Spanning-tree games

We introduce and study a game variant of the classical spanning-tree problem. Our spanning-tree game is played between two players, min and max, who alternate turns in jointly constructing a spanning tree of a given connected weighted graph G. Starting with the empty graph, in each turn a player chooses an edge that does not close a cycle in the forest that has been generated so far and adds it to that forest. The game ends when the chosen edges form a spanning tree in G. The goal of min is to minimize the weight of the resulting spanning tree and the goal of max is to maximize it. A strategy for a player is a function that maps each forest in G to an edge that is not yet in the forest and does not close a cycle. We show that while in the classical setting a greedy approach is optimal, the game setting is more complicated: greedy strategies, namely ones that choose in each turn the lightest (min) or heaviest (max) legal edge, are not necessarily optimal, and calculating their values is NP-hard. We study the approximation ratio of greedy strategies. We show that while a greedy strategy for min guarantees nothing, the performance of a greedy strategy for max is satisfactory: it guarantees that the weight of the generated spanning tree is at least ^w(MST(G⁾⁾, where w(MST(G)) 2 is the weight of a maximum spanning tree in G, and its approximation ratio with respect to an optimal strategy for max is 1.5+ _w(_MST ¹ (G)), assuming weights in [0,1]. We also show that these bounds are tight. Moreover, in a stochastic setting, where weights for the complete graph K_n are chosen at random from [0,1], the expected performance of greedy strategies is asymptotically optimal. Finally, we study some variants of the game and study an extension of our results to games on general matroids.