An Efficient Characterization of Submodular Spanning Tree Games

Abstract

Cooperative games are an important class of problems in game theory, where the goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns a value share to each player. A crucial aspect of such games is submodularity (or convexity). Indeed, convex instances of cooperative games exhibit several nice properties, e.g. regarding the existence and computation of allocations realizing some of the most important solution concepts proposed in the literature. For this reason, a relevant question is whether one can give a polynomial time characterization of submodular instances, for prominent cooperative games that are in general non-convex.

In this paper, we focus on a fundamental and widely studied cooperative game, namely the spanning tree game. An efficient recognition of submodular instances of this game was not known so far, and explicitly mentioned as an open question in the literature. We here settle this open problem by giving a polynomial time characterization of submodular spanning tree games.

This work was supported by the NSERC Discovery Grant Program and an Early Researcher Award by the Province of Ontario.

Z. K. Koh—This work was done while the author was at the University of Waterloo.

Appendix

1.1 Proof of Lemma 2(a)

We will prove the contrapositive. Let C be a bad hole in $G_i$ for some $i<k$. Consider the following cases:

Case 1: C contains r. Let u, v be the vertices adjacent to r in C. Let P be the path obtained by deleting r, u, v from C. Let $u',v'$ be the endpoints of P where $uu',vv'\in E(C)$. Note that $u'=v'$ if P is a singleton. Let $S=V(P)\cup r$. Then,

$$\begin{aligned} \textsf {mst}(S)\ge & {} w(P) + w_{i+1} \\ \textsf {mst}(S\cup u)= & {} w(P) + w(ru) + w(uu') \\ \textsf {mst}(S\cup v)= & {} w(P) + w(rv) + w(vv') \\ \textsf {mst}(S\cup \left\{ u,v\right\} )\ge & {} w(P) + w(ru) + w(uu') + w(rv) + w(vv') - w_i \end{aligned}$$

which yields: $\textsf {mst}(S\cup u) + \textsf {mst}(S\cup v) - \textsf {mst}(S) - \textsf {mst}(S\cup \left\{ u,v\right\} ) \le w_i - w_{i+1} < 0.$

Case 2: Cdoes not contain r. Let $s=\arg \min _{x\in V(C)}w(rx)$. Let u, v be the vertices adjacent to s in C. Let P be the path obtained by deleting s, u, v from C. Let $u',v'$ be the endpoints of P where $uu',vv'\in E(C)$. Let $S=V(P)\cup \left\{ r,s\right\}$. Observe that if r is adjacent to two non-adjacent vertices of C, then we are done because there is a bad hole containing r. We are left with the following subcases:

Subcase 2.1: r is adjacent to at most one vertex of C. We have

$$\begin{aligned} \textsf {mst}(S)\ge & {} w(P) + w(rs) + w_{i+1} \\ \textsf {mst}(S\cup u)= & {} w(P) + w(rs) + w(su) + w(uu') \\ \textsf {mst}(S\cup v)= & {} w(P) + w(rs) + w(sv) + w(vv') \\ \textsf {mst}(S\cup \left\{ u,v\right\} )\ge & {} w(P) + w(rs) + w(su) + w(uu') + w(sv) + w(vv') - w_i \end{aligned}$$

which yields: $\textsf {mst}(S\cup u) + \textsf {mst}(S\cup v) - \textsf {mst}(S) - \textsf {mst}(S\cup \left\{ u,v\right\} ) \le w_i - w_{i+1} < 0.$

Subcase 2.2: r is adjacent to two vertices of C. Suppose $rs,ru\in E_i$. Then,

$$\begin{aligned} \textsf {mst}(S)\ge & {} w(P) + w(rs) + w_{i+1} \\ \textsf {mst}(S\cup u)= & {} w(P) + w(rs) + \min \left\{ w(ru),w(su)\right\} + w(uu') \\ \textsf {mst}(S\cup v)= & {} w(P) + w(rs) + w(sv) + w(vv') \\ \textsf {mst}(S\cup \left\{ u,v\right\} )\ge & {} w(P) + w(rs) + \min \left\{ w(ru),w(su)\right\} + w(uu') + w(sv) + w(vv') - w_i \end{aligned}$$

which yields:

   $\square$

1.2 Proof of Lemma 2(b)

We will prove the contrapositive. Let D be a bad induced diamond in $G_i$ for some $i<k$. Consider the following cases:

