scholarly journals Translating between Horn Representations and their Characteristic Models

1995 ◽  
Vol 3 ◽  
pp. 349-372 ◽  
Author(s):  
R. Khardon
Keyword(s):  
Alternative Model ◽  
Decision Problem ◽  
Relational Databases ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Database Theory ◽  
Hypergraph Transversal ◽  
The Cost ◽  
Special Case

Characteristic models are an alternative, model based, representation for Horn expressions. It has been shown that these two representations are incomparable and each has its advantages over the other. It is therefore natural to ask what is the cost of translating, back and forth, between these representations. Interestingly, the same translation questions arise in database theory, where it has applications to the design of relational databases. This paper studies the computational complexity of these problems. Our main result is that the two translation problems are equivalent under polynomial reductions, and that they are equivalent to the corresponding decision problem. Namely, translating is equivalent to deciding whether a given set of models is the set of characteristic models for a given Horn expression. We also relate these problems to the hypergraph transversal problem, a well known problem which is related to other applications in AI and for which no polynomial time algorithm is known. It is shown that in general our translation problems are at least as hard as the hypergraph transversal problem, and in a special case they are equivalent to it.

THE PROBLEM OF PARTITIONING WITH DUPLICATIONS AND ITS APPLICATIONS

1994 ◽  
Vol 03 (03) ◽  
pp. 395-405
Author(s):  
J. HARALAMBIDES ◽  
S. TRAGOUDAS
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Vlsi Layout ◽  
Optimal Polynomial ◽  
Partitioning Problem ◽  
Polynomially Solvable ◽  
The Cost ◽  
Parallel Graph ◽  
Special Case

The problem of partitioning the elements of a graph G=(V, E) into two equal size sets A and B that share at most d elements such that the total number of edges (u, v), u∈A−B, v∈B−A is minimized, arises in the areas of Hypermedia Organization, Network Integrity, and VLSI Layout. We formulate the problem in terms of element duplication, where each element c∈A∩B is substituted by two copies c′∈A and c″∈B As a result, edges incident to c′ or c″ need not count in the cost of the partition. We show that this partitioning problem is NP-hard in general, and we present a solution which utilizes an optimal polynomial time algorithm for the special case where G is a series-parallel graph. We also discuss special other cases where the partitioning problem or variations are polynomially solvable.

k-Efficient domination: Algorithmic perspective

2022 ◽  
Author(s):  
Mohsen Alambardar Meybodi
Keyword(s):  
Decision Problem ◽  
Dominating Set ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Sparse Graphs ◽  
Fpt Algorithm ◽  
Domination Problem ◽  
Efficient Dominating Set ◽  
Np Complete

A set [Formula: see text] of a graph [Formula: see text] is called an efficient dominating set of [Formula: see text] if every vertex [Formula: see text] has exactly one neighbor in [Formula: see text], in other words, the vertex set [Formula: see text] is partitioned to some circles with radius one such that the vertices in [Formula: see text] are the centers of partitions. A generalization of this concept, introduced by Chellali et al. [k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122], is called [Formula: see text]-efficient dominating set that briefly partitions the vertices of graph with different radiuses. It leads to a partition set [Formula: see text] such that each [Formula: see text] consists a center vertex [Formula: see text] and all the vertices in distance [Formula: see text], where [Formula: see text]. In other words, there exist the dominators with various dominating powers. The problem of finding minimum set [Formula: see text] is called the minimum [Formula: see text]-efficient domination problem. Given a positive integer [Formula: see text] and a graph [Formula: see text], the [Formula: see text]-efficient Domination Decision problem is to decide whether [Formula: see text] has a [Formula: see text]-efficient dominating set of cardinality at most [Formula: see text]. The [Formula: see text]-efficient Domination Decision problem is known to be NP-complete even for bipartite graphs [M. Chellali, T. W. Haynes and S. Hedetniemi, k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122]. Clearly, every graph has a [Formula: see text]-efficient dominating set but it is not correct for efficient dominating set. In this paper, we study the following: [Formula: see text]-efficient domination problem set is NP-complete even in chordal graphs. A polynomial-time algorithm for [Formula: see text]-efficient domination in trees. [Formula: see text]-efficient domination on sparse graphs from the parametrized complexity perspective. In particular, we show that it is [Formula: see text]-hard on d-degenerate graphs while the original dominating set has Fixed Parameter Tractable (FPT) algorithm on d-degenerate graphs. [Formula: see text]-efficient domination on nowhere-dense graphs is FPT.

