scholarly journals MDD Propagation for Sequence Constraints

2014 ◽  
Vol 50 ◽  
pp. 697-722 ◽  
Author(s):  
D. Bergman ◽  
A. A. Cire ◽  
W. Van Hoeve
Keyword(s):  
Constraint Programming ◽  
Search Tree ◽  
Tree Size ◽  
Np Hard ◽  
Maximum Width ◽  
Fixed Parameter Tractable ◽  
Filtering Algorithm ◽  
Sequence Constraint ◽  
Fixed Parameter ◽  
Sequence Constraints

We study propagation for the Sequence constraint in the context of constraint programming based on limited-width MDDs. Our first contribution is proving that establishing MDD-consistency for Sequence is NP-hard. Yet, we also show that this task is fixed parameter tractable with respect to the length of the sub-sequences. In addition, we propose a partial filtering algorithm that relies on a specific decomposition of the constraint and a novel extension of MDD filtering to node domains. We experimentally evaluate the performance of our proposed filtering algorithm, and demonstrate that the strength of the MDD propagation increases as the maximum width is increased. In particular, MDD propagation can outperform conventional domain propagation for Sequence by reducing the search tree size and solving time by several orders of magnitude. Similar improvements are observed with respect to the current best MDD approach that applies the decomposition of Sequence into Among constraints.

scholarly journals On Succinct Encodings for the Tournament Fixing Problem

2019 ◽  
Author(s):  
Sushmita Gupta ◽  
Saket Saurabh ◽  
Ramanujan Sridharan ◽  
Meirav Zehavi
Keyword(s):  
Polynomial Time ◽  
Complete Graph ◽  
Np Hard ◽  
Feedback Arc Set ◽  
Fixed Parameter Tractable ◽  
Competitive Environments ◽  
Fixed Parameter ◽  
Sat Encoding

Single-elimination tournaments are a popular format in competitive environments. The Tournament Fixing Problem (TFP), which is the problem of finding a seeding of the players such that a certain player wins the resulting tournament, is known to be NP-hard in general and fixed-parameter tractable when parameterized by the feedback arc set number of the input tournament (an oriented complete graph) of expected wins/loses. However, the existence of polynomial kernelizations (efficient preprocessing) for TFP has remained open. In this paper, we present the first polynomial kernelization for TFP parameterized by the feedback arc set number of the input tournament. We achieve this by providing a polynomial-time routine that computes a SAT encoding where the number of clauses is bounded polynomially in the feedback arc set number.

scholarly journals Dividing Splittable Goods Evenly and With Limited Fragmentation

Algorithmica ◽  
2019 ◽  
Vol 82 (5) ◽  
pp. 1298-1328
Author(s):  
Peter Damaschke
Keyword(s):  
Related Problem ◽  
Linear Time ◽  
Np Hard ◽  
Weighted Graphs ◽  
Fixed Parameter Tractable ◽  
Averaging Problem ◽  
Fixed Parameter ◽  
Splitting Problem ◽  
Structural Characterizations ◽  
Balanced Solutions

Abstract A splittable good provided in n pieces shall be divided as evenly as possible among m agents, where every agent can take shares from at most F pieces. We call F the fragmentation and mainly restrict attention to the cases $$F=1$$F=1 and $$F=2$$F=2. For $$F=1$$F=1, the max–min and min–max problems are solvable in linear time. The case $$F=2$$F=2 has neat formulations and structural characterizations in terms of weighted graphs. First we focus on perfectly balanced solutions. While the problem is strongly NP-hard in general, it can be solved in linear time if $$m\ge n-1$$m≥n-1, and a solution always exists in this case, in contrast to $$F=1$$F=1. Moreover, the problem is fixed-parameter tractable in the parameter $$2m-n$$2m-n. (Note that this parameter measures the number of agents above the trivial threshold $$m=n/2$$m=n/2.) The structural results suggest another related problem where unsplittable items shall be assigned to subsets so as to balance the average sizes (rather than the total sizes) in these subsets. We give an approximation-preserving reduction from our original splitting problem with fragmentation $$F=2$$F=2 to this averaging problem, and some approximation results in cases when m is close to either n or n / 2.

scholarly journals Multivariate Investigation of NP-Hard Problems: Boundaries Between Parameterized Tractability and Intractability

2020 ◽  
Author(s):  
Uéverton Souza ◽  
Fábio Protti ◽  
Maise Da Silva ◽  
Dieter Rautenbach
Keyword(s):  
Polynomial Time ◽  
Parameterized Complexity ◽  
Complexity Analysis ◽  
Np Hard ◽  
Fixed Parameter Tractable ◽  
Complementary Approach ◽  
Fixed Parameter ◽  
Hard Problems ◽  
Induced Matchings ◽  
Np Hard Problems

