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### On Problems as Hard as CNF-SAT

Marek Cygan, Holger Dell, Daniel Lokshtanov, Dániel Marx, Jesper Nederlof, Yoshio Okamoto, Ramamohan Paturi, Saket Saurabh, Magnus Wahlström
2016 ACM Transactions on Algorithms
Graph Coloring, Hamiltonian Path, Dominating Set and 3-CNF-Sat.  ...  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.  ...  In addition to these two basic problems, we can name new problems by adding one of the following modifiers to their names (which we do by example just for CNF-Sat). • k-CNF-Sat is the problem in which  ...

### From Non-Clausal to Clausal MinSAT [chapter]

Chu-Min Li, Felip Manyà, Joan Ramon Soler, Amanda Vidal
2021 Frontiers in Artificial Intelligence and Applications
We tackle the problem of solving MinSAT for multisets of propositional formulas that are not necessarily in clausal form.  ...  Our approach reduces non-clausal to clausal MinSAT, since this allows us to rely on the much developed clause-based MinSAT solvers.  ...  These problems are significant because many practical questions can be solved by first encoding them as a SAT, MaxSAT or MinSAT problems, and then finding a solution by solving the resulting encoding with  ...

### Complexity and Approximability of Parameterized MAX-CSPs

Holger Dell, Eun Jung Kim, Michael Lampis, Valia Mitsou, Tobias Mömke
2017 Algorithmica
W[1]-hardness results.  ...  In standard CSPs, we want to decide whether this fraction equals one.  ...  There exists an FPT-AS which, given > 0 and an instance φ of MAX-THRESHOLD, computes a (1 − )-approximate solution and runs in time f (fvs * , ) · poly(n), where fvs * is the size of the smallest feedback  ...

### Extended Conjunctive Normal Form and An Efficient Algorithm for Cardinality Constraints

Zhendong Lei, Shaowei Cai, Chuan Luo
2020 Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence
SAT and MaxSAT are expressed in CNF, which is difficult to deal with cardinality constraints.  ...  Satisfiability (SAT) and Maximum Satisfiability (MaxSAT) are two basic and important constraint problems with many important applications.  ...  Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence  ...

### On the Exact Complexity of Evaluating Quantified k -CNF

Chris Calabro, Russell Impagliazzo, Ramamohan Paturi
2012 Algorithmica
) and show that s(∞) (the limit of s(k) as k → ∞) is at most s(Π23-sat).  ...  We relate the exponential complexities 2 s(k)n of k-sat and the exponential complexity 2 s(Π 2 3-sat)n of Π23-sat (evaluating formulas of the form ∀x∃yφ(x, y) where φ is a 3-CNF in x variables and y variables  ...  As with cnf-sat, we will consider various syntactic restrictions of quantified k-sat to arrive at a minimally complex set of formulas which are as hard as the standard existentially quantified 3-CNFs as  ...

### Solving Over-Constrained Problems with SAT [chapter]

Josep Argelich, Felip Manyà
2005 Lecture Notes in Computer Science
We present a new generic problem solving approach for overconstrained problems based on Max-SAT.  ...  We first define a clausal form formalism that deals with blocks of clauses and distinguish between hard and soft blocks.  ...  The Soft-SAT problem is the problem of finding a solution to a Soft CNF formula.  ...

### Hypergraph Acyclicity and Propositional Model Counting [chapter]

Florent Capelli, Arnaud Durand, Stefan Mengel
2014 Lecture Notes in Computer Science
We show that the propositional model counting problem #SAT for CNF-formulas with hypergraphs that allow a disjoint branches decomposition can be solved in polynomial time.  ...  We show that this class of hypergraphs is incomparable to hypergraphs of bounded incidence cliquewidth which were the biggest class of hypergraphs for which #SAT was known to be solvable in polynomial  ...  In fact, most classical decision results based on tractability of resolution (such as for 2−SAT) or unit propagation (Horn−SAT) do not extend to counting as the respective counting problems are hard (see  ...

