Computing the Nucleolus of Weighted Cooperative Matching Games in Polynomial Time

Abstract

We provide an efficient algorithm for computing the nucleolus for an instance of a weighted cooperative matching game. This resolves a long-standing open question posed in [Faigle, Kern, Fekete, Hochstättler, Mathematical Programming, 1998].

This work was done in part while the second author was visiting the Simons Institute for the Theory of Computing. Supported by DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF grant #CCF-1740425.

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG).

Appendices

Appendix

A Proof of Theorem 2

Proof

First, we show that $P_1(\varepsilon _1)\subseteq \overline{P}_1(\varepsilon _1)$. Consider $x\in P_1(\varepsilon _1)$. By Lemma 5(i) we have

Lemma 5(ii) shows that for all $e\in E^+$, and holds by the universality of $M^*$. It remains to show that

Suppose for contradiction there exists $e \in E^*$ such that . By the definition of $E^*$, there exists a universal matching $M'$ containing e. Since $M'$ is universal, . But by our choice of e,

contradicting that x is in $P_1(\varepsilon _1)$. Thus we showed that $(x,\varepsilon _1)$ is feasible for ($\overline{P}_1$), i.e. we showed that $P_1(\varepsilon _1)\subseteq \overline{P}_1(\varepsilon _1)$.

To complete the proof we show that $\overline{P}_1(\overline{\varepsilon }_1)\subseteq P_1(\overline{\varepsilon }_1)$. Let x be an allocation in $\overline{P}_1(\overline{\varepsilon }_1)$. Due to the description of the linear program ($P_1$), it is enough to show that for every matching $M\in \mathcal {M}$ we have

Since it suffices to consider only the matchings M, which are unions of matchings on the graphs $G[S_i^*]$, $i \in [k]$. Let $t_i:=|M\cap E(S_i^*)|$. By Lemma 6 applied to $x^*$ there exists $M' \subseteq M^*$ such that and $|M'\cap E(S_i^*)|=t_i$, for all $i\in [k]$. Then due to constraints (2) in ($\overline{P}_1$) we have

where the last inequality follows since $M' \subseteq M^*$ and for all $e \in E^*$.

Thus, we showed that $P_1(\varepsilon _1)\subseteq \overline{P}_1(\varepsilon _1)$ and $\overline{P}_1(\overline{\varepsilon }_1)\subseteq P_1(\overline{\varepsilon }_1)$. Recall, that $\varepsilon _1$ and $\overline{\varepsilon }_1$ are the optimal values of the linear programs ($P_1$) and ($\overline{P}_1$) respectively. Thus, we have $\varepsilon _1=\overline{\varepsilon }_1$ and $P_1(\varepsilon _1)=\overline{P}_1(\overline{\varepsilon }_1)$.    $\blacksquare$

B Example of a Matching Game With Empty Core

Consider the example in Fig. 1. This graph $G=(V,E)$ is a 5-cycle with two adjacent edges 15 and 12 of weight 2, and the remaining three edges of weight 1. Since the maximum weight matching value is $\nu (G) = 3$, but the maximum weight fractional matching value is $\frac{7}{2}$, the core of this game is empty. The allocation $x^*$ defined by $x^*(1) = \frac{7}{5}$ and $x^*(2) = x^*(3) = x^*(4) = x^*(5) = \frac{2}{5}$ lies in the leastcore. Each edge has the same excess, $-\frac{1}{5}$, and any coalition of four vertices yields a minimum excess coalition with excess $-\frac{2}{5}$. Hence the leastcore value of this game is $\varepsilon _1=-\frac{2}{5}$.

In fact, we can see that $x^*$ is the nucleolus of this game. To certify this we can use the result of Schmeidler [38] that the nucleolus lies in the intersection of the leastcore and the prekernel. For this example, the prekernel condition that for all $i\ne j \in V$,

$$\begin{aligned} \max \{ x(S\cup \{i\}) -\nu (S\cup \{i\}) : S\subseteq V\backslash \{j\}\} \\ = \max \{ x(S\cup \{j\}) -\nu (S\cup \{j\}):S\subseteq V\backslash \{i\}\} \end{aligned}$$

reduces to the condition that the excess values of non-adjacent edges are equal. Since G is an odd cycle, this implies that all edges has equal excess, i.e.

Combining the four equations above with the leastcore condition that $x(V) = \nu (G)$ we obtain a system of equations with the unique solution $x^*$. Hence the intersection of the leastcore and prekernel is precisely $\{x^*\}$, and so by Schmeidler, $x^*$ is the nucleolus.

Fig. 1.

Matching game with empty core