Non-approximability results for optimization problems on bounded degree instances

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 <italic>B</italic> clauses, is hard to approximate to within a factor $7/8 + <italic>O</italic>(1/\sqrt{<italic>B</italic>})$, unless $<italic>RP</italic> = <italic>NP</italic>$. H\aa stad [18] proved that the problem is approximable to within a factor $7/8 + 1/64<italic>B</italic>$ in polynomial time, and that is hard to approximate to within a factor $7/8 +1/(\log <italic>B</italic>)^{&OHgr;(1)}$. 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 <italic>B</italic> is hard to approximate to within a factor $\ln <italic>B</italic> - <italic>O</italic>(\ln\ln <italic>B</italic>)$ unless $<italic>P</italic>=<italic>NP</italic>$. The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of $(1+\ln <italic>B</italic>)$-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 <italic>B</italic> is hard to approximate to within a factor $<italic>B</italic>^{1/19}$. The problem can be approximated to within a factor <italic>B</italic> [19], and it is the Vertex Cover problem for <italic>B</italic>=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 <italic>B</italic> is hard to approximate to within a factor $<italic>B</italic>/2^{O(sqrt{\log <italic>B</italic>})}$, unless <italic>P</italic> = <italic>NP</italic>. 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 $<italic>B</italic>^{&OHgr; (1)}$ unless <italic>P</italic>=<italic>NP</italic> [1]. An algorithm achieving factor $<italic>O</italic> (<italic>B</italic>)$ is also known [21, 2, 30, 16}.