Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (2008) Phase Transitions and Complexity of Weighted Satisfiability and Other Intractable Parameterized Problems Yong Gao Department of Computer Science Irving K. Barber School of Arts and Sciences University of British Columbia Okanagan Kelowna, Canada V1V 1V7 [email protected] Abstract such as backdoor. In particularly, the issue of the worst- case intractability versus the practical hardness of the back- The study of random instances of NP complete and coNP door detection problem has been raised: while the backdoor complete problems has had much impact on our understand- ing of the nature of hard problems. In this work, we initiate detection problem is NP-complete and/or fixed-parameter an effort to extend this line of research to random instances intractable for many types of backdoors, SAT-solvers such of intractable parameterized problems. as SATZ can exploit the existence of small-sized back- We propose random models for a representative intractable doors effectively (Dilkina, Gomes, and Sabharwal 2007; parameterized problem, the weighted d-CNF satisfiability, Szeider 2005). and its generalization to the constraint satisfaction problem. The study of parameterized proof complexity of the The exact threshold for the phase transition of the proposed propositional satisfiability problem has been initiated in models is determined. Lower bounds on the time complex- (Dantchev, Martin, and Szeider 2007), where lower bounds ity of variants of the DPLL algorithm for these parameter- on the parameterized resolution proof are established for ized problems are also established. In particularly, we show CNF formulas that encode some first-order combinatorial that random instances of the weighted 2-CNF satisfiability, principle. already an intractable parameterized problem, are typically easy in both of the satisfiable and unsatisfiable regions by The study of random instances of NP-complete prob- exploiting an interesting connection between the unsatisfia- lems and coNP-complete problems such as SAT has had bility of a weighted 2-CNF formula and the existence of a much impact on our understanding of the nature of hard Hamiltonian-cycle-like global structure. problems (Achlioptas, Beame, and Molloy 2001; Beame et al. 2002; Cheeseman, Kanefsky, and Taylor 1991) as well 1. Introduction as the strength and weakness of well-founded algorithms and heuristics (Gent and Walsh 1998; Gomes et al. 2005; The theory of parameterized complexity and fixed- Gao and Culberson 2007). parameter algorithms is becoming an active research area In this work, we initiate an effort to extend this line of in recent years (Downey and Fellows 1999; Neidermeier research to intractable parameterized problem in the param- 2006). Parameterized complexity provides a new perspec- eterized complexity hierarchy. We propose random models tive on hard algorithmic problems, while fixed-parameter al- for a representative parameterized intractable problem, the gorithms have found applications in a variety of areas such weighted d-CNF satisfiability, and its variants. We establish as computational biology, cognitive modeling, graph theory, the exact threshold of the phase transition of the proposed and various optimization problems. model. This is in contrast to the threshold behavior of most Parameterized algorithmic problems also arise in many of the NP complete problems, such as the satisfiability and areas of artificial intelligence research. See, for example, the graph coloring, where the exact threshold is extremely hard survey of Gottlob and Szeider (Gottlob and Szeider 2007). to nail down. backdoor Recently, some problems related to detecting For the weighted 2-CNF satisfiability problem, which is sets for instances of the propositional satisfiability problem already W[1]-complete, we show that random instances are (SAT) have been studied from the perspective of parameter typically easy in both of the satisfiable and unsatisfiable re- complexity (Szeider 2005; Nishimura, Ragde, and Szeider gions. We prove that random (unsatisfiable) instances of 2004). A backdoor (to instances of a hard problem) is a sub- the weighted 2-CNF satisfiability problem have a “fixed- set of variables that, once fixed, will result in one or more parameter-sized” resolution proof by exploiting an interest- polynomial time sovable subproblems. The existence of ing link between the unsatisfiability of a weighted 2-CNF small sized backdoors naturally leads to fixed-parameter al- formula and the existence of a Hamiltonian cycle in a prop- gorithms. For example, in Bayesian networks and constraint erly defined directed graph. This is somewhat surprising networks, a feedback vertex set of the (directed) network is and indicates that the typical-case behavior of random in- Copyright c 2008, Association for the Advancement of Artificial stances of parameterized problems are quite different from Intelligence (www.aaai.org). All rights reserved. that of NP-complete problems. First, it