### A criterion for "easiness" of certain SAT problems [article]

Bernd R. Schuh
2017 arXiv   pre-print
Also individual instances can be checked for easiness with respect to a given SAT problem.  ...  A generalized 1-in-3SAT problem is defined and found to be in complexity class P when restricted to a certain subset of CNF expressions.  ...  Similarly we can argue that the 1-in-kSAT problem, i.e. the same problem without restrictions on the number of literals per clause, is NP-hard, as well.  ...

### Backdoors into Two Occurrences

Jan Johannsen
2020 Journal on Satisfiability, Boolean Modeling and Computation
The question whether there exists a CNF(2)-backdoor set of size k is hard for the class W[2], for both weak and strong backdoors, and in both cases it becomes fixed-parameter tractable when restricted  ...  to inputs in d-CNF for a fixed d.  ...  The research leading to the results in this paper was initiated at Dagstuhl Seminar 12471 "SAT Interactions".  ...

### Solving Over-Constrained Problems with SAT Technology [chapter]

Josep Argelich, Felip Manyà
2005 Lecture Notes in Computer Science
We present a new generic problem solving approach for overconstrained problems based on Max-SAT.  ...  We first define a clausal form formalism that deals with blocks of clauses instead of individual clauses, and that allows one to declare each block either as hard (i.e., must be satisfied by any solution  ...  The Soft-SAT problem is the problem of finding a solution to a Soft CNF formula.  ...

### Satisfiability of mixed Horn formulas

Stefan Porschen, Ewald Speckenmeyer
2007 Discrete Applied Mathematics
A satisfiability formulation of problems on level graphs, ENDM 9 (2001)] and graph colorability formulas.  ...  We further show that the NP-hard optimization problem minimum weight SAT for mixed Horn formulas can be solved in time O(2 0.5284n ) if non-negative weights are assigned to the variables.  ...  a maximum Horn model, which is an NP-hard optimization problem.  ...

### Logical Analysis of Hash Functions [chapter]

Dejan Jovanović, Predrag Janičić
2005 Lecture Notes in Computer Science
In addition, one can finely tune the hardness of generated formulae. This can be very important for different applications, including testing (complete or incomplete) sat solvers.  ...  satisfiability problem.  ...  We are grateful to anonymous reviewers for useful comments on the first version of this paper.  ...

### The Boolean Satisfiability Problem [article]

Yu Geng
2020 Figshare
The Boolean Satisfiability Problem is also the first problem proven to be NP-complete.  ...  (Thus being able to solve a NP-complete problem is equivalent to being able to solve every problem in NP).  ...  13 3-SAT Hardness As n increases hardness transition grows sharper m / n 14 Transition at m/n ' 4.3 m / n Threshold Conjecture • For every k, there exists a c* such that -For m/n <  ...

### Backdoors to Satisfaction [article]

Serge Gaspers, Stefan Szeider
2011 arXiv   pre-print
We also discuss recent results on backdoor sets for problems that are beyond NP.  ...  In this survey we review parameterized complexity results for problems that arise in the context of backdoor sets, such as the problem of finding a backdoor set of size at most k, parameterized by k.  ...  Acknowledgment We thank Ryan Williams for his comments on an earlier version of this survey.  ...

### Symmetry Breaking for Maximum Satisfiability [article]

Joao Marques-Silva, Ines Lynce, Vasco Manquinho
2008 arXiv   pre-print
In SAT, the identification of symmetry breaking predicates (SBPs) is a well-known, often effective, technique for solving hard problems.  ...  As with SAT and PB, symmetry breaking predicates for MaxSAT and variants are shown to be effective for a representative number of problem domains, allowing solving problem instances that current state  ...  For the weighted partial MaxSAT problem, the formula is the conjunction of a weighted CNF formula (soft clauses) and a classical CNF formula (hard clauses).  ...
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