Oliver Kullmann

New methods for 3-SAT decision and worst-case analysis

Abstract We prove the worst-case upper bound 1.5045..n for the time complexity of 3-SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2- and 3-clauses, called “blocked clauses”, generalizing the extension rule of “Extended Resolution.” Our methods for estimating the size of trees lead to a refined measure of formula complexity of 3-clause-sets and can be applied also to arbitrary trees.

Theory and Applications of Satisfiability Testing -- SAT 2009

Invited Talks.- SAT Modulo Theories: Enhancing SAT with Special-Purpose Algorithms.- Symbolic Techniques in Propositional Satisfiability Solving.- Applications of SAT.- Efficiently Calculating Evolutionary Tree Measures Using SAT.- Finding Lean Induced Cycles in Binary Hypercubes.- Finding Efficient Circuits Using SAT-Solvers.- Encoding Treewidth into SAT.- Complexity Theory.- The Complexity of Reasoning for Fragments of Default Logic.- Does Advice Help to Prove Propositional Tautologies?.- Structures for SAT.- Backdoors in the Context of Learning.- Solving SAT for CNF Formulas with a One-Sided Restriction on Variable Occurrences.- On Some Aspects of Mixed Horn Formulas.- Variable Influences in Conjunctive Normal Forms.- Resolution and SAT.- Clause-Learning Algorithms with Many Restarts and Bounded-Width Resolution.- An Exponential Lower Bound for Width-Restricted Clause Learning.- Improved Conflict-Clause Minimization Leads to Improved Propositional Proof Traces.- Boundary Points and Resolution.- Translations to CNF.- Sequential Encodings from Max-CSP into Partial Max-SAT.- Cardinality Networks and Their Applications.- New Encodings of Pseudo-Boolean Constraints into CNF.- Efficient Term-ITE Conversion for Satisfiability Modulo Theories.- Techniques for Conflict-Driven SAT Solvers.- On-the-Fly Clause Improvement.- Dynamic Symmetry Breaking by Simulating Zykov Contraction.- Minimizing Learned Clauses.- Extending SAT Solvers to Cryptographic Problems.- Solving SAT by Local Search.- Improving Variable Selection Process in Stochastic Local Search for Propositional Satisfiability.- A Theoretical Analysis of Search in GSAT.- The Parameterized Complexity of k-Flip Local Search for SAT and MAX SAT.- Hybrid SAT Solvers.- A Novel Approach to Combine a SLS- and a DPLL-Solver for the Satisfiability Problem.- Building a Hybrid SAT Solver via Conflict-Driven, Look-Ahead and XOR Reasoning Techniques.- Automatic Adaption of SAT Solvers.- Restart Strategy Selection Using Machine Learning Techniques.- Instance-Based Selection of Policies for SAT Solvers.- Width-Based Restart Policies for Clause-Learning Satisfiability Solvers.- Problem-Sensitive Restart Heuristics for the DPLL Procedure.- Stochastic Approaches to SAT Solving.- (1,2)-QSAT: A Good Candidate for Understanding Phase Transitions Mechanisms.- VARSAT: Integrating Novel Probabilistic Inference Techniques with DPLL Search.- QBFs and Their Representations.- Resolution and Expressiveness of Subclasses of Quantified Boolean Formulas and Circuits.- A Compact Representation for Syntactic Dependencies in QBFs.- Beyond CNF: A Circuit-Based QBF Solver.- Optimisation Algorithms.- Solving (Weighted) Partial MaxSAT through Satisfiability Testing.- Nonlinear Pseudo-Boolean Optimization: Relaxation or Propagation?.- Relaxed DPLL Search for MaxSAT.- Branch and Bound for Boolean Optimization and the Generation of Optimality Certificates.- Exploiting Cycle Structures in Max-SAT.- Generalizing Core-Guided Max-SAT.- Algorithms for Weighted Boolean Optimization.- Distributed and Parallel Solving.- PaQuBE: Distributed QBF Solving with Advanced Knowledge Sharing.- c-sat: A Parallel SAT Solver for Clusters.We develop techniques to calculate important measures in evolutionary biology by encoding to CNF formulas and using powerful SAT solvers. Comparing evolutionary trees is a necessary step in tree reconstruction algorithms, locating recombination and lateral gene transfer, and in analyzing and visualizing sets of trees. We focus on two popular comparison measures for trees: the hybridization number and the rooted subtree-prune-and-regraft (rSPR) distance. Both have recently been shown to be NP-hard, and effcient algorithms are needed to compute and approximate these measures. We encode these as a Boolean formula such that two trees have hybridization number k (or rSPR distance k) if and only if the corresponding formula is satisfiable. We use state-of-the-art SAT solvers to determine if the formula encoding the measure has a satisfying assignment. Our encoding also provides a rich source of real-world SAT instances, and we include a comparison of several recent solvers (minisat, adaptg2wsat, novelty+p, Walksat, March KS and SATzilla).

Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference

A formula (in conjunctive normal form) is said to be minimal unsatisfiable if it is unsatisfiable and deleting any clause makes it satisfiable. The deficiency of a formula is the difference of the number of clauses and the number of variables. It is known that every minimal unsatisfiable formula has positive deficiency. Until recently, polynomial-time algorithms were known to recognize minimal unsatisfiable formulas with deficiency 1 and 2. We state an algorithm which recognizes minimal unsatisfiable formulas with any fixed deficiency in polynomial time.

On a generalization of extended resolution

Abstract Motivated by improved SAT algorithms ((O. Kullmann, DIMACS Series, vol. 35, Amer. Math. Soc., Providence, RI, 1997; O. Kullmann, Theoret. Comput. Sci. (1999); O. Kullmann, Inform. Comput., submitted); yielding new worst-case upper bounds) a natural parameterized generalization GER of Extended Resolution (ER) is introduced. ER can simulate polynomially GER, but GER allows special cases for which exponential lower bounds can be proven.

Investigations on autark assignments

Abstract The structure of the monoid of autarkies and the monoid of autark subsets for clause-sets F is investigated, where autarkies are partial (truth) assignments satisfying some subset F′⊆F (called an autark subset), while not interacting with the clauses in F⧹F′. Generalising minimally unsatisfiable clause-sets, the notion of lean clause-sets is introduced, which do not have non-trivial autarkies, and it is shown that a clause-set is lean iff every clause can be used by some resolution refutation. The largest lean sub-clause-set and the largest autark subset yield a (2-)partition for every clause-set. As a special case of autarkies we introduce the notion of linear autarkies, which can be found in polynomial time by means of linear programming. Clause-sets without non-trivial linear autarkies we call linearly lean, and clause-sets satisfiable by a linear autarky we call linearly satisfiable. As before, the largest linearly lean sub-clause-set and the largest linearly autark subset yield a (2-)partition for every clause-set, but this time the decomposition is computable in polynomial time. The class of linearly satisfiable clause-sets generalises the notion of matched clause-sets introduced in a recent paper by J. Franco and A. Van Gelder, and, as shown by H. van Maaren, contains also (“modulo Unit-clause elimination”) all satisfiable q-Horn clause-sets, introduced by E. Boros, Y. Crama and P. Hammer. The class of linearly lean clause-sets is stable under “crossing out variables” and union, and has some interesting combinatorial properties with respect to the deficiency δ=c−n, the difference of the number of clauses and the number of variables: So for example (non-empty) linearly lean clause-sets fulfill δ⩾1, where this property has been known before only for minimally unsatisfiable clause-sets.

Cube and conquer: guiding CDCL SAT solvers by lookaheads

Satisfiability (SAT) is considered as one of the most important core technologies in formal verification and related areas. Even though there is steady progress in improving practical SAT solving, there are limits on scalability of SAT solvers. We address this issue and present a new approach, called cube-and-conquer, targeted at reducing solving time on hard instances. This two-phase approach partitions a problem into many thousands (or millions) of cubes using lookahead techniques. Afterwards, a conflict-driven solver tackles the problem, using the cubes to guide the search. On several hard competition benchmarks, our hybrid approach outperforms both lookahead and conflict-driven solvers. Moreover, because cube-and-conquer is natural to parallelize, it is a competitive alternative for solving SAT problems in parallel.

Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer

The boolean Pythagorean Triples problem has been a longstanding open problem in Ramsey Theory: Can the set N =

The seventh QBF solvers evaluation (QBFEVAL’10)

\{1, 2, ...\}

Upper and Lower Bounds on the Complexity of Generalised Resolution and Generalised Constraint Satisfaction Problems

of natural numbers be divided into two parts, such that no part contains a triple

The Combinatorics of Conflicts between Clauses

(a,b,c)