Two-stage Stochastic Assignment Games

TL;DR

This paper investigates a two-stage stochastic assignment game, demonstrating polynomial solvability when the distribution is explicit, establishing integrality properties, and connecting the problem to multistage vertex cover, with implications for approximation and complexity.

Contribution

The paper introduces a polynomial-time approach for the stochastic assignment game with explicit distributions and reveals integrality properties that aid in approximation and complexity analysis.

Findings

Problem is polynomial-time solvable with explicit distribution.

Polyhedron associated with the problem is integral.

Connection established to multistage vertex cover problem.

Abstract

In this paper, we study a two-stage stochastic version of the assignment game, which is a fundamental cooperative game. Given an initial setting, the set of players may change in the second stage according to some probability distribution, and the goal is to find core solutions that are minimally modified. When the probability distribution is given explicitly, we observe that the problem is polynomial time solvable, as it can be modeled as an LP. More interestingly, we prove that the underlying polyhedron is integral, and exploit this in two ways. First, integrality of the polyhedron allows us to show that the problem can be well approximated when the distribution is unknown, which is a hard setting. Second, we can establish an intimate connection to the well-studied multistage vertex cover problem. Here, it is known that the problem is NP-hard even when there are only 2 stages and the graph in each stage is bipartite. As a byproduct of our result, we can prove that the problem is polynomial-time solvable if the bipartition is the same in each stage.