Approximate Core Allocations for Edge Cover Games

TL;DR

This paper investigates the approximate core in edge cover cooperative games, establishing a tight 3/4 ratio that can be computed efficiently and relates to the graph's shortest odd cycle.

Contribution

It proves the 3/4-core of edge cover games is always non-empty, computable in polynomial time, and characterizes the ratio in terms of graph properties.

Findings

The 3/4-core ratio is the best possible and equals the LP integrality gap.

The approximate core ratio correlates with the shortest odd cycle length.

The core can be efficiently computed using linear program duality.

Abstract

We study the approximate core for edge cover games, which are cooperative games stemming from edge cover problems. In these games, each player controls a vertex on a network $G = (V, E; w)$, and the cost of a coalition $S\subseteq V$ is equivalent to the minimum weight of edge covers in the subgraph induced by $S$. We prove that the 3/4-core of edge cover games is always non-empty and can be computed in polynomial time by using linear program duality approach. This ratio is the best possible, as it represents the integrality gap of the natural LP for edge cover problems. Moreover, our analysis reveals that the ratio of approximate core corresponds with the length of the shortest odd cycle of underlying graphs.