scholarly journals On the rectangular knapsack problem: approximation of a specific quadratic knapsack problem

2020 ◽  
Vol 92 (1) ◽  
pp. 107-132 ◽  
Author(s):  
Britta Schulze ◽  
Michael Stiglmayr ◽  
Luís Paquete ◽  
Carlos M. Fonseca ◽  
David Willems ◽  
...  
Keyword(s):  
Knapsack Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Approximation Ratio ◽  
Experimental Results ◽  
Cardinality Constraint ◽  
Quadratic Knapsack Problem ◽  
Problem Instances ◽  
Quadratic Knapsack ◽  
Special Case

Abstract In this article, we introduce the rectangular knapsack problem as a special case of the quadratic knapsack problem consisting in the maximization of the product of two separate knapsack profits subject to a cardinality constraint. We propose a polynomial time algorithm for this problem that provides a constant approximation ratio of 4.5. Our experimental results on a large number of artificially generated problem instances show that the average ratio is far from theoretical guarantee. In addition, we suggest refined versions of this approximation algorithm with the same time complexity and approximation ratio that lead to even better experimental results.

scholarly journals IT IS NL-COMPLETE TO DECIDE WHETHER A HAIRPIN COMPLETION OF REGULAR LANGUAGES IS REGULAR

2011 ◽  
Vol 22 (08) ◽  
pp. 1813-1828 ◽  
Author(s):  
VOLKER DIEKERT ◽  
STEFFEN KOPECKI
Keyword(s):  
Polynomial Time ◽  
Dna Computing ◽  
Regular Language ◽  
Decision Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Complexity Bound ◽  
The One ◽  
Context Free ◽  
Hairpin Formation

The hairpin completion is an operation on formal languages which is inspired by the hairpin formation in biochemistry. Hairpin formations occur naturally within DNA-computing. It has been known that the hairpin completion of a regular language is linear context-free, but not regular, in general. However, for some time it is was open whether the regularity of the hairpin completion of a regular language is decidable. In 2009 this decidability problem has been solved positively in [5] by providing a polynomial time algorithm. In this paper we improve the complexity bound by showing that the decision problem is actually NL-complete. This complexity bound holds for both, the one-sided and the two-sided hairpin completions.

THE EDIT-DISTANCE BETWEEN A REGULAR LANGUAGE AND A CONTEXT-FREE LANGUAGE

2013 ◽  
Vol 24 (07) ◽  
pp. 1067-1082 ◽  
Author(s):  
YO-SUB HAN ◽  
SANG-KI KO ◽  
KAI SALOMAA
Keyword(s):  
Regular Language ◽  
Edit Distance ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Optimal Alignment ◽  
Worst Case ◽  
Context Free Language ◽  
Context Free ◽  
Free Language

The edit-distance between two strings is the smallest number of operations required to transform one string into the other. The distance between languages L1and L2is the smallest edit-distance between string wi∈ Li, i = 1, 2. We consider the problem of computing the edit-distance of a given regular language and a given context-free language. First, we present an algorithm that finds for the languages an optimal alignment, that is, a sequence of edit operations that transforms a string in one language to a string in the other. The length of the optimal alignment, in the worst case, is exponential in the size of the given grammar and finite automaton. Then, we investigate the problem of computing only the edit-distance of the languages without explicitly producing an optimal alignment. We design a polynomial time algorithm that calculates the edit-distance based on unary homomorphisms.

scholarly journals Maximin-Aware Allocations of Indivisible Goods

2019 ◽  
Author(s):  
Hau Chan ◽  
Jing Chen ◽  
Bo Li ◽  
Xiaowei Wu
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Indivisible Goods ◽  
Know How

We study envy-free allocations of indivisible goods to agents in settings where each agent is unaware of the goods allocated to other agents. In particular, we propose the maximin aware (MMA) fairness measure, which guarantees that every agent, given the bundle allocated to her, is aware that she does not envy at least one other agent, even if she does not know how the other goods are distributed among other agents. We also introduce two of its relaxations, and discuss their egalitarian guarantee and existence. Finally, we present a polynomial-time algorithm, which computes an allocation that approximately satisfies MMA or its relaxations. Interestingly, the returned allocation is also 1/2-approximate EFX when all agents have sub- additive valuations, which improves the algorithm in [Plaut and Roughgarden, 2018].

