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### Complexity of Partial Satisfaction [chapter]

K. J. Lieberherr, E. Specker
1990 Ernst Specker Selecta
that satisfies at least the fraction h of its clauses, where h = (x/5 -1)/2 ~ 0.618 (the reciprocal of the "golden ratio").  ...  It is shown that, for any rational h' > h, the set of all 2-satisfiable cnfs that have truth assignments satisfying at least the fraction h' of their clauses ~s an NP-complete set gEY WORDS AND PHRASES  ...  This paper deals with the "maximum satisfiability" problem of  : Given a conjunctive-normal-form expression (cnf), with repeated clauses allowed, find a truth assignment that satisfies a maximum number  ...

### Complexity of partial satisfaction

Karl Lieberherr, Ernst Specker
1979 20th Annual Symposium on Foundations of Computer Science (sfcs 1979)
that satisfies at least the fraction h of its clauses, where h = (x/5 -1)/2 ~ 0.618 (the reciprocal of the "golden ratio").  ...  It is shown that, for any rational h' > h, the set of all 2-satisfiable cnfs that have truth assignments satisfying at least the fraction h' of their clauses ~s an NP-complete set gEY WORDS AND PHRASES  ...  This paper deals with the "maximum satisfiability" problem of  : Given a conjunctive-normal-form expression (cnf), with repeated clauses allowed, find a truth assignment that satisfies a maximum number  ...

### Complexity of Partial Satisfaction

K. J. Lieberherr, E. Specker
1981 Journal of the ACM
that satisfies at least the fraction h of its clauses, where h = (x/5 -1)/2 ~ 0.618 (the reciprocal of the "golden ratio").  ...  It is shown that, for any rational h' > h, the set of all 2-satisfiable cnfs that have truth assignments satisfying at least the fraction h' of their clauses ~s an NP-complete set gEY WORDS AND PHRASES  ...  This paper deals with the "maximum satisfiability" problem of  : Given a conjunctive-normal-form expression (cnf), with repeated clauses allowed, find a truth assignment that satisfies a maximum number  ...

### A SAT-Based Approach for Index Calculus on Binary Elliptic Curves [chapter]

Monika Trimoska, Sorina Ionica, Gilles Dequen
2020 Lecture Notes in Computer Science
With xor operations being at the core of many cryptographic problems, recent research in this area has focused on handling xor clauses efficiently.  ...  We extend this solver by adding a novel symmetry breaking technique and optimizing the time complexity of the point decomposition step by a factor of m! for the (m + 1) th summation polynomial.  ...  We thank the anonymous reviewers of the Africacrypt conference for their comments.  ...

### Page 4392 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews
In contrast, we give an example of an aperiodic monoid outside DA for which EQUATION SATISFIABILITY is com- putable in polynomial time and discuss the relative complexity of the two problems.  ...  Our linear time algorithm for solving the clauses with bounded size in fact solves the (,/logn) bounded self-duality problem in O(n?  ...

### Page 1155 of Mathematical Reviews Vol. , Issue 98B [page]

1998 Mathematical Reviews
We prove a dichotomy the- orem for these satisfiability problems with symmetric polynomial clauses: SAT(£;,---,£,) is either polynomial or NP-complete.  ...  {For the entire collection see MR 98a:68008. } 98b:68073 68Q25 03B05 03D15 68Q15 Creignou, Nadia (F-CAEN; Caen); More, Malika (F-CAEN; Caen) Complexity of satisfiability problems with symmetric polynomial  ...

### Polynomial Time SAT Decision, Hypergraph Transversals and the Hermitian Rank [chapter]

Nicola Galesi, Oliver Kullmann
2005 Lecture Notes in Computer Science
Combining graph theory and linear algebra, we study SAT problems of low "linear algebra complexity", considering formulas with bounded hermitian rank.  ...  We show polynomial time SAT decision of the class of formulas with hermitian rank at most one, applying methods from hypergraph transversal theory.  ...  Acknowledgements Part of this work was done while we were visiting the ICTP in Trieste participating in the Thematic Institute of the Complex Systems Network of Excellence (EXYSTENCE): "Algorithms and  ...

