scholarly journals Maximum Exact Satisfiability: NP-completeness Proofs and Exact Algorithms

2004 ◽  
Vol 11 (19) ◽  
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.

scholarly journals Algorithms for Necklace Maps

2015 ◽  
Vol 25 (01) ◽  
pp. 15-36 ◽  
Author(s):  
Bettina Speckmann ◽  
Kevin Verbeek
Keyword(s):  
Exact Algorithm ◽  
Dimensional Problem ◽  
Order Problem ◽  
Fixed Parameter Tractable ◽  
One Dimensional ◽  
Algorithmic Question ◽  
Fixed Parameter ◽  
Fixed Order ◽  
The One ◽  
Decision Version

Necklace maps visualize quantitative data associated with regions by placing scaled symbols, usually disks, without overlap on a closed curve (the necklace) surrounding the map regions. Each region is projected onto an interval on the necklace that contains its symbol. In this paper we address the algorithmic question how to maximize symbol sizes while keeping symbols disjoint and inside their intervals. For that we reduce the problem to a one-dimensional problem which we solve efficiently. Solutions to the one-dimensional problem provide a very good approximation for the original necklace map problem. We consider two variants: Fixed-Order, where an order for the symbols on the necklace is given, and Any-Order where any symbol order is possible. The Fixed-Order problem can be solved in O(n log n) time. We show that the Any-Order problem is NP-hard for certain types of intervals and give an exact algorithm for the decision version. This algorithm is fixed-parameter tractable in the thickness K of the input. Our algorithm runs in O(n log n + n2K4K) time which can be improved to O(n log n + nK2K) time using a heuristic. We implemented our algorithm and evaluated it experimentally.

COMPARING AND AGGREGATING PARTIAL ORDERS WITH KENDALL TAU DISTANCES

2013 ◽  
Vol 05 (02) ◽  
pp. 1360003 ◽  
Author(s):  
FRANZ J. BRANDENBURG ◽  
ANDREAS GLEIßNER ◽  
ANDREAS HOFMEIER
Keyword(s):  
Partial Order ◽  
Nearest Neighbor ◽  
Partial Orders ◽  
Rank Aggregation ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
Kendall Tau Distance ◽  
Np Complete ◽  
The Difference ◽  
Ranking Information

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.

scholarly journals FPT-ALGORITHMS FOR MINIMUM-BENDS TOURS

2011 ◽  
Vol 21 (02) ◽  
pp. 189-213 ◽  
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.

scholarly journals A Multivariate Complexity Analysis of Lobbying in Multiple Referenda

2014 ◽  
Vol 50 ◽  
pp. 409-446 ◽  
Author(s):  
R. Bredereck ◽  
J. Chen ◽  
S. Hartung ◽  
S. Kratsch ◽  
R. Niedermeier ◽  
...  
Keyword(s):  
Computational Complexity ◽  
Complexity Analysis ◽  
Central Question ◽  
Logarithmic Factor ◽  
Fixed Parameter Tractable ◽  
Complete Problem ◽  
Fixed Parameter ◽  
Matrix Modification ◽  
Simple Matrix ◽  
Limited Nondeterminism

Assume that each of n voters may or may not approve each of m issues. If an agent (the lobby) may influence up to k voters, then the central question of the NP-hard Lobbying problem is whether the lobby can choose the voters to be influenced so that as a result each issue gets a majority of approvals. This problem can be modeled as a simple matrix modification problem: Can one replace k rows of a binary n x m-matrix by k all-1 rows such that each column in the resulting matrix has a majority of 1s? Significantly extending on previous work that showed parameterized intractability (W[2]-completeness) with respect to the number k of modified rows, we study how natural parameters such as n, m, k, or the "maximum number of 1s missing for any column to have a majority of 1s" (referred to as "gap value g") govern the computational complexity of Lobbying. Among other results, we prove that Lobbying is fixed-parameter tractable for parameter m and provide a greedy logarithmic-factor approximation algorithm which solves Lobbying even optimally if m < 5. We also show empirically that this greedy algorithm performs well on general instances. As a further key result, we prove that Lobbying is LOGSNP-complete for constant values g>0, thus providing a first natural complete problem from voting for this complexity class of limited nondeterminism.

scholarly journals Parameterized Intractability of Even Set and Shortest Vector Problem

2021 ◽  
Vol 68 (3) ◽  
pp. 1-40
Author(s):  
Arnab Bhattacharyya ◽  
Édouard Bonnet ◽  
László Egri ◽  
Suprovat Ghoshal ◽  
Karthik C. S. ◽  
...  
Keyword(s):  
Linear Codes ◽  
Constant Factor ◽  
Nonzero Vector ◽  
Fixed Parameter Tractable ◽  
Vector Problem ◽  
Shortest Vector Problem ◽  
Open Questions ◽  
Fixed Parameter ◽  
Distance Problem ◽  
Open Question

