Abstract
par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degree&r dquo;or bounded “width.” Specifically:
We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor $7/8 + O(1/\sqrt{B})$, unless $RP = NP$. H\aa stad [18] proved that the problem is approximable to within a factor $7/8 + 1/64B$ in polynomial time, and that is hard to approximate to within a factor $7/8 +1/(\log B)^{Ω(1)}$. Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well.
We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor $\ln B - O(\ln\ln B)$ unless $P=NP$. The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of $(1+\ln B)$-approximate algorithms [20, 23, 9]
We present a new PCP construction, based on applying parallel repetition to the ``inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor $B^{1/19}$. The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B=2. The relationship between hardness of approximation and set size seems to have not been explored before.
We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor $B/2^{O(sqrt{\log B})}$, unless P = NP. This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor $B^{Ω (1)}$ unless P=NP [1]. An algorithm achieving factor $O (B)$ is also known [21, 2, 30, 16}.