### Algorithms for Modular Counting of Roots of Multivariate Polynomials [chapter]

Parikshit Gopalan, Venkatesan Guruswami, Richard J. Lipton
2006 Lecture Notes in Computer Science
The modular root counting problem is given a modulus r, to determine N r (P ) = N (P ) mod r. We study the complexity of computing N r (P ), when the polynomial is given as a sum of monomials.  ...  We show an equivalence between maximum-likelihood decoding for Reed-Solomon codes and a root-finding problem for symmetric polynomials.  ...  A natural question is what is the complexity of the feasibility problem for symmetric polynomials in the sparse representation.  ...

### Algorithms for Modular Counting of Roots of Multivariate Polynomials

Parikshit Gopalan, Venkatesan Guruswami, Richard J. Lipton
2007 Algorithmica
The modular root counting problem is given a modulus r, to determine N r (P ) = N (P ) mod r. We study the complexity of computing N r (P ), when the polynomial is given as a sum of monomials.  ...  We show an equivalence between maximum-likelihood decoding for Reed-Solomon codes and a root-finding problem for symmetric polynomials.  ...  A natural question is what is the complexity of the feasibility problem for symmetric polynomials in the sparse representation.  ...

### Complexity Classifications for Propositional Abduction in Post's Framework [article]

Nadia Creignou, Johannes Schmidt, Michael Thomas
2010 arXiv   pre-print
Thus, we get a detailed picture of the complexity of the propositional abduction problem, hence highlighting sources of intractability.  ...  Further, we address the problem of counting the explanations and draw a complete picture for the counting complexity.  ...  The complexity of Abd(B) We commence with the symmetric abduction problem Abd(B) The results of this section are summarized in Figure 1 .  ...

### Complexity of Partial Satisfaction II

Karl Lieberherr, Ernst Specker
2012 Elemente der Mathematik
The fraction τ ψ of the clauses of a ψ-formula can be satisfied in polynomial time, while the set of ψ-formulas which have an assignment satisfying the fraction τ (τ > τ ψ , τ rational) of the clauses  ...  For several maximum ψ-satisfiability problems we explicitly determine algebraic numbers τ ψ (0 < τ ψ < 1), which separate NP-complete from polynomial problems.  ...  The problem with 3-satisfiable formulas is that they are not closed under symmetrization.  ...

Jiří Šíma
1994 Neural Computation
For this purpose we will employ the technique of polynomial time reduction from satisfiability problem (Balcazar et al. 1988).  ...  First, we prove the following lemma concerning the complexity of the decision problem of whether the instance (Z,C) of MSAT is satisfiable. Lemma. MSAT is NP-complete. Proof.  ...

### The Minimum Reload s-t Path/Trail/Walk Problems [chapter]

Laurent Gourvès, Adria Lyra, Carlos Martinhon, Jérôme Monnot
2009 Lecture Notes in Computer Science
Polynomial algorithms and proofs of NP-hardness are given for particular cases: when the triangle inequality is satisfied or not, when reload costs are symmetric (i.e. ri,j = rj,i) or asymmetric.  ...  This paper deals with problems on non-oriented edge-colored graphs. The aim is to find a route between two given vertices s and t. This route can be a walk, a trail or a path.  ...  When c = 2, we do not know the complexity of the minimum symmetric reload s-t path and the minimum asymmetric reload s-t trail problems if the matrix of reload costs does not satisfy the triangle inequality  ...

### Page 5109 of Mathematical Reviews Vol. , Issue 911 [page]

1991 Mathematical Reviews
(H-AOS) Lower bounds to the complexity of symmetric Boolean functions. Theoret. Comput. Sci. 74 (1990), no. 3, 313-323.  ...  Finally, L; = L(K3, R3), where K; is the set of all satisfiable Horn formulas, i.e., CNFs with clauses containing at most one positive literal, and F, R3F> if and only if F, is obtained from F, by log-bandwidth  ...

### Backdoors to Satisfaction [article]

Serge Gaspers, Stefan Szeider
2011 arXiv   pre-print
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.  ...  A backdoor set is a set of variables of a propositional formula such that fixing the truth values of the variables in the backdoor set moves the formula into some polynomial-time decidable class.  ...  Acknowledgment We thank Ryan Williams for his comments on an earlier version of this survey.  ...
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