that has played a key role in the advancement of new algorithmic paradigms including parallel algorithms, randomized algorithms, and, more recently, online algorithms in sponsored-search advertising
We do not yet have a full algorithmic understanding even for various basic stochastic versions of the problem. We advance this goal by improving upon the bounds of [2] and [1] for the matching problem in graphs and in uniform hypergraphs
Summary
Stochastic optimization deals with problems where there is uncertainty inherent in the input [14]; this classical sub-area of optimization has received much attention in computer science over the last decade, especially from the viewpoints of approximation algorithms and of (efficiently) handling various models for the input (see, e.g., [3, 5, 10, 11, 12, 13, 15, 16]). The edges incident upon any vertex v can only be probed for up to tv times; i.e., we cannot exceed the hard constraint of the patience of any vertex Under these constraints, the goal is to find a matching of maximum expected weight, where the expectation is taken both over the stochastic existence of the edges, and over any internal randomization of our algorithm. The goal is to find a matching of maximum expected weight, where the expectation is taken both over the stochastic existence of the edges, and over any internal randomization of our algorithm It is not yet known if this problem is N P -hard. There is a 2.675–approximation algorithm for the weighted stochastic matching problem on a general graph if patience constraints are allowed to be violated by 1. For any ε > 0, Pr[X ≥ (1 + ε)μ] ≤ exp Pr[X ≤ (1 − ε)μ] ≤ exp ε2 −μ
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