Tuesday, May 02, 2006

Game theoretic characterization of treewidth

There is no doubt that the notion of treewidth introduced by Robertson and Seymour is one of the most important concepts in math and computer science introduced in the past thirty years. Roughly speaking, treewidth of a graph G measures how much G looks like a tree. For example, the treewidths of a tree, of a cycle, and of a clique (with n vertices) are respectively 1, 2, and n.

Some of the readers might not be aware of an intuitive way of understanding the concept of treewidths that is provided by the cops-and-robber games, as pointed out by Seymour and Thomas in their paper "Graph searching and a min-max theorem for treewidth", Journal of Combinatorial Theory B, Vol. 58 (1993). The following presentation follows the second paper cited at the bottom. The k-cops-and-robber games, where k > 1, takes a graph G = (V,E) as a parameter. Each game has two players: the cops and the robber. The game goes as follows:

  1. In round 0, the cops choose a subset X0 of V of size at most k. The robber chooses a vertex v0 of V - X0.
  2. In round i+1, the cops choose a subset Xi+1 of V of size at most k. If possible, the robber chooses a vertex vi+1 of V - Xi+1 such that there is a (possibly empty) path from vi to vi+1 which does not pass through Xi ∩ Xi+1. Otherwise, the cops win the game right there.

If this game has no end (i.e. the robber can evade capture), then the robber wins. The cops are said to have a winning strategy on the k-cops-and-robber game on G if they can play in such a way that they will eventually win the game. Otherwise, we say that the robber has a winning strategy on this game.

It is helpful to think of the set Xi+1 - Xi as set of vertices above which the cops are hovering in helicopters in round i+1. [This tells us that cops can move from vertices to vertices unrestrictedly.] Also, the set Xi+1 ∩ Xi can be thought of as the vertices where the cops have landed from helicopters. So, it takes two moves for a cop to be at a particular vertex v (i.e. on the ground): she must first ride a helicopter to v, and second land the helicopter at v. On the other hand, a cop may take off to another vertex (but still is in a helicopter) in just one move. In addition, the robber can see the cops in the helicopter, and may elude capture by running at great speed to another vertex as long as he does not run through a cop on the ground during his escape.

Theorem (Seymour and Thomas): A graph G has treewidth ≤ k-1 iff the cops have a winning strategy in the k-cops-and-robber game on G.

Equivalently, a graph G has treewidth k-1 iff k is the smallest number k' such that the cops have a winning strategy in the k'-cops-and-robber game on G. Let's try this game on a tree. The cops can win the game using 2 cops only. The idea is to gradually shrink the set of vertices the robber can go to in the next round by alternately moving the two cops in a "depth-first traversal" manner. On the other hand, it is clear that the robber player has a winning strategy on a tree if he is chased by only a poor lone cop.

In recent years, some researchers in databases and verification have proposed variants of the cops-and-robber games so as to explain some phenomena in those fields. Let me conclude by mentioning two such papers:

  1. Gottlob, Leone, and Scarcello. Robbers, marshals, and guards: game theoretic and logical characterization of hypertree width, JCSS 66, 2003.
  2. Berwanger, Dawar, Hunter, and Kreutzer. DAG-width and Parity Games, STACS 2006.