极大极小值的可逼近性和参数化复杂性
Approximability and parameterized complexity of minmax values
摘要：
我们考虑以策略形式近似多人博弈的极大极小值。博格斯等人收紧了最近的界限，我们观察到，以10 logn位数的精度来近似该值（对于任何常数n>0）是NP难的，其中n是游戏的大小。另一方面，以c log log n位数的精度近似该值（对于任何常数c >= 1）可以在准多项式时间内完成。我们考虑问题的参数化复杂性，参数是计算最小最大值的玩家的纯策略k的数量。我们表明，如果有三个球员，k=2，只有两个可能的合理的回报，最小最大值是一个有理数，可以精确地计算在线性时间。在一般情况下，我们表明，该值可以近似与任何多项式的数字的精度在时间n^（O（k））。另一方面，我们证明了极大极小值近似是W[1]-困难的，因此不太可能是固定参数的。具体地说，我们证明了如果k-CLIQUE需要时间n^（Omega（k）），那么minmax值计算也需要时间。
英文摘要：
We consider approximating the minmax value of a multi-player game in strategic form. Tightening recent bounds by Borgs et al., we observe that approximating the value with a precision of epsilon log n digits (for any constant epsilon>0 is NP-hard, where n is the size of the game. On the other hand, approximating the value with a precision of c log log n digits (for any constant c >= 1) can be done in quasi-polynomial time. We consider the parameterized complexity of the problem, with the parameter being the number of pure strategies k of the player for which the minmax value is computed. We show that if there are three players, k=2 and there are only two possible rational payoffs, the minmax value is a rational number and can be computed exactly in linear time. In the general case, we show that the value can be approximated with any polynomial number of digits of accuracy in time n^(O(k)). On the other hand, we show that minmax value approximation is W[1]-hard and hence not likely to be fixed parameter tractable. Concretely, we show that if k-CLIQUE requires time n^(Omega(k)) then so does minmax value computation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