is the existence of 265 a global structure (a directed Hamiltonian cycle) that leads As in the case of the theory of NP-completeness, the sat- to easy instances, as compared to the case of NP-complete isfiability problem plays an important role in the theory of problems where small structures are usually the signature parameterized complexity (Downey and Fellows 1999). A of easy instances. Secondly, unlike the case of traditional CNF formula (over a set of Boolean variables) is a conjunc- satisfiability, the weighted 2-CNF satisfiability problem is tion of disjunctions of literals. A d-clause is a disjunction of already fixed-parameter intractable. As a comparison, for d-literals. weighted d-CNF satisfiability problem with d > 2, we es- Definition 1. An assignment to a set of n Boolean variables tablish a non-fixed parameter tractable lower bound on the n search tree size of the parameterized version of the ordered is a vector in f0; 1g . The weight of an assignment is the DPLL algorithm. number of the variables that are set to 1 (true) by the as- We observe that there are certain relations between find- signment, i.e., the number of 1’s in the assignment. ing a solution to an instance of the parameterized problem A representative W [1]-complete problem is the following under consideration the tasks of backdoor detection with re- weighted d-CNF satisfiability problem (weighted d-SAT) spect to an (extremely) naive sub-solver that simply checks (Downey and Fellows 1999): whether the all-zero assignment is satisfying, and hope that further investigation along this line may help shed further Problem 1. Weighted d-SAT light on the (typical-case) hardness of the backdoor detec- Input: A CNF formula consisting of d-clauses, and a tion problem and the empirical observation on the effective- positive integer k. ness of practical satisfiability solvers in exploiting small- sized backdoors. Question: Is there a satisfying assignment of weight k? The results reported in the current paper are all of theo- retical nature and the objective is to lay the necessary foun- Unlike the traditional satisfiability problem, weighted dation for a study of the phase transitions and typical-case 2-SAT is already W[1]-complete. The anti-monotone hardness of fixed-parameter intractable problems. We be- weighted d-SAT problem (the problem where each clause lieve that in the future study, empirical studies are indispens- contains negative literatures only) is also W[1]-complete. able, especially for situations where theoretical analyses are The weighted satisfiability problem (weighted SAT) is hard to come by. See, for example, the work in (Gent and similar to weighted d-SAT except that there is no restriction Walsh 1998; Gomes et al. 2005; Gao and Culberson 2007) on the length of a clause in the formula. Weighted SAT is a and the references therein for the many empirical studies on representative W [2]-complete problem. the phase transitions of NP-hard problems. Parameterized Proof Systems and Parameterized 2. Preliminaries DPLL Algorithms Parameterized complexity A formal definition of a parameterized tree-like resolution parameterized decision problem An instance of a is a pair proof system for the weighted d-SAT problem is given in (I; k) where I is a problem instance and k is the problem pa- (Dantchev, Martin, and Szeider 2007). Basically, a param- rameter. Usually, the parameter k either specifies the “size” eterized resolution can be regarded as a classical resolution of the solution or is related to some structural property of that have access (for free) to all clauses with more than k the underlying problem, such as the treewidth of a graph. negated variables, where k is the parameter of the weighted For example, in an instance (I; k) of the parameterized ver- SAT problem. tex cover (VC) problem, I is a graph and k is the size of the vertex cover. The question is to decide whether I has a Accordingly, one can consider the parameterized version vertex cover of size k. of the DPLL algorithm for weighted SAT. It proceeds in the A parameterized problem is fixed-parameter tractable same way as the standard DPLL algorithm with the excep- (FPT) if any instance (I; k) of the problem can be solved tion that a node in the search tree fails if in f(k)jIjO(1) time, where f(k) is a computable function of 1. either a clause has been falsified by the partial assignment, k only. It is known that the parameterized VC problem can be solved in O(1:28k + kn) time where n is the number of 2. or the number of variables assigned to true in the partial vertices of the input graph. assignment has exceeded k. Parameterized problems are inter-related by parameter- In addition to the parameterized version of the general ized reductions, resulting in a classification of parameterized DPLL algorithm, we may also consider the parameterized problems into a hierarchy of complexity classes version of ordered-DPLL and the oblivious-DPLL algo- FPT ⊆ W [1] ⊆ W [2] · · · ⊆ XP: rithms studied in (Beame et al.