Published: June 10, 2012

**Keywords:**approximation algorithms, hardness of approximation, Unique Games conjecture, Horn-SAT, exact hitting set

**Categories:**complexity theory, lower bounds, inapproximability, Unique Games, UG-hardness, SAT, 3SAT, Horn-3SAT, hitting set

**ACM Classification:**G.1.6

**AMS Classification:**90C59

**Abstract:**
[Plain Text Version]

We study the approximability of two natural Boolean
constraint satisfaction problems: Horn satisfiability and exact hitting set.
Under the Unique Games conjecture, we
prove the following *optimal* inapproximability and approximability results for finding an
assignment satisfying as many constraints as possible given a
*near-satisfiable* instance.

1. Given an instance of Max Horn-3SAT that admits an assignment satisfying $(1 - \epsilon)$ of its constraints for some small constant $\epsilon > 0$, it is hard to find an assignment satisfying more than $(1 - 1/O (\log (1 / \epsilon)))$ of the constraints. This matches a linear programming based algorithm due to Zwick (STOC 1998), resolving the natural open question raised in that work concerning the optimality of the approximation bound.

Given a $(1 - \epsilon)$-satisfiable instance of Max Horn-2SAT for some constant $\epsilon > 0$, it is possible to find a $(1 - 2\epsilon)$-satisfying assignment efficiently. This improves the algorithm given in Khanna et al. (2000) which finds a $(1 - 3\epsilon)$-satisfying assignment, and also matches the $(1 - c\epsilon)$ hardness for any $c < 2$ derived from vertex cover (under UGC).

2. An instance of Max $1$-in-$k$-HS consists of a universe $U$ and a collection $\mathcal{C}$ of subsets of $U$ of size at most $k$, and the goal is to find a subset of $U$ that intersects the maximum number of sets in $\mathcal{C}$ at a unique element. We prove that Max $1$-in-$k$-HS is hard to approximate within a factor of $O(1/\log k)$ for every fixed integer $k$. This matches (up to constant factors) an easy factor $\Omega(1/\log k)$ approximation algorithm for the problem, and resolves a question posed in Guruswami and Trevisan (APPROX 2005).

It is crucial for the above hardness that sets of size *up to*
$k$ are allowed; indeed, when all sets have size $k$, there is a
simple factor $1/e$-approximation algorithm.

Our hardness results are proved by constructing integrality gap instances for a semidefinite programming relaxation for the problems, and using Raghavendra's result (STOC 2008) to conclude that no algorithm can do better than the SDP assuming the UGC. In contrast to previous such constructions where the instances had a good SDP solution by design and the main task was bounding the integral optimum, the challenge in our case is the construction of appropriate SDP vectors and the integral optimum is easy to bound. Our algorithmic results are based on rounding appropriate linear programming relaxations.