Case 1: D contains r. Note that r is a tip of D. Let s be the other tip and u, v be the non-tip vertices of D. Let $S=\left\{ r,s\right\}$. Then,

$$\begin{aligned} \textsf {mst}(S)\ge & {} w_{i+1} \\ \textsf {mst}(S\cup u)= & {} w(ru) + w(su) \\ \textsf {mst}(S\cup v)= & {} w(rv) + w(sv) \\ \textsf {mst}(S\cup \left\{ u,v\right\} )\ge & {} w(ru) + w(su) + w(rv) + w(sv) - w_i \end{aligned}$$

which yields: $\textsf {mst}(S\cup u) + \textsf {mst}(S\cup v) - \textsf {mst}(S) - \textsf {mst}(S\cup \left\{ u,v\right\} ) \le w_i - w_{i+1} < 0.$

Case 2: D does not contain r. Let s, t be the tips of D where $w(rs)\le w(rt)$. Note that $rt\notin E_i$. Let u, v be the non-tip vertices of D where $w(ru)\le w(rv)$. Let $S=\left\{ r,s,t\right\}$. Consider the following subcases:

Subcase 2.1: r is adjacent to at most one vertex of D. Note that $rv\notin E_i$. So,

$$\begin{aligned} \textsf {mst}(S)\ge & {} w(rs) + w_{i+1} \\ \textsf {mst}(S\cup u)= & {} \min \left\{ w(rs),w(ru)\right\} + w(su) + w(tu) \\ \textsf {mst}(S\cup v)= & {} \min \left\{ w(rs), w(rv)\right\} + w(sv) + w(tv) \\ \textsf {mst}(S\cup \left\{ u,v\right\} )\ge & {} \min \left\{ w(rs),w(ru)\right\} + w(su) + w(tu) + w(sv) + w(tv) - w_i \end{aligned}$$

which yields: $\textsf {mst}(S\cup u) + \textsf {mst}(S\cup v) - \textsf {mst}(S) - \textsf {mst}(S\cup \left\{ u,v\right\} ) \le w_i - w_{i+1} < 0.$

Subcase 2.2: r is adjacent to two vertices of D. Observe that if $ru,rv\in E_i$, then we are done because there is a bad induced diamond containing r. So, let $rv\notin E_i$. This implies $rs,ru\in E_i$. We may also assume $w(su)<\max \left\{ w(rs),w(ru)\right\}$. Otherwise, $\left\{ rs,ru,su,sv,uv\right\}$ is a bad induced diamond containing r. Then,

$$\begin{aligned} \textsf {mst}(S)\ge & {} w(rs) + w_{i+1} \\ \textsf {mst}(S\cup u)= & {} \min \left\{ w(rs),w(ru)\right\} + w(su) + w(tu) \\ \textsf {mst}(S\cup v)= & {} w(rs) + w(sv) + w(tv) \\ \textsf {mst}(S\cup \left\{ u,v\right\} )\ge & {} \min \left\{ w(rs),w(ru)\right\} + w(su) + w(tu) + w(sv) + w(tv) - w_i \end{aligned}$$

which yields: $\textsf {mst}(S\cup u) + \textsf {mst}(S\cup v) - \textsf {mst}(S) - \textsf {mst}(S\cup \left\{ u,v\right\} ) \le w_i - w_{i+1} < 0.$

Subcase 2.3: r is adjacent to three vertices of D. Let $w(rv)=w_j$ for some $j\le i$. Consider the induced diamond $\left\{ ru,rv,tu,tv,uv\right\}$. If it is well-covered, then we are done because it is bad and contains r. So let $\max \left\{ w(su),w(sv)\right\} \le w(uv) < w(rv)$. We may also assume $w(su)<\max \left\{ w(rs),w(ru)\right\}$. Otherwise, $\left\{ rs,ru,su,sv,uv\right\}$ is a bad induced diamond in $G_{j-1}$ which contains r. Then,

$$\begin{aligned} \textsf {mst}(S)\ge & {} w(rs) + w_{i+1} \\ \textsf {mst}(S\cup u)= & {} \min \left\{ w(rs),w(ru)\right\} + w(su) + w(tu) \\ \textsf {mst}(S\cup v)= & {} \min \left\{ w(rs),w(rv)\right\} + w(sv) + w(tv) \\ \textsf {mst}(S\cup \left\{ u,v\right\} )\ge & {} \min \left\{ w(rs),w(ru)\right\} + w(su) + w(tu) + w(sv) + w(tv) - w_i \end{aligned}$$

which yields:

   $\square$