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On the Core of the $b$-Matching Game

Rohith Reddy Gangam, Shayan Taherijam, Vijay V. Vazirani

TL;DR

The paper proves that testing core membership for the edge-unconstrained $b$-matching game is co-NP-hard, answering a long-standing open question in this domain. It delivers a complete core characterization and a polynomial-time Core test for star-graph instances, while establishing NP-hardness results for unstable coalitions and fair core imputations. The hardness results are extended from star graphs to general bipartite graphs via careful reductions, showing a sharp delineation of computational boundaries for core-related questions in matching games. These findings clarify the feasibility landscape for core-based profit sharing and fair imputation computations in graph-structured cooperative games.

Abstract

The core is a quintessential solution concept for profit sharing in cooperative game theory. An imputation allocates the worth of the given game among its agents. The imputation lies in the core of the game if, for each sub-coalition, the amount allocated to its agents is at least the worth of this sub-coalition. Hence, under a core imputation, each of exponentially many sub-coalitions gets satisfied. The following computational question has received much attention: Given an imputation, does it lie in the core? Clearly, this question lies in co-NP, since a co-NP certificate for this problem would be a sub-coalition which is not satisfied under the imputation. This question is in P for the assignment game [SS71] and has been shown to be co-NP-hard for several natural games, including max-flow [FZCD02] and MST [FKFH97]. The one natural game for which this question has remained open is the b-matching game when the number of times an edge can be matched is unconstrained; in case each edge can be matched at most once, it is co-NP-hard [BKPW18]. At the outset, it was not clear which way this open question would resolve: on the one hand, for all but one game, this problem was shown co-NP-hard and on the other hand, proximity to the assignment problem and the deep structural properties of matching could lead to a positive result. In this paper, we show that the problem is indeed co-NP-hard.

On the Core of the $b$-Matching Game

TL;DR

The paper proves that testing core membership for the edge-unconstrained -matching game is co-NP-hard, answering a long-standing open question in this domain. It delivers a complete core characterization and a polynomial-time Core test for star-graph instances, while establishing NP-hardness results for unstable coalitions and fair core imputations. The hardness results are extended from star graphs to general bipartite graphs via careful reductions, showing a sharp delineation of computational boundaries for core-related questions in matching games. These findings clarify the feasibility landscape for core-based profit sharing and fair imputation computations in graph-structured cooperative games.

Abstract

The core is a quintessential solution concept for profit sharing in cooperative game theory. An imputation allocates the worth of the given game among its agents. The imputation lies in the core of the game if, for each sub-coalition, the amount allocated to its agents is at least the worth of this sub-coalition. Hence, under a core imputation, each of exponentially many sub-coalitions gets satisfied. The following computational question has received much attention: Given an imputation, does it lie in the core? Clearly, this question lies in co-NP, since a co-NP certificate for this problem would be a sub-coalition which is not satisfied under the imputation. This question is in P for the assignment game [SS71] and has been shown to be co-NP-hard for several natural games, including max-flow [FZCD02] and MST [FKFH97]. The one natural game for which this question has remained open is the b-matching game when the number of times an edge can be matched is unconstrained; in case each edge can be matched at most once, it is co-NP-hard [BKPW18]. At the outset, it was not clear which way this open question would resolve: on the one hand, for all but one game, this problem was shown co-NP-hard and on the other hand, proximity to the assignment problem and the deep structural properties of matching could lead to a positive result. In this paper, we show that the problem is indeed co-NP-hard.

Paper Structure

This paper contains 9 sections, 14 theorems, 26 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. Star Graphs
  5. Efficient Characterization of the Core
  6. Unstable Coalitions under General Profit Shares
  7. General Bipartite Graphs
  8. Core imputations
  9. Fair core imputations

Key Result

Theorem 1

The core of a $b$-matching game on star graphs is completely characterizable.

Figures (5)

  • Figure 1: A star graph $G=(U,V,E)$.
  • Figure 2: Graphs used in proof of \ref{['cl:marginal_utility']}
  • Figure 3: Graph $G^*$ used in proof of \ref{['thm:profit_share_for_star']}
  • Figure 4: (Left) $G^*;\quad\quad$(Right) Add blue vertices and edges to $G^*$ to get $G$
  • Figure 5: (Left) $G;\quad$(Right) We add partner vertices, shown in red, for every vertex in $G$ to get $G'$

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 1
  • proof
  • Lemma 1
  • ...and 33 more