Stabilization of Capacitated Matching Games

TL;DR

This paper investigates the computational complexity of stabilizing capacitated matching games by blocking vertices, revealing polynomial-time solutions under certain restrictions and NP-hardness in the general case, and explores the differences from the unit-capacity scenario.

Contribution

It extends the understanding of the vertex-stabilizer problem from unit-capacity to arbitrary capacities, identifying complexity boundaries and the failure of equivalences present in the unit-capacity case.

Findings

Vertex-stabilizer problem with a restriction remains polynomial-time solvable.

Without the restriction, the problem becomes NP-hard and hard to approximate.

The equivalence between stability in different game models does not hold in the capacitated case.

Abstract

An edge-weighted, vertex-capacitated graph G is called stable if the value of a maximum-weight capacity-matching equals the value of a maximum-weight fractional capacity-matching. Stable graphs play a key role in characterizing the existence of stable solutions for popular combinatorial games that involve the structure of matchings in graphs, such as network bargaining games and cooperative matching games. The vertex-stabilizer problem asks to compute a minimum number of players to block (i.e., vertices of G to remove) in order to ensure stability for such games. The problem has been shown to be solvable in polynomial-time, for unit-capacity graphs. This stays true also if we impose the restriction that the set of players to block must not intersect with a given specified maximum matching of G. In this work, we investigate these algorithmic problems in the more general setting of arbitrary capacities. We show that the vertex-stabilizer problem with the additional restriction of avoiding a given maximum matching remains polynomial-time solvable. Differently, without this restriction, the vertex-stabilizer problem becomes NP-hard and even hard to approximate, in contrast to the unit-capacity case. Finally, in unit-capacity graphs there is an equivalence between the stability of a graph, existence of a stable solution for network bargaining games, and existence of a stable solution for cooperative matching games. We show that this equivalence does not extend to the capacitated case.