COMPARING AND AGGREGATING PARTIAL ORDERS WITH KENDALL TAU DISTANCES

Comparing and ranking information is an important topic in social and information sciences, and in particular on the web. Its objective is to measure the difference of the preferences of voters on a set of candidates and to compute a consensus ranking. Commonly, each voter provides a total order or a bucket order of all candidates, where bucket orders allow ties. In this work we consider the generalization of total and bucket orders to partial orders and compare them by the nearest neighbor and the Hausdorff Kendall tau distances. For total and bucket orders these distances can be computed in [Formula: see text] time. We show that the computation of the nearest neighbor Kendall tau distance is NP-hard, 2-approximable and fixed-parameter tractable for a total and a partial order. The computation of the Hausdorff Kendall tau distance for a total and a partial order is shown to be coNP-hard. The rank aggregation problem is known to be NP-complete for total and bucket orders, even for four voters and solvable in [Formula: see text] time for two voters. We show that it is NP-complete for two partial orders and the nearest neighbor Kendall tau distance. For the Hausdorff Kendall tau distance it is in [Formula: see text], but not in NP or coNP unless NP = coNP, even for four voters.

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Maximum Exact Satisfiability: NP-completeness Proofs and Exact Algorithms

BRICS Report Series ◽

10.7146/brics.v11i19.21844 ◽

2004 ◽

Vol 11 (19) ◽

Cited By ~ 3

Author(s):

Bolette Ammitzbøll Madsen ◽

Peter Rossmanith

Keyword(s):

Time Complexity ◽

Exact Algorithm ◽

Conjunctive Normal Form ◽

Exact Algorithms ◽

Fixed Parameter Tractable ◽

Complete Problem ◽

Satisfiability Problems ◽

Fixed Parameter ◽

Np Complete ◽

Open Question

Inspired by the Maximum Satisfiability and Exact Satisfiability problems we present two Maximum Exact Satisfiability problems. The first problem called Maximum Exact Satisfiability is: given a formula in conjunctive normal form and an integer k, is there an assignment to all variables in the formula such that at least k clauses have exactly one true literal. The second problem called Restricted Maximum Exact Satisfiability has the further restriction that no clause is allowed to have more than one true literal. Both problems are proved NP-complete restricted to the versions where each clause contains at most two literals. In fact Maximum Exact Satisfiability is a generalisation of the well-known NP-complete problem MaxCut. We present an exact algorithm for Maximum Exact Satisfiability where each clause contains at most two literals with time complexity O(poly(L) . 2^{m/4}), where m is the number of clauses and L is the length of the formula. For the second version we give an algorithm with time complexity O(poly(L) . 1.324718^n) , where n is the number of variables. We note that when restricted to the versions where each clause contains exactly two literals and there are no negations both problems are fixed parameter tractable. It is an open question if this is also the case for the general problems.

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FPT-ALGORITHMS FOR MINIMUM-BENDS TOURS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195911003615 ◽

2011 ◽

Vol 21 (02) ◽

pp. 189-213 ◽

Cited By ~ 4

Author(s):

VLADIMIR ESTIVILL-CASTRO ◽

APICHAT HEEDNACRAM ◽

FRANCIS SURAWEERA

Keyword(s):

Traveling Salesman Problem ◽

Traveling Salesman ◽

Fixed Parameter Tractable ◽

Line Segments ◽

Fpt Algorithms ◽

Fixed Parameter ◽

Axis Parallel ◽

Np Complete ◽

Number Of Bends ◽

Straight Lines

This paper discusses the κ-BENDS TRAVELING SALESMAN PROBLEM. In this NP-complete problem, the inputs are n points in the plane and a positive integer κ, and we are asked whether we can travel in straight lines through these n points with at most κ bends. There are a number of applications where minimizing the number of bends in the tour is desirable because bends are considered very costly. We prove that this problem is fixed-parameter tractable (FPT). The proof is based on the kernelization approach. We also consider the RECTILINEAR κ-BENDS TRAVELING SALESMAN PROBLEM, which requires that the line-segments be axis-parallel. 1 Note that a rectilinear tour with κ bends is a cover with κ-line segments, and therefore a cover by lines. We introduce two types of constraints derived from the distinction between line-segments and lines. We derive FPT-algorithms with different techniques and improved time complexity for these cases.

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Complexidade Parametrizada de Cliques e Conjuntos Independentes em Grafos Prismas Complementares

10.5753/etc.2018.3166 ◽

2018 ◽

Author(s):

Priscila Camargo ◽

Alan D. A. Carneiro ◽

Uéverton S. Santos

Keyword(s):

Disjoint Union ◽

Perfect Matching ◽

Independent Set ◽

Point Of View ◽

Polynomial Kernel ◽

Input Graph ◽

Fixed Parameter Tractable ◽

Fixed Parameter ◽

Np Complete ◽

Complementary Prism

The complementary prism GG¯ arises from the disjoint union of the graph G and its complement G¯ by adding the edges of a perfect matching joining pairs of corresponding vertices of G and G¯. The classical problems of graph theory, clique and independent set were proved NP-complete when the input graph is a complemantary prism. In this work, we study the complexity of both problems in complementary prisms graphs from the parameterized complexity point of view. First, we prove that these problems have a kernel and therefore are Fixed-Parameter Tractable (FPT). Then, we show that both problems do not admit polynomial kernel.

