john@cpss11.cps.msu.edu (Hyun Duk John) (03/25/91)
Dear Sirs : Have you ever seen the problem stated below ? If so, please give me the information. Your help would be much appreciated. Mar. 24, 1991 >>> Hyun-Duk John <<< **************** START of the Problem ************************** << Problem : Traveling Stars >> A network routing problem can be simplified and/or modified to more generalized problems. Here, one of the generalized version of the network routing problem named ``Traveling Stars'' is described. The famous problem ``8-puzzle'' can be viewed as an instance of the traveling stars problem. The problem can also be viewed as a permutation problem. Only decision problems are described here, but algorithms are of interest too. == Problem -. Given an undirected graph G = (V,E) and m stars, where m <= |V|. -. A star occupies a vertex and can be moved to an adjacent vertex one by one. (Only one star can be moved at a time) -. No two stars can share a vertex at any moment. -. Initially, each star i is located in its starting vertex s(i), and is destined to its terminal vertex t(i). -. Questions : 1. Can all the stars be moved from their starting vertices to terminal vertices ? 2. Is there a way of moving stars from their starting vertices to terminal vertices in less than or equal to k steps ? (A movement of a single star from a vertex to its adjacent vertex is called a step.) << A Formal Expression of the Problem >> == For more formal approach, we define some terminologies. -. A `state' is a sequence of vertices (v-1,v-2,...,v-m) such that for all i <> j, v-i <> v-j. -. A `transition' is an ordered pair of states (W,X) where W = (w-1,w-2,...,w-m) and X = (x-1,x-2,...,x-m), in which there exists one and only one i, such that for all j <> i, w-j = x-j and (w-i,x-i) is in E. -. A `move' of size k is a sequence of states (W-0,W-1,W-2,...,W-k) where every pair (W-i,W-(i+1)) is a transition. We denote the move as [W-0,W-k]. == Instance : -. Undirected graph G = (V,E). -. Two states S and T of size m. -. An integer k. == Question : -. Is there a move [S,T] ? -. Is there a move [S,T] of size <= k ? *********************** END ********************************