A Note on MAX-Ek-LIN2 and the Unique Games

Most combinatorial optimization problems are NP-hard, i.e. they cannot be solved in polynomial time unless P = NP. Two typical approaches to deal with these problems are either to devise approximation algorithms with good performance, or to prove that such algorithms cannot exist. In 2002 S. Khot stated the Unique Games Conjecture (UGC), which generalizes the PCP Theorem. The UGC would imply significant hardness results for several optimization problems (e.g., MAX CUT or VERTEX COVER). Loosely speaking, the UGC states that for a class of games (called unique) in which the optimal solution takes values between 0 and 1, it is NP-hard to decide whether the optimal is close to 0 or close to 1. This conjecture has become one of the most outstanding open problems in complexity and approximation theory. In this article we study a problem which turns out to be closely related to the UGC: MAX-E2-LIN2 in bipartite graphs. The input of MAXE2-LIN2 is a graph G with two types of edges. Namely, each edge requires its end-vertices to be colored with either the same color or different colors. The objective is to 2-color the vertices of G maximizing the number of satisfied edges. The problem is known to be APX-complete in general graphs. In this article we prove that the problem remains APX-complete in bipartite graphs and, using the Parallel Repetition Theorem, we discuss the consequences that this result could have in the framework of unique games and the UGC.