Exact complexity of exact-fourcolorability and of the winner problem for young elections

Abstract

We classify two problems: E:uct-Four-Colorabili ty 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. Keywords: Computational complexity; graph colorability; completeness; Boolean hierarchy; voting schemes.