On Unique Games with Negative Weights

TL;DR

This paper introduces a generalized version of the Unique Game problem allowing negative weights, explores its special cases, and investigates the validity of the Unique Game Conjecture within this framework, establishing some conditions under which it holds.

Contribution

The paper defines GUGP with negative weights, analyzes its special cases, and relates the conjecture's validity to existing conjectures, expanding understanding of unique games with negative weights.

Findings

Unique Game Conjecture holds for GUGP-PWT(1)

Unique Game Conjecture holds for GUGP-PWT(1/2) if 2-to-1 Conjecture is true

Open problem on conjecture validity for 0<ρ<1

Abstract

In this paper, the author defines Generalized Unique Game Problem (GUGP), where weights of the edges are allowed to be negative. Two special types of GUGP are illuminated, GUGP-NWA, where the weights of all edges are negative, and GUGP-PWT($\rho$), where the total weight of all edges are positive and the negative-positive ratio is at most $\rho$. The author investigates the counterpart of the Unique Game Conjecture on GUGP-PWT($\rho$). The author shows that Unique Game Conjecture on GUGP-PWT(1) holds true, and Unique Game Conjecture on GUGP-PWT(1/2) holds true, if the 2-to-1 Conjecture holds true. The author poses an open problem whether Unique Game Conjecture holds true on GUGP-PWT($\rho$) with $0<\rho<1$.