On problems as hard as CNF-SAT

The field of exact exponential time algorithms for non-deterministic polynomial-time hard problems has thrived since the mid-2000s. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, non-trivial exponential time algorithms have been found for a myriad of problems, including GRAPH COLORING, HAMILTONIAN PATH, DOMINATING SET, and 3-CNF-SAT. In some instances, improving these algorithms further seems to be out of reach. The CNF-SAT problem is the canonical example of a problem for which the trivial exhaustive search algorithm runs in time O(2(n)), where n is the number of variables in the input formula. While there exist non-trivial algorithms for CNF-SAT that run in time o(2(n)), no algorithm was able to improve the growth rate 2 to a smaller constant, and hence it is natural to conjecture that 2 is the optimal growth rate. The strong exponential time hypothesis (SETH) by Impagliazzo and Paturi [JCSS 2001] goes a little bit further and asserts that, for every epsilon < 1, there is a (large) integer k such that k-CNF-SAT cannot be computed in time 2(epsilon n). In this article, we show that, for every epsilon < 1, the problems HITTING SET, SET SPLITTING, and NAE-SAT cannot be computed in time O(2(epsilon n)) unless SETH fails. Here n is the number of elements or variables in the input. For these problems, we actually get an equivalence to SETH in a certain sense. We conjecture that SETH implies a similar statement for SET COVER and prove that, under this assumption, the fastest known algorithms for STEINER TREE, CONNECTED VERTEX COVER, SET PARTITIONING, and the pseudo-polynomial time algorithm for SUBSET SUM cannot be significantly improved. Finally, we justify our assumption about the hardness of SET COVER by showing that the parity of the number of solutions to SET COVER cannot be computed in time O(2(epsilon n)) for any epsilon < 1 unless SETH fails

ON SIMPLE BUT HARD RANDOM INSTANCES OF PROPOSITIONAL THEORIES AND LOGIC PROGRAMS

In the last decade, Answer Set Programming (ASP) and Satisfiability (SAT) have been used to solve combinatorial search problems and practical applications in which they arise. In each of these formalisms, a tool called a solver is used to solve problems. A solver takes as input a specification of the problem – a logic program in the case of ASP, and a CNF theory for SAT – and produces as output a solution to the problem. Designing fast solvers is important for the success of this general-purpose approach to solving search problems. Classes of instances that pose challenges to solvers can help in this task. In this dissertation we create challenging yet simple benchmarks for existing solvers in ASP and SAT.We do so by providing models of simple logic programs as well as models of simple CNF theories. We then randomly generate logic programs as well as CNF theories from these models. Our experimental results show that computing answer sets of random logic programs as well as models of random CNF theories with carefully chosen parameters is hard for existing solvers. We generate random logic programs with 2-literals, and our experiments show that it is hard for ASP solvers to obtain answer sets of purely negative and constraint-free programs, indicating the importance of these programs in the development of ASP solvers. An easy-hard-easy pattern emerges as we compute the average number of choice points generated by ASP solvers on randomly generated 2-literal programs with an increasing number of rules. We provide an explanation for the emergence of this pattern in these programs. We also theoretically study the probability of existence of an answer set for sparse and dense 2-literal programs. We consider simple classes of mixed Horn formulas with purely positive 2- literal clauses and purely negated Horn clauses. First we consider a class of mixed Horn formulas wherein each formula has m 2-literal clauses and k-literal negated Horn clauses. We show that formulas that are generated from the phase transition region of this class are hard for complete SAT solvers. The second class of Mixed Horn Formulas we consider are obtained from completion of a certain class of random logic programs. We show the appearance of an easy-hard-easy pattern as we generate formulas from this class with increasing numbers of clauses, and that the formulas generated in the hard region can be used as benchmarks for testing incomplete SAT solvers

Do Hard SAT-Related Reasoning Tasks Become Easier in the Krom Fragment?

Many reasoning problems are based on the problem of satisfiability (SAT). While SAT itself becomes easy when restricting the structure of the formulas in a certain way, the situation is more opaque for more involved decision problems. We consider here the CardMinSat problem which asks, given a propositional formula $\phi$ and an atom $x$, whether $x$ is true in some cardinality-minimal model of $\phi$. This problem is easy for the Horn fragment, but, as we will show in this paper, remains $\Theta_2$-complete (and thus $\mathrm{NP}$-hard) for the Krom fragment (which is given by formulas in CNF where clauses have at most two literals). We will make use of this fact to study the complexity of reasoning tasks in belief revision and logic-based abduction and show that, while in some cases the restriction to Krom formulas leads to a decrease of complexity, in others it does not. We thus also consider the CardMinSat problem with respect to additional restrictions to Krom formulas towards a better understanding of the tractability frontier of such problems

Many Hard Examples in Exact Phase Transitions with Application to Generating Hard Satisfiable Instances

This paper first analyzes the resolution complexity of two random CSP models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it is proved that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CNF formulas hard for resolution, which is a central task of Proof-Complexity theory, but also propose models with both many hard instances and exact phase transitions. Then, the implications of such models are addressed. It is shown both theoretically and experimentally that an application of Model RB/RD might be in the generation of hard satisfiable instances, which is not only of practical importance but also related to some open problems in cryptography such as generating one-way functions. Subsequently, a further theoretical support for the generation method is shown by establishing exponential lower bounds on the complexity of solving random satisfiable and forced satisfiable instances of RB/RD near the threshold. Finally, conclusions are presented, as well as a detailed comparison of Model RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively, exhibit three different kinds of phase transition behavior in NP-complete problems.Comment: 19 pages, corrected mistakes in Theorems 5 and