Abstract
In this paper, the authors define 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 authors investigate the counterpart of the Unique Game Conjecture on GUGP-PWT(ρ). The authors prove Unique Game Conjecture holds true on GUGP-PWT(1) by reducing the parallel repetition of Max 3-Cut Problem to GUGP-PWT(1), and Unique Game Conjecture holds true on GUGP-PWT(1/2) if the 2-to-1 Conjecture holds true. The authors pose an open problem whether Unique Game Conjecture holds true on GUGP-PWT(ρ) with 0 < ρ < 1.
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Cui, P., Liu, T., Xu, K. (2011). On Unique Games with Negative Weights. In: Wang, W., Zhu, X., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2011. Lecture Notes in Computer Science, vol 6831. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22616-8_37
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DOI: https://doi.org/10.1007/978-3-642-22616-8_37
Publisher Name: Springer, Berlin, Heidelberg
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