EXACT COMPLEXITY OF EXACT-FOUR-COLORABILITY AND OF THE WINNER PROBLEM FOR YOUNG ELECTIONS

摘要

We classify two problems: Exact-Four-Colorability and the winner problem for Young elections. Regarding the former problem, Wagner raised the question of whether it is DP-complete to determine if the chromatic number of a given graph is exactly four. We prove a general result that in particular solves Wagner's question in the affirmative. In 1977, Young proposed a voting scheme that extends the Condorcet Principle based on the fewest possible number of voters whose removal yields a Condorcet winner. We prove that both the winner and the ranking problem for Young elections is complete for P_‖~(NP), the class of problems solvable in polynomial time by parallel access to NP. Analogous results for Lewis Carroll's 1876 voting scheme were recently established by Hemaspaandra et al. In contrast, we prove that the winner and ranking problems in Fishburn's homogeneous variant of Carroll's voting scheme can be solved efficiently by linear programming.