Design of Approximation Algorithms

16.4 Reductions from label cover 427clauses are satisfiable, for some constant ρ < 1. By following the same reduction as above, weobtain an instance of the maximization version of the label cover problem in which each i ∈ V 1(corresponding to a variable x i ) has degree 5, and each j ∈ V 2 (corresponding to a clause C j )has degree exactly 3. Thus the instance is (5,3)-regular, and it is still hard to distinguish inpolynomial time between instances in which all the edges are satisfiable, and instances in whichat most an α fraction of the edges are satisfiable for some constant α < 1. It is still also thecase that given an edge (u, v) ∈ E, and a label l 2 for v ∈ V 2 , there is at most one label l 1 for usuch that (l 1 , l 2 ) ∈ R uv . Following the rest of the reduction above then gives the following.Theorem 16.26: If for a particular constant α < 1 there is a α-approximation algorithm forthe maximization version of the label cover problem on (5,3)-regular instances, then P = NP.From the hardness of the label cover problem on (5, 3)-regular instances, we can derivethe hardness of the maximization version of the label cover problem on d-regular instances ford = 15. Given a (d 1 , d 2 )-regular instance (V 1 , V 2 , E) with labels L 1 and L 2 , we create a newinstance (V 1 ′,V 2 ′,E′ ) in which V 1 ′ = V 1 × V 2 , V 2 ′ = V 2 × V 1 , L ′ 1 = L 1, and L ′ 2 = L 2. For any(u, v) ∈ V ′1 and (v′ , u ′ ) ∈ V ′2 , we create an edge ((u, v), (v′ , u ′ )) ∈ E ′ exactly when (u, v ′ ) ∈ Eand (u ′ , v) ∈ E; then |E ′ | = |E| 2 . If we label (u, v) ∈ V 1 ′ with label l 1 and (v ′ , u ′ ) ∈ V 2 ′ withlabel l 2 , then (l 1 , l 2 ) is in the relation R ′ ((u,v),(v ′ ,u ′ )) for the edge ((u, v), (v′ , u ′ )) if and onlyif (l 1 , l 2 ) ∈ R uv ′. By construction, for any fixed (u ′ , v) ∈ E, there is a copy of the originalinstance: each edge (u, v ′ ) in the original instance corresponds to an edge ((u, v), (v ′ , u ′ )) in thenew instance. If (u, v) ∈ V ′1 is labelled with l 1 and (v ′ , u ′ ) ∈ V 2 is labelled with l 2 then theedge is satisfied in the new instance if and only if labelling u ∈ V 1 with l 1 and v ′ ∈ V 2 with l 2satisfies the edge (u, v ′ ) in the original instance. Thus if all edges are satisfiable in the originalinstance, then all edges will be satisfiable in the new instance, whereas if at most an α fractionof the edges are satisfiable in the original instance, then for each fixed (u ′ , v) ∈ E, at most an αfraction of the corresponding set of edges ((u, v), (v ′ , u ′ )) are satisfiable. Since the edges of thenew instance can be partitioned according to (u ′ , v), it follows that if at most an α fraction ofthe edges of the original instance can be satisfied, then at most an α fraction of edges of the newinstance can be satisfied. To see that the new instance is d-regular, fix any vertex (u, v) ∈ V 1 ′.Then since the original instance is (d 1 , d 2 )-regular, there are d 1 possible vertices v ′ ∈ V 2 suchthat (u, v ′ ) is an edge in the original instance, and d 2 possible vertices u ′ ∈ V 1 such that (u ′ , v)is an edge in the original instance. Hence there are d = d 1 d 2 edges ((u, v), (v ′ , u ′ )) incident onthe vertex (u, v) ∈ V 1 ′ . The argument that each vertex in V′2 has degree d = d 1d 2 is similar.Finally, suppose in the original instance it was the case that given (u, v) ∈ E and a label l 2 forv ∈ V 2 , there is at most one label l 1 for u such that (l 1 , l 2 ) ∈ R uv . Then in the new instance,given an edge ((u, v), (v ′ , u ′ )) ∈ E ′ and a label l 2 for (v ′ , u ′ ) ∈ V 2 ′ , then again there can be atmost one l 1 for (u, v) ∈ V 1 ′ such that (l 1, l 2 ) is in the relation for the edge. We thus have thefollowing result.Theorem 16.27: If for a particular constant α < 1 there is a α-approximation algorithm forthe maximization version of the label cover problem on 15-regular instances, then P = NP.We can prove a theorem with a much stronger hardness bound by using a technique similarto the one we used for the independent set problem, and reducing the maximization versionof the label cover problem to itself. Given an instance I of the maximization version of thelabel cover problem with vertex sets V 1 and V 2 , label sets L 1 and L 2 , edges E, and relationsR uv for all (u, v) ∈ E, we create a new instance I ′ of the problem with vertex sets V ′1 = V kV 1 × V 1 × · · · × V 1 and V ′2 = V k2 , with label sets L′ 1 = Lk 1 and L′ 2 = Lk 2E ′ = E k for any positive integer k. Consider an edge (u, v) ∈ E ′ ; let u ∈ V ′11 =, and with edge setbe a vertex suchElectronic web edition. Copyright 2011 by David P. Williamson and David B. Shmoys.To be published by Cambridge University Press