scholarly journals Polynomial Time Instances for the IKHO Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Romeo Rizzi ◽  
Luca Nardin
Keyword(s):  
Dynamic Programming ◽  
Polynomial Time ◽  
Broad Class ◽  
Structural Parameters ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Heuristic Optimization ◽  
Negative Fact

The Interactive Knapsacks Heuristic Optimization (IKHO) problem is a particular knapsacks model in which, given an array of knapsacks, every insertion in a knapsack affects also the other knapsacks, in terms of weight and profit. The IKHO model was introduced by Isto Aho to model instances of the load clipping problem. The IKHO problem is known to be APX-hard and, motivated by this negative fact, Aho exhibited a few classes of polynomial instances for the IKHO problem. These instances were obtained by limiting the ranges of two structural parameters, c and u, which describe the extent to which an insertion in a knapsack in uences the nearby knapsacks. We identify a new and broad class of instances allowing for a polynomial time algorithm. More precisely, we show that the restriction of IKHO to instances where is bounded by a constant can be solved in polynomial time, using dynamic programming.

scholarly journals Tree Reconstruction from Triplet Cover Distances

10.37236/3388 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Katharina T. Huber ◽  
Mike Steel
Keyword(s):  
Polynomial Time ◽  
Classical Result ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Interior Vertex ◽  
Evolutionary Genomics ◽  
Tree Reconstruction ◽  
Finite Tree ◽  
Path Distance ◽  
Special Case

It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict subset $\mathcal{L}$ of leaf pairs to suffice for reconstructing the tree and its edge weights, given just the distances between the leaf pairs in $\mathcal{L}$. It is known that any set ${\mathcal L}$ with this property for a tree in which all interior vertices have degree 3 must form a cover  for $T$ - that is, for each interior vertex $v$ of $T$, ${\mathcal L}$ must contain a pair of leaves from each pair of the three components of  $T-v$.  Here we provide a partial converse of this result by showing that if a set ${\mathcal L}$ of leaf pairs forms a cover  of a certain type for such a tree $T$ then $T$ and its edge weights can be uniquely determined from the distances between the pairs of leaves in ${\mathcal L}$. Moreover,  there is a polynomial-time algorithm for achieving this reconstruction. The result establishes a special case of a recent question concerning 'triplet covers', and is relevant to a problem arising in evolutionary genomics.

scholarly journals Complexity of Strong Approximation on the Sphere

2019 ◽  
Author(s):  
Naser T Sardari
Keyword(s):  
Polynomial Time ◽  
Strong Approximation ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Ramanujan Graphs ◽  
Other Hand ◽  
Np Complete

Abstract By assuming some widely believed arithmetic conjectures, we show that the task of accepting a number that is representable as a sum of $d\geq 2$ squares subjected to given congruence conditions is NP-complete. On the other hand, we develop and implement a deterministic polynomial-time algorithm that represents a number as a sum of four squares with some restricted congruence conditions, by assuming a polynomial-time algorithm for factoring integers and Conjecture 1.1. As an application, we develop and implement a deterministic polynomial-time algorithm for navigating Lubotzky, Phillips, Sarnak (LPS) Ramanujan graphs, under the same assumptions.

Efficient Processing of XML Tree Pattern Queries

2006 ◽  
Vol 10 (5) ◽  
pp. 738-743 ◽  
Author(s):  
Yangjun Chen ◽  
◽  
Dunren Che ◽  
Keyword(s):  
Dynamic Programming ◽  
Polynomial Time ◽  
Main Idea ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Tree Pattern ◽  
Efficient Processing

In this paper, we present a polynomial-time algorithm for TPQ (tree pattern queries) minimization without XML constraints involved. The main idea of the algorithm is a dynamic programming strategy to find all the matching subtrees within a TPQ. A matching subtree implies a redundancy and should be removed in such a way that the semantics of the original TPQ is not damaged. Our algorithm consists of two parts: one for subtree recognization and the other for subtree deletion. Both of them needs only O(<I>n</I>2) time, where <I>n</I> is the number of nodes in a TPQ.

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