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A* is a commonly used technique for finding shortest paths for navigation in video games. We propose partially precomputed A* (PPA*), which is much faster than A* at runtime, and uses much less memory than completely precalculating all shortest paths with an algorithm such as Floyd-Warshall. At runtime, PPA* is very similar to A*, so it is simple and safe to integrate into existing video game code...
We give a subexponential time approximation algorithm for the Unique Games problem. The algorithms run in time that is exponential in an arbitrarily small polynomial of the input size, nε. The approximation guarantee depends on ε, but not on the alphabet size or the number of variables. We also obtain a subexponential algorithms with improved approximations for SMALL-SET EXPANSION and MULTICUT. For...
Prisoner's dilemma (PD) game has been used as a prototypical model for studying social choice situations with self-interested agents. Although in a single shot PD game, both players playing defect is a Nash equilibrium, in social settings, cooperation among self-interested agents is usually observed. This phenomenon of emergence of cooperation can be captured by repeated PD games in graphs consisting...
We present a new algorithm for Unique Games which is based on purely spectral techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our algorithm is able to recover a good assignment. The approximation guarantee depends only on the completeness of the game, and not on the alphabet size, while...
With the work of Khot and Vishnoi as a starting point, we obtain integrality gaps for certain strong SDP relaxations of Unique Games. Specifically, we exhibit a Unique Games gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner products of up to exp(??(log log n)1/4) vectors. For a stronger relaxation obtained from the basic semidefinite program...
We study the inapproximability of Vertex Cover and Independent Set on degree d graphs. We prove that: (1) Vertex Cover is Unique Games-hard to approximate to within a factor 2 - (2 + od(1)) log log d/log d. This exactly matches the algorithmic result of Halperin up to the od(1) term. (2) Independent Set is Unique Games-hard to approximate to within a factor O(d/log2d). This improves the d/logO(1)(d))...
A permutation constraint satisfaction problem (permCSP) of arity k is specified by a subset LambdasubeSk of permutations on {1,2,...,k}. An instance of such a permCSP consists of a set of variables V and a collection of constraints each of which is an ordered k-tuple of V. The objective is to find a global ordering sigma of the variables that maximizes the number of constraint tuples whose ordering...
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