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Remarks on k-Clique, k-Independent Set and 2-Contamination in Complementary Prisms

International Journal of Foundations of Computer Science ◽

10.1142/s0129054121500027 ◽

2021 ◽

pp. 1-16

Author(s):

Priscila P. Camargo ◽

Uéverton S. Souza ◽

Julliano R. Nascimento

Keyword(s):

Disjoint Union ◽

Independent Set ◽

Point Of View ◽

Polynomial Kernel ◽

Fixed Parameter Tractable ◽

Graph Problems ◽

Fixed Parameter ◽

Np Complete ◽

Complementary Prism ◽

Contamination Problem

Complementary prism graphs arise from the disjoint union of a graph [Formula: see text] and its complement [Formula: see text] by adding the edges of a perfect matching joining pairs of corresponding vertices of [Formula: see text] and [Formula: see text]. Classical graph problems such as Clique and Independent Set were proved to be NP-complete on such a class of graphs. In this work, we study the complexity of both problems on complementary prism graphs from the parameterized complexity point of view. First, we prove that both problems admit a kernel and therefore are fixed-parameter tractable (FPT) when parameterized by the size of the solution, [Formula: see text]. Then, we show that [Formula: see text]-Clique and [Formula: see text]-Independent Set on complementary prisms do not admit polynomial kernel when parameterized by [Formula: see text], unless [Formula: see text]. Furthermore, we address the [Formula: see text]-Contamination problem in the context of complementary prisms. This problem consists in completely contaminating a given graph [Formula: see text] using a minimum set of initially infected vertices. For a vertex to be contaminated, it is enough that at least two of its neighbors are contaminated. The propagation of the contamination follows this rule until no more vertex can be contaminated. It is known that the minimum set of initially contaminated vertices necessary to contaminate a complementary prism of connected graphs [Formula: see text] and [Formula: see text] has cardinality at most [Formula: see text]. In this paper, we show that the tight upper bound for this invariant on complementary prisms is [Formula: see text], improving a result of Duarte et al. (2017).

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THE NATURAL PARTIAL ORDER ON LINEAR SEMIGROUPS WITH NULLITY AND CO-RANK BOUNDED BELOW

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972714000793 ◽

2014 ◽

Vol 91 (1) ◽

pp. 104-115 ◽

Cited By ~ 1

Author(s):

SUREEPORN CHAOPRAKNOI ◽

TEERAPHONG PHONGPATTANACHAROEN ◽

PONGSAN PRAKITSRI

Keyword(s):

Semigroup Forum ◽

Partial Order ◽

Lower Bounds ◽

Partial Orders ◽

Natural Partial Order ◽

Linear Transformations ◽

Bounded Below

AbstractHiggins [‘The Mitsch order on a semigroup’, Semigroup Forum 49 (1994), 261–266] showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are interested in the study of the natural partial order on nonregular semigroups. Of particular interest are the nonregular semigroups of linear transformations with lower bounds on the nullity or the co-rank. In this paper, we determine when they exist, characterise the natural partial order on these nonregular semigroups and consider questions of compatibility, minimality and maximality. In addition, we provide many examples associated with our results.

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A Genetic Algorithm Approach for Group Recommender System Based on Partial Rankings

Journal of Intelligent Systems ◽

10.1515/jisys-2017-0561 ◽

2018 ◽

Vol 29 (1) ◽

pp. 653-663 ◽

Cited By ~ 1

Author(s):

Ritu Meena ◽

Kamal K. Bharadwaj

Keyword(s):

Genetic Algorithm ◽

Recommender Systems ◽

Rank Aggregation ◽

Partial Ranking ◽

Novel Approach ◽

Kendall Tau Distance ◽

Ultimate Objective ◽

Aggregation Technique ◽

The Individual ◽

Group Recommender

Abstract Many recommender systems frequently make suggestions for group consumable items to the individual users. There has been much work done in group recommender systems (GRSs) with full ranking, but partial ranking (PR) where items are partially ranked still remains a challenge. The ultimate objective of this work is to propose rank aggregation technique for effectively handling the PR problem. Additionally, in real applications, most of the studies have focused on PR without ties (PRWOT). However, the rankings may have ties where some items are placed in the same position, but where some items are partially ranked to be aggregated may not be permutations. In this work, in order to handle problem of PR in GRS for PRWOT and PR with ties (PRWT), we propose a novel approach to GRS based on genetic algorithm (GA) where for PRWOT Spearman foot rule distance and for PRWT Kendall tau distance with bucket order are used as fitness functions. Experimental results are presented that clearly demonstrate that our proposed GRS based on GA for PRWOT (GRS-GA-PRWOT) and PRWT (GRS-GA-PRWT) outperforms well-known baseline GRS techniques.

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