An Efficient Characterization of Submodular Spanning Tree Games

Zhuan Khye Koh,Laura Sanità

doi:10.1007/978-3-030-17953-3_21

Abstract

Cooperative games are an important class of problems in game theory, where the goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns a value share to each player. A crucial aspect of such games is submodularity (or convexity). Indeed, convex instances of cooperative games exhibit several nice properties, e.g. regarding the existence and computation of allocations realizing some of the most important solution concepts proposed in the literature. For this reason, a relevant question is whether one can give a polynomial time characterization of submodular instances, for prominent cooperative games that are in general non-convex.

Highlights

Cooperative games are among the most studied classes of problems in game theory, with plenty of applications in economics, mathematics, and computer science
This paper focuses on one of the most fundamental cooperative games, namely the spanning tree game
An important step forward was made by Kobayashi and Okamoto [9], who gave a characterization of submodularity for instances of the spanning tree game where the edge weights are restricted to take only two values

Summary

Introduction

Cooperative games are among the most studied classes of problems in game theory, with plenty of applications in economics, mathematics, and computer science. An important step forward was made by Kobayashi and Okamoto [9], who gave a characterization of submodularity for instances of the spanning tree game where the edge weights are restricted to take only two values. Whether a polynomial-time characterization of submodularity exists for spanning tree games is left as an open question They stated twice in their paper: “We feel that recognizing a submodular minimum-cost spanning tree game is coNPcomplete, but we are still far from proving such a result.”. We can efficiently test the submodularity of our instance by checking whether the inequality (∗) is satisfied on this family of subsets of vertices Combining these two ingredients yields a polynomial-time characterization of submodularity for spanning tree games, as described in Sect.

Preliminaries and notation

Violated cycles

Candidate edges and expensive neighborhood

Characterization of submodularity

S-wide spanning trees