Equitable Core Imputations for Max-Flow, MST and $b$-Matching Games

TL;DR

This paper addresses fair allocation in three central combinatorial games—max-flow, MST/min-cost branching, and $b$-matching—through the core concept and its equitable refinements. It shows that while arbitrary core imputations can be unfair, the Owen set provides a tractable conduit to leximin/leximax fairness, with polynomial-time methods to compute Owen-set imputations across all three games. It develops dual-based constructions, Picard-Queyranne structural tools, and LP/ellipsoid/separation-based algorithms to obtain leximin/leximax Owen-set imputations, alongside NP-hardness results for core imputations beyond the Owen set. The work highlights the practical relevance of fairness in automated decision-making and points to efficient, implementable procedures for fair cost/profit sharing in key network-optimization contexts. It also delineates clear boundaries between tractable Owen-set computations and intractable core-imputation problems, guiding future algorithmic development in cooperative game theory for network applications.

Abstract

We study fair allocation of profit (or cost) for three central problems from combinatorial optimization: Max-Flow, MST and $b$-matching. The essentially unequivocal choice of solution concept for this purpose would be the core, because of its highly desirable properties. However, recent work [Vaz24] observed that for the assignment game, an arbitrary core imputation makes no fairness guarantee at the level of individual agents. To rectify this deficiency, special core imputations, called equitable core imputations, were defined - there are two such imputations, leximin and leximax - and efficient algorithms were given for finding them. For all three games, we start by giving examples to show that an arbitrary core imputation can be excessively unfair to certain agents. This led us to seek equitable core imputations for our three games as well. However, the ubiquitous tractable vs intractable schism separates the assignment game from our three games, making our task different from that of [Vaz24]. As is usual in the presence of intractability, we resorted to defining the Owen set for each game and algorithmically relating it to the set of optimal dual solutions of the underlying combinatorial problem. We then give polynomial time algorithms for finding equitable imputations in the Owen set. The motivation for this work is two-fold: the emergence of automated decision-making, with a special emphasis on fairness, and the plethora of industrial applications of our three games.

Figures (16)

• Figure 1: The graph for Example \ref{['ex.flow-no-core']}.
• Figure 2: Algorithm to decide if an imputation is in the Owen set of a max-flow game
• Figure 3: Algorithm to find the leximin Owen set imputation in a max-flow game.
• Figure 4: Illustration of claim \ref{['cl:orient']}
• Figure 5: The green edges have cost 0 and the black edges are unit cost. The blue edge has cost 2.

Theorems & Definitions (85)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Claim 4.1
• proof
• Lemma 4.2
• proof
• Theorem 5.1