Motion Planning on a Graph

In this paper, all graphs are simple, i.e., without self-loops or parallel edges, unless otherwise stated. A weighted graph G is a triple (V (G), E(G), costG), where V (G) is the vertex set of G, E(G) is the edge set of G, and costG is a function from E(G) to nonnegative reals, called the cost function of G. For each edge e of G, we call costG(e) the cost of e. Let G be a (weighted) graph. For U ⊆ V (G), G \ U will denote the subgraph of G induced by V (G) \ U . When G is weighted, we assume that the subgraph inherits the cost function of G appropriately restricted. A walk in G is a sequence of vertices of G such that each pair of vertices consecutively appearing in the sequence are adjacent in G. If a walk starts from u and ends with v, we call u the initial vertex and v the final vertex of the walk. We say that a walk visits a vertex v if v appears in the sequence. A walk is a path if no vertex is repeated in the sequence. Although a walk is primarily a sequence of vertices, we can also view it as a sequence of (oriented versions of) edges. In particular, the length of walk W , denoted by |W |, is defined to be the number of edges in the walk. For each walk W of G, we extend the cost notation so that costG(W ) denotes the cost of W , i.e., the sum of the costs of all the edges in W . We start by formalizing the rules of our game. Let G be a weighted graph. A configuration γ on G is a a pair (Oγ , rγ), where Oγ is a subset of V (G) and rγ is a vertex in V (G) \Oγ . Informally, we interpret configuration γ by regarding each vertex in Oγ to have an obstacle, each vertex in V (G) \Oγ to have a hole, and vertex rγ to have the robot. Note that the definition of a configuration requires that the robot to coincide with a hole. We use the convention to denote V (G) \Oγ , the set of vertices with holes, by Hγ . ∗Department of Computer Science and Engineering, UCSD, La Jolla, CA 92093. †IBM T.J. Watson Research Center, Yorktown Heights, NY 10598. ‡IBM Tokyo Research Laboratory, Kanagawa 242, Japan. Part of the work done at the IBM T.J. Watson Research Center.