Some friends of yours have grown tired of the game "Six Degrees of Kevin Bacon" (after all, they ask, isn't it just breadth-first search?) and decide to invent a game with a little more punch, algorithmically speaking. Here's how it works.
You start with a set X of n actresses and a set Y of n actors, and two players P0 and P1. Player P0 names an actress x1 ε X, player P1 names an actor y1 who has appeared in a movie with x1, player P0 names an actress x2 who has appeared in a movie with y1, and so on. Thus, P0 and P1 collectively generate a sequence x1, y1, x2, y2,... such that each actor/actress in the sequence has costarred with the actress/actor immediately preceding. A player Pi (i = 0, 1) loses when it is Pi's turn to move, and he/she cannot name a member of his/her set who hasn't been named before.
Suppose you are given a specific pair of such sets X and Y , with complete information on who has appeared in a movie with whom. A strategy for Pi, in our setting, is an algorithm that takes a current sequence x1, y1, x2, y2, … and generates a legal next move for Pt (assuming it's Pi's turn to move). Give a polynomial-time algorithm that decides which of the two players can force a win, in a particular instance of this game.
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