In this thesis we present a multivariate investigation of the complexity of some NP-hard problems, i.e., we first develop a systematic complexity analysis of these problems, defining its subproblems and mapping which one belongs to each side of an “imaginary boundary” between polynomial time solvability and intractability. After that, we analyze which sets of aspects of these problems are sources of their intractability, that is, subsets of aspects for which there exists an algorithm to solve the associated problem, whose non-polynomial time complexity is purely a function of those sets. Thus, we use classical and parameterized complexity in an alternate and complementary approach, to show which subproblems of the given problems are NP-hard and latter to diagnose for which sets of parameters the problems are fixed-parameter tractable, or in FPT. This thesis exhibits a classical and parameterized complexity analysis of different groups of NP-hard problems. The addressed problems are divided into four groups of distinct nature, in the context of data structures, combinatorial games, and graph theory: (I) and/or graph solution and its variants; (II) flooding-filling games; (III) problems on P3-convexity; (IV) problems on induced matchings.

scholarly journals Krausz dimension and its generalizations in special graph classes

2013 ◽  
Vol Vol. 15 no. 1 (Graph Theory) ◽  
Author(s):  
Olga Glebova ◽  
Yury Metelsky ◽  
Pavel Skums
Keyword(s):  
Open Problem ◽  
Chordal Graphs ◽  
Np Hard ◽  
Induced Subgraphs ◽  
Fixed Parameter Tractable ◽  
Graph Classes ◽  
Fixed Parameter ◽  
Minimum Number ◽  
International Audience ◽  
Split Graphs

Graph Theory International audience A Krausz (k,m)-partition of a graph G is a decomposition of G into cliques, such that any vertex belongs to at most k cliques and any two cliques have at most m vertices in common. The m-Krausz dimension kdimm(G) of the graph G is the minimum number k such that G has a Krausz (k,m)-partition. In particular, 1-Krausz dimension or simply Krausz dimension kdim(G) is a well-known graph-theoretical parameter. In this paper we prove that the problem "kdim(G)≤3" is polynomially solvable for chordal graphs, thus partially solving the open problem of P. Hlineny and J. Kratochvil. We solve another open problem of P. Hlineny and J. Kratochvil by proving that the problem of finding Krausz dimension is NP-hard for split graphs and complements of bipartite graphs. We show that the problem of finding m-Krausz dimension is NP-hard for every m≥1, but the problem "kdimm(G)≤k" is is fixed-parameter tractable when parameterized by k and m for (∞,1)-polar graphs. Moreover, the class of (∞,1)-polar graphs with kdimm(G)≤k is characterized by a finite list of forbidden induced subgraphs for every k,m≥1.

scholarly journals Towards fixed-parameter tractable algorithms for abstract argumentation

2012 ◽  
Vol 186 ◽  
pp. 1-37 ◽  
Author(s):  
Wolfgang Dvořák ◽  
Reinhard Pichler ◽  
Stefan Woltran
Keyword(s):  
Fixed Parameter Tractable ◽  
Abstract Argumentation ◽  
Fixed Parameter

scholarly journals Interval Completion Is Fixed Parameter Tractable

2009 ◽  
Vol 38 (5) ◽  
pp. 2007-2020 ◽  
Author(s):  
Yngve Villanger ◽  
Pinar Heggernes ◽  
Christophe Paul ◽  
Jan Arne Telle
Keyword(s):  
Fixed Parameter Tractable ◽  
Fixed Parameter

When is Red-Blue Nonblocker Fixed-Parameter Tractable?

2018 ◽  
pp. 515-528
Author(s):  
Serge Gaspers ◽  
Joachim Gudmundsson ◽  
Michael Horton ◽  
Stefan Rümmele
Keyword(s):  
Fixed Parameter Tractable ◽  
Fixed Parameter

scholarly journals Threshold Treewidth and Hypertree Width

2020 ◽  
Author(s):  
Robert Ganian ◽  
Andre Schidler ◽  
Manuel Sorge ◽  
Stefan Szeider
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Structural Parameters ◽  
Exact Methods ◽  
Fixed Parameter Tractable ◽  
Computational Costs ◽  
Fixed Parameter ◽  
Account Information ◽  
The Given ◽  
Theoretical Results

Treewidth and hypertree width have proven to be highly successful structural parameters in the context of the Constraint Satisfaction Problem (CSP). When either of these parameters is bounded by a constant, then CSP becomes solvable in polynomial time. However, here the order of the polynomial in the running time depends on the width, and this is known to be unavoidable; therefore, the problem is not fixed-parameter tractable parameterized by either of these width measures. Here we introduce an enhancement of tree and hypertree width through a novel notion of thresholds, allowing the associated decompositions to take into account information about the computational costs associated with solving the given CSP instance. Aside from introducing these notions, we obtain efficient theoretical as well as empirical algorithms for computing threshold treewidth and hypertree width and show that these parameters give rise to fixed-parameter algorithms for CSP as well as other, more general problems. We complement our theoretical results with experimental evaluations in terms of heuristics as well as exact methods based on SAT/SMT encodings.

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