The -Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over , which can be stated as follows: given a generator matrix and an integer , determine whether the code generated by has distance at most , or, in other words, whether there is a nonzero vector such that has at most nonzero coordinates. The question of whether -Even Set is fixed parameter tractable (FPT) parameterized by the distance has been repeatedly raised in the literature; in fact, it is one of the few remaining open questions from the seminal book of Downey and Fellows [1999]. In this work, we show that -Even Set is W [1]-hard under randomized reductions. We also consider the parameterized -Shortest Vector Problem (SVP) , in which we are given a lattice whose basis vectors are integral and an integer , and the goal is to determine whether the norm of the shortest vector (in the norm for some fixed ) is at most . Similar to -Even Set, understanding the complexity of this problem is also a long-standing open question in the field of Parameterized Complexity. We show that, for any , -SVP is W [1]-hard to approximate (under randomized reductions) to some constant factor.

scholarly journals On Minimum Representations of Matched Formulas

2014 ◽  
Vol 51 ◽  
pp. 707-723
Author(s):  
O. Cepek ◽  
S. Gursky ◽  
P. Kucera
Keyword(s):  
Normal Form ◽  
Boolean Function ◽  
Minimization Problem ◽  
Conjunctive Normal Form ◽  
Boolean Formula ◽  
Polynomial Hierarchy ◽  
Partial Assignment ◽  
Complete Problem ◽  
Boolean Minimization ◽  
Np Complete

A Boolean formula in conjunctive normal form (CNF) is called matched if the system of sets of variables which appear in individual clauses has a system of distinct representatives. Each matched CNF is trivially satisfiable (each clause can be satisfied by its representative variable). Another property which is easy to see, is that the class of matched CNFs is not closed under partial assignment of truth values to variables. This latter property leads to a fact (proved here) that given two matched CNFs it is co-NP complete to decide whether they are logically equivalent. The construction in this proof leads to another result: a much shorter and simpler proof of the fact that the Boolean minimization problem for matched CNFs is a complete problem for the second level of the polynomial hierarchy. The main result of this paper deals with the structure of clause minimum CNFs. We prove here that if a Boolean function f admits a representation by a matched CNF then every clause minimum CNF representation of f is matched.

scholarly journals Complexidade Parametrizada de Cliques e Conjuntos Independentes em Grafos Prismas Complementares

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.

On Random Betweenness Constraints

2010 ◽  
Vol 19 (5-6) ◽  
pp. 775-790 ◽  
Author(s):  
ANDREAS GOERDT
Keyword(s):  
Normal Form ◽  
High Probability ◽  
Conjunctive Normal Form ◽  
Satisfiability Problem ◽  
Linear Ordering ◽  
Satisfiability Problems ◽  
Ordering Constraints ◽  
Random Instances ◽  
Np Complete

Ordering constraints are formally analogous to instances of the satisfiability problem in conjunctive normal form, but instead of a boolean assignment we consider a linear ordering of the variables in question. A clause becomes true given a linear ordering if and only if the relative ordering of its variables obeys the constraint considered.The naturally arising satisfiability problems are NP-complete for many types of constraints. We look at random ordering constraints. Previous work of the author shows that there is a sharp unsatisfiability threshold for certain types of constraints. The value of the threshold, however, is essentially undetermined. We pursue the problem of approximating the precise value of the threshold. We show that random instances of the betweenness constraint are satisfiable with high probability if the number of randomly picked clauses is ≤0.92n, where n is the number of variables considered. This improves the previous bound, which is <0.82n random clauses. The proof is based on a binary relaxation of the betweenness constraint and involves some ideas not used before in the area of random ordering constraints.

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

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).

scholarly journals Popular Matching in Roommates Setting Is NP-hard

2021 ◽  
Vol 13 (2) ◽  
pp. 1-20
Author(s):  
Sushmita Gupta ◽  
Pranabendu Misra ◽  
Saket Saurabh ◽  
Meirav Zehavi
Keyword(s):  
Computational Complexity ◽  
Np Hard ◽  
Np Complete ◽  
Open Question ◽  
Popular Matching

An input to the P OPULAR M ATCHING problem, in the roommates setting (as opposed to the marriage setting), consists of a graph G (not necessarily bipartite) where each vertex ranks its neighbors in strict order, known as its preference. In the P OPULAR M ATCHING problem the objective is to test whether there exists a matching M * such that there is no matching M where more vertices prefer their matched status in M (in terms of their preferences) over their matched status in M *. In this article, we settle the computational complexity of the P OPULAR M ATCHING problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly and explicitly asked over the last decade.

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