On Unique Games with Negative Weights

In this paper, the author defines Generalized Unique Game Problem (GUGP), where weights of the edges are allowed to be negative. Focuses are made on two special types of GUGP, GUGP-NWA, where the weights of all edges are negative, and GUGP-PWT(ρ), where the total weight of all edges are positive and the negative/positive ratio is at most ρ. The author investigates the counterparts of the Unique Game Conjecture for the two types of generalized unique game problems. The author proves Unique Game Conjecture holds false on GUGP-NWA by giving a factor 2 approximation algorithm for Max GUGPNWA, and Unique Game Conjecture holds true on GUGP-PWT(1) by reducing the parallel repetition of Max 3-Cut Problem to GUGP-PWT(1). The author poses an open problem whether Unique Game Conjecture holds true on GUGP-PWT(ρ) with 0 < ρ < 1.