On Unique Games with Negative Weights

arXiv:1102.5605v4 [cs.CC] 14 Mar 2013 On Unique Games with Negative Weights Peng Cui1 Key Laboratory of Data Engineering and Knowledge Engineering, MOE, School of Information Resource Management, Renmin University of China, Beijing 100872, P. R. China. cuipeng@ruc.edu.cn 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(ρ), where the total weight of all edges are positive and the negative-positive ratio is at most ρ. The author investigates the counterpart of the Unique Game Conjecture on GUGP-PWT(ρ). 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(ρ) with 0 < ρ < 1. 1 Introduction The Unique Game Conjecture (UGC) is put forward by Khot on STOC 2002 as a pow- erful tool to prove lower bound of inapproximabilty for combinatorial optimization problems[6]. It has been shown by researchers a positive resolution of this conjecture would imply improved even best possible hardness results for many famous problems, to name a few, Max Cut, Vertex Cover, Multicut, Min 2CNF Deletion, making an im- portant challenge to prove or refute the conjecture. Some variations of UGC have been mentioned. Rao proves a strong parallel repe- tition theorem which shows Weak Unique Game Conjecture is equivalent to UGC[10]. Khot et al. show that Unique Game Conjecture on Max 2LIN(q) is equivalent to UGC[7]. Khot poses the d-to-1 Conjectures in his original paper for d ≥2[6]. O’Donell et al. show a tight hardness for approximating satisfiable constraint satisfaction problem on 3 Boolean variables assuming the d-to-1 Conjecture for any fixed d[8]. Dinur et al. use the 2-to-2 Conjecture to derive the hardness results of Approximate Coloring Problem, and prove that the 2-to-1 Conjecture implies their 2-to-2 Conjecture[2]. Guruswami et al. use the 2-to-1 Conjecture to derive the hardness result of Maximum k-Colorable Sub- graph Problem[4]. It is unknown whether UGC implies any of the d-to-1 Conjectures, or vice versa. Recently, the authors of [1] designed a subexponential time algorithm for Unique Games Problem (UGP), which challenges the position of UGC as a tool to prove lower bound of inapproximabilty. While their results stop short of refuting UGC, they do suggest that UGP is significantly easier than NP-hard problems. On the other side, the authors of [9] determined a new point of (c, s)-approximation NP-hardness of UGP, compared to [5], and their new result, together with the result of [3], determines the two-dimensional region of all known (c, s)-approximation NP-hardness of UGP. 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 illumi- nated, 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. GUGP-PWT(ρ) over 1 ≥ρ ≥0 makes a possible phase transition from a 2-Prover 1-Round Game Problem with (1−ζ, δ)-approximation NP-hardness to UGP. The author shows that UGC on GUGP-PWT(1) holds true, UGC on GUGP-PWT(1/2) holds true, if the 2-to-1 Conjecture holds true, and the (1 −ζ, δ)-approximation NP-hardness of GUGP-PWT(ρ) possesses the compactness property when ρ →0. Section 2 demonstrates some definitions. The author shows the main results for GUGP-NWA and GUGP-PWT(ρ) in Section 3. Section 4 is some discussions. 2 Preliminaries In 2-Prover 1-Round Game Problem (2P1R), we are given a bipartite graphG = (V, W; E), with each edge e having a weight we ∈Q+. We are also given two sets of labels, k1 and k2, which we identify with [k1] = {1, · · · , k1} and [k2] = {1, · · · , k2}. Each edge e = (u, v) in the graph is equipped with a relation Re ⊆[k1] × [k2]. The solution of the problem is a labeling f1 : V →[k1] and f2 : W →[k2] which assigns a label to each vertex of G. An edge e = (u, v) is said to be satisfied under f1 and f2 if (f1(u), f2(v)) ∈Re, else is said to be unsatisfied. The object of the problem is to find a labeling maximizing the total weight of the satisfied edges. The value of the instance, Val(G), is defined as the maximum total weight of the satisfied edges divided by the total weight of all edges. Unique Game Problem (UGP) can be viewed as a special type of 2P1R. In UGP, we are given a graph G = (V, E), a weight function we ∈Q+ for e ∈E, and a set of labels, [k]. Each edge e = (u, v) in the graph is equipped with a permutation πe : [k] →[k]. The solution of the problem is a labeling f : V →[k] which assigns a label to each vertex of G. An edge e = (u, v) is said to be satisfied under f if πe(f(u)) = f(v), else is said to b

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