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.

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).

Multiplicative Parameterization Above a Guarantee

2021 ◽  
Vol 13 (3) ◽  
pp. 1-16
Author(s):  
Fedor V. Fomin ◽  
Petr A. Golovach ◽  
Daniel Lokshtanov ◽  
Fahad Panolan ◽  
Saket Saurabh ◽  
...  
Keyword(s):  
Minimum Size ◽  
Input Graph ◽  
Fixed Parameter Tractable ◽  
Long Cycle ◽  
Fixed Parameter ◽  
Parameterized Problem ◽  
Fixed Constant ◽  
Fixed Parameter Algorithm ◽  
Np Hardness ◽  
Additive Form

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k + g(I) . Here, g(I) is usually a lower bound on the minimum size of a solution. Since its introduction in 1999 for M AX SAT and M AX C UT (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above (or, rather, times) a guarantee: Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k · g(I) . In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, which is the most natural guarantee for this problem, and provide a fixed-parameter algorithm. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ε > 0, multiplicative parameterization above g(I) 1+ε of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of fixed-parameter algorithms as well as kernelization algorithms for additional problems parameterized multiplicatively above girth.

On the Complexity of Reverse Minus and Signed Domination on Graphs

2015 ◽  
Vol 15 (01n02) ◽  
pp. 1550008
Author(s):  
CHUAN-MIN LEE ◽  
CHENG-CHIEN LO
Keyword(s):  
Planar Graphs ◽  
Decision Problem ◽  
Linear Time ◽  
Point Of View ◽  
Chordal Graphs ◽  
Fixed Parameter Tractable ◽  
Signed Domination ◽  
Domination Problem ◽  
Np Complete ◽  
Domination Problems

Motivated by the concept of reverse signed domination, we introduce the reverse minus domination problem on graphs, and study the reverse minus and signed domination problems from the algorithmic point of view. In this paper, we show that both the reverse minus and signed domination problems are polynomial-time solvable for strongly chordal graphs and distance-hereditary graphs, and are linear-time solvable for trees. For chordal graphs and bipartite planar graphs, however, we show that the decision problem corresponding to the reverse minus domination problem is NP-complete. For doubly chordal graphs and bipartite planar graphs, we show that the decision problem corresponding to the reverse signed domination problem is NP-complete. Furthermore, we show that even when restricted to bipartite planar graphs or doubly chordal graphs, the reverse signed domination problem is not fixed parameter tractable.

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.

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 On the General Position Number of Complementary Prisms

2021 ◽  
Vol 178 (3) ◽  
pp. 267-281
Author(s):  
P. K. Neethu ◽  
S.V. Ullas Chandran ◽  
Manoj Changat ◽  
Sandi Klavžar
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
General Position ◽  
Disjoint Union ◽  
Perfect Matching ◽  
Split Graph ◽  
Block Graphs ◽  
Disconnected Graphs ◽  
Complementary Prism ◽  
Sharp Lower Bound

The general position number gp(G) of a graph G is the cardinality of a largest set of vertices S such that no element of S lies on a geodesic between two other elements of S. The complementary prism G G ¯ of G is the graph formed from the disjoint union of G and its complement G ¯ by adding the edges of a perfect matching between them. It is proved that gp(G G ¯ ) ≤ n(G) + 1 if G is connected and gp(G G ¯ ) ≤ n(G) if G is disconnected. Graphs G for which gp(G G ¯ ) = n(G) + 1 holds, provided that both G and G ¯ are connected, are characterized. A sharp lower bound on gp(G G ¯ ) is proved. If G is a connected bipartite graph or a split graph then gp(G G ¯ ) ∈ {n(G), n(G)+1}. Connected bipartite graphs and block graphs for which gp(G G ¯ ) = n(G) + 1 holds are characterized. A family of block graphs is constructed in which the gp-number of their complementary prisms is arbitrary smaller than their order.

scholarly journals Recognizing Generating Subgraphs Revisited

2021 ◽  
Vol 32 (01) ◽  
pp. 93-114
Author(s):  
Vadim E. Levit ◽  
David Tankus
Keyword(s):  
Weight Function ◽  
Independent Set ◽  
Independent Sets ◽  
Weight Functions ◽  
Polynomial Algorithms ◽  
Maximal Independent Set ◽  
Input Graph ◽  
Bipartite Subgraph ◽  
Np Complete ◽  
Maximal Independent Sets

A graph [Formula: see text] is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function [Formula: see text] is defined on its vertices. Then [Formula: see text] is [Formula: see text]well-covered if all maximal independent sets are of the same weight. For every graph [Formula: see text], the set of weight functions [Formula: see text] such that [Formula: see text] is [Formula: see text]-well-covered is a vector space, denoted as WCW(G). Deciding whether an input graph [Formula: see text] is well-covered is co-NP-complete. Therefore, finding WCW(G) is co-NP-hard. A generating subgraph of a graph [Formula: see text] is an induced complete bipartite subgraph [Formula: see text] of [Formula: see text] on vertex sets of bipartition [Formula: see text] and [Formula: see text], such that each of [Formula: see text] and [Formula: see text] is a maximal independent set of [Formula: see text], for some independent set [Formula: see text]. If [Formula: see text] is generating, then [Formula: see text] for every weight function [Formula: see text]. