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Stabilization of Capacitated Matching Games

Part of the Lecture Notes in Computer Science book series (LNCS,volume 13904)


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.


  • Matching
  • Game theory
  • Network bargaining

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  1. 1.

    It is stated in [8] (Theorem 2.3.9) that a stable allocation for capacitated CMG exists iff G is stable, but our example shows this statement is not correct.

  2. 2.

    It was stated in [9, corollary 1] that \(M\) is maximum if and only if \(M'\) is maximum, but this example shows this to be false.

  3. 3.

    In fact, this is the way the M-vertex-stabilizer problem is defined in [15]. We instead use the original definition in [1, 6] which assumes M to be maximum.

  4. 4.

    [2] assumes that the graph is bipartite, but bipartiteness is not needed in their proof.


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The second and third authors are supported by the NWO VIDI grant VI.Vidi.193.087. The second author thanks the 2021 Hausdorff Research Institute for Mathematics Program Discrete Optimization, during which part of this work was developed.

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Correspondence to Lucy Verberk .

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Gerstbrein, M., Sanità, L., Verberk, L. (2023). Stabilization of Capacitated Matching Games. In: Del Pia, A., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2023. Lecture Notes in Computer Science, vol 13904. Springer, Cham.

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