Abstract
An edge-weighted graph [Formula: see text] is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in network bargaining games and cooperative matching games, because they characterize instances that admit stable outcomes. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from G yields a stable graph, for any weighted graph G. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph G yields a stable graph, does not admit any constant-factor approximation algorithm, unless P = NP. In this setting, we develop an O(Δ)-approximation algorithm for the problem, where Δ is the maximum degree of a node in G.
Highlights
Several interesting game theory problems are defined on networks, where the vertices represent players and the edges model the way players can interact with each other
Popular examples are cooperative matching games, introduced by Shapley and Shubik [17], and network bargaining games, defined by Kleinberg and Tardos [13], both extensively studied in the game theory community
Instances of such games are described by a graph G = (V, E) with edge weights w ∈ RE≥0, where V represents a set of players, and the value of a maximum-weight matching, denoted as ν(G), is the total value that the players could get by interacting with each other
Summary
Several interesting game theory problems are defined on networks, where the vertices represent players and the edges model the way players can interact with each other. Ahmadian et al [1] and Ito et al [10] have shown independently that finding a minimum-cardinality vertex-stabilizer is a polynomial-time solvable problem These (exact and approximate) algorithmic results, developed for unweighted instances, do not generalize when dealing with arbitrary edge-weights, since they heavily rely on the structure of maximum matchings in unweighted graphs. Biró et al [5] and Könemann et al [14] studied a variant of the problem where the goal is to compute a minimum-cardinality set of blocking pairs, that are edges whose removal from the graph yield the existence of a fractional vertex cover of size at most ν(G) (but note that the resulting graph might not be stable).
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