We present three results on the complexity of MINIMAX APPROVAL VOTING. First, we study MINIMAX APPROVAL VOTING parameterized by the Hamming distance d from the solution to the votes. We show MINIMAX APPROVAL VOTING admits no algorithm running in time O ⋆ (2 o ( d log d ) , unless the Exponential Time Hypothesis (ETH) fails. This means that the O ⋆ ( d 2 d ) algorithm of Misra et al. (AAMAS 2015) is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time O ⋆ ((3/ε) 2 d ), which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for MINIMAX APPROVAL VOTING, which runs in time n O(1/ε2·log(1/ε)) · poly( m ), almost matching the running time of the fastest known PTAS for CLOSEST STRING due to Ma and Sun (SIAM J. Comp. 2009).

For some two-player games (e.g. Go), no accurate and inexpensive heuristic is known for evaluating leaves of a search tree. For other games (e.g. chess), a heuristic is known (sum of piece values). For other games (e.g. Hex), only a local heuristic — one that compares children reliably, but non-siblings poorly — is known (cell voltage drop in the Shannon/Anshelevich electric circuit model). In this paper we introduce a search algorithm for a two-player perfect information game with a reasonable local heuristic. Sibling Conspiracy Number Search (SCNS) is an anytime best-first version of Conspiracy Number Search based not on evaluation of leaf states of the search tree, but — for each node — on relative evaluation scores of all children of that node. SCNS refines CNS search value intervals, converging to Proof Number Search. SCNS is a good framework for a game player. We tested SCNS in the domain of Hex, with promising results. We implemented an 11-by-11 SCNS Hex bot, DeepHex. We competed DeepHex against current Hex bot champion MoHex, a Monte-Carlo Tree Search player, and previous Hex bot champion Wolve, an Alpha-Beta Search player. DeepHex widely outperforms Wolve at all time levels, and narrowly outperforms MoHex once time reaches 4min/move.