Design of Approximation Algorithms

16.5 Reductions from unique games 439Lemma 16.46: For any ϵ such that 0 ≤ ϵ ≤ 1, given any feasible solution to the multicutinstance of cost at most ϵ|E ′ |, there is a solution to the MAX 2LIN(k) instance that satisfies atleast (1 − 2ϵ)|E| edges.Proof. Suppose that if we remove the edges in the multicut solution from G ′ , the graph ispartitioned into l components. We randomly index the corresponding partition of the vertexset V ′ from 1 to l, so that we have V 1 ′,. . . , V l ′ . We use this partition to determine a labelling forthe MAX 2LIN(k) instance: in particular, for each u ∈ V , there is some part V c ′ of least indexc such that some vertex (u, i) ∈ V c ′ for some label i ∈ L and no vertex (u, j) ∈ Vc ′ for c ′ < c.′Because the partition is given by a multicut, we know that there can be no other (u, j) ∈ V c′for j ≠ i. We then label u with i. We say that the part V ′c defines u.In order to analyze the number of edges satisfied by this labeling, consider an edge (u, v) ∈ E.Consider the k corresponding edges in E ′ , and let ϵ uv be the fraction of these k edges that arein the multicut. Then for a 1 − ϵ uv fraction of these edges, both endpoints (u, i) and (v, j) areinside the same part of the partition. Suppose some such part V ′c contains both endpoints (u, i)and (v, j) of an edge ((u, i), (v, j)) and the part defines both u and v. Then the labeling of uand v will satisfy (u, v) ∈ E since then u is labelled with i, v is labelled with j, and the edge((u, i), (v, j)) implies that i − j = c uv (modk), so that the labels satisfy the edge (u, v). Wecall such a part of the partition a good part; there are (1 − ϵ uv )k good parts of the partition.We now analyze the probability that a good part of the partition defines both u and v. Thisprobability gives us a lower bound on the probability that the edge (u, v) is satisfied. To dothis, we analyze the probability that some other part (a bad part) defines a label for u or v.Since ϵ uv k edges corresponding (u, v) are in the multicut, there are at most 2ϵ uv k parts of thepartition for which one of the following three things is true: it contains a vertex (u, i) but novertex (v, j); it contains a vertex (v, j) but no vertex (u, i); or it contains both (u, i) and (v, j),but there is no edge ((u, i), (v, j)). If any such part is ordered first, a good part will not definethe labels for u and v. Suppose there are b ≤ 2ϵ uv k bad parts. Thus the probability that edge(u, v) ∈ E is not satisfied by the labelling is at most the probability that of the b + (1 − ϵ uv )ktotal good and bad parts of the partition, one of the bad parts is ordered first. This is at mostbb + (1 − ϵ uv )k ≤2ϵ uv k2ϵ uv k + (1 − ϵ uv )k =2ϵ uv1 + ϵ uv≤ 2ϵ uv .Therefore, the overall expected number of edges that are not satisfied by the random labellingis at most 2 ∑ (u,v)∈E ϵ uv. By the definition of ϵ uv , there are k ∑ (u,v)∈E ϵ uv edges of|E ′ | in the multicut. Thus if the multicut has cost k ∑ ∑(u,v)∈E ϵ uv ≤ ϵ|E ′ | = ϵk|E|, then(u,v)∈E ϵ uv ≤ ϵ|E|. Then the expected number of edges not satisfied is at most 2ϵ|E|, sothat the expected number of satisfied edges is at least (1 − 2ϵ)|E|.Although the lemma above gives a randomized algorithm for obtaining a solution to theMAX 2LIN(k) instance from the solution to the multicut instance, it is not hard to convert thisto a deterministic algorithm. We leave this as an exercise to the reader.Corollary 16.47: There is a deterministic polynomial-time algorithm such that given any feasiblesolution to the multicut instance of cost at most ϵ|E ′ | finds a solution to the MAX 2LIN(k)instance that satisfies at least (1 − 2ϵ)|E| edges.From these two lemmas, we can derive the following corollaries.Corollary 16.48: Given the unique games conjecture, for any constant α ≥ 1, there is noα-approximation algorithm for the multicut problem unless P = NP.Electronic web edition. Copyright 2011 by David P. Williamson and David B. Shmoys.To be published by Cambridge University Press