Variations on an Ordering Theme with Constraints

Graphs with bounded thinness were defined in 2007 as a generalization of interval graphs. In this paper we introduce the concept of proper thinness, such that graphs with bounded proper thinness generalize proper interval graphs. We study the complexity of problems related to the computation of these parameters, describe the behavior of the thinness and proper thinness under three graph operations, and relate thinness and proper thinness to other graph invariants in the literature. Finally, we describe a wide family of problems that can be solved in polynomial time for graphs with bounded thinness, generalizing for example list matrix partition problems with bounded size matrix, and enlarge this family of problems for graphs with bounded proper thinness, including domination problems.

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“…The proof is based on a reduction from the following problem, which is known to be NP-complete [18].…”

Section: Algorithmic Aspectsmentioning

confidence: 99%

On the thinness and proper thinness of a graph

Bonomo

Estrada

2019

Discrete Applied Mathematics

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“…Hence every solution of the learning problem made of these two sentences will be such that C and A occur before B in t 1 . In Section 4, this technique enables us to reduce the problem of betweenness [12] to our grammar lifting problem. This problem is known to be NP-complete.…”

Section: An Examplementioning

confidence: 99%

“…Proof. We give a reduction of the betweenness problem [12] to our grammar induction problem. The betweenness problem is as follows.…”

Section: Learning Pivotal Categories From Compact Closed Categories Tmentioning

confidence: 99%

Complexity of Grammar Induction for Quantum Types

Delpeuch

2014

Electron. Proc. Theor. Comput. Sci.

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Most categorical models of meaning use a functor from the syntactic category to the semantic category. When semantic information is available, the problem of grammar induction can therefore be defined as finding preimages of the semantic types under this forgetful functor, lifting the information flow from the semantic level to a valid reduction at the syntactic level. We study the complexity of grammar induction, and show that for a variety of type systems, including pivotal and compact closed categories, the grammar induction problem is NP-complete. Our approach could be extended to linguistic type systems such as autonomous or bi-closed categories.

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“…Guttmann and Maucher [32] showed that there are in fact only 13 distinct Π-CSP's of arity 3 up to symmetry, of which 11 are nontrivial. They are listed in Table 1 together with their complexity.…”

Section: Ordering Cspsmentioning

confidence: 99%

Constraint Satisfaction Problems Parameterized above or below Tight Bounds: A Survey

Gutin

Yeo

2012

The Multivariate Algorithmic Revolution and Beyond

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Abstract. We consider constraint satisfaction problems parameterized above or below tight bounds. One example is MaxSat parameterized above m/2: given a CNF formula F with m clauses, decide whether there is a truth assignment that satisfies at least m/2 + k clauses, where k is the parameter. Among other problems we deal with are MaxLin2-AA (given a system of linear equations over F2 in which each equation has a positive integral weight, decide whether there is an assignment to the variables that satisfies equations of total weight at least W/2 + k, where W is the total weight of all equations), Max-r-Lin2-AA (the same as MaxLin2-AA, but each equation has at most r variables, where r is a constant) and Max-r-Sat-AA (given a CNF formula F with m clauses in which each clause has at most r literals, decide whether there is a truth assignment satisfying at least m i=1 (1 − 2 r i ) + k clauses, where k is the parameter, ri is the number of literals in Clause i, and r is a constant). We also consider Max-r-CSP-AA, a natural generalization of both Max-r-Lin2-AA and Max-r-Sat-AA, order (or, permutation) constraint satisfaction problems of arities 2 and 3 parameterized above the average value and some other problems related to MaxSat. We discuss results, both polynomial kernels and parameterized algorithms, obtained for the problems mainly in the last few years as well as some open questions.

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Variations on an Ordering Theme with Constraints

Cited by 32 publications

References 6 publications

On the thinness and proper thinness of a graph

On the thinness and proper thinness of a graph

Complexity of Grammar Induction for Quantum Types

Constraint Satisfaction Problems Parameterized above or below Tight Bounds: A Survey

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