The Complexity of Approximation - Lalitha Pragada

Theorem 8.26 Approximating the optimal solution to Clique within any constant ratio is NP-hard.

Proof: Develop a Gap-amplifying procedure, show that it turns any constant-ratio approximation into an approximation scheme, then appeal to theorem 8.23 to conclude that no constant-ratio approximation can exist.

Let G be any graph on n vertices Let G be any graph on n vertices. Consider the new graph G2 on n2 vertices, where each vertex of G has been replaced by a copy of G itself and vertices in two copies corresponding to two vertices joined in the original are connected with all possible n2 edges connecting a vertex in one copy to a vertex in the other.

We can claim that G has a clique of size k iff G2 has a clique of size k2. The k copies of the clique of G corresponding to the k clique vertices in G form a clique of size k2 in G2.

The “If” part is harder since we have no a priori constraint on the composition of the clique in G2. However, two copies of G in the larger graph are either fully connected to each other or not at all.

So, if two vertices in different copies belong to the large clique, then the two copies must be fully connected and an edge exists in G between the vertices corresponding to the copies. If two vertices in the same copy belong to the large clique, then these two vertices are connected by an edge in G.

Thus, every edge used in large clique corresponds to an edge in G Thus, every edge used in large clique corresponds to an edge in G. So, if the large clique has vertices in k or more distinct copies, then G has a clique of size k or more . If large clique has vertices in at most k distinct copies then it must include at least k vertices from same copy. So, G has a clique of size at least k.

Assume we have an approximation algorithm A For clique with absolute ratio E. Given some graph G with largest clique of size k we compute G2 ;run A on G2. We get a clique of size at least E k2 and then recover from this clique of size at least squareroot(E k2) = k*squareroot(E ). This new procedure , call it A ‘ runs in polynomial time if A does and has ratio RA ‘ = Squareroot(RA )

So more generally, j applications of this scheme yield procedure A j with absolute ratio j RA Given any approximation ratio E we can apply the scheme [log E ] times to obtain a log RA procedure with desired ratio.

Since [log E ] is a constant and since each log RA application can run in a polynomial time we can say that we derived a polynomial-time approximation scheme for clique. But clique is OPTNP-hard and thus, according to theorem 8.23 it cannot be in PTAS . That’s the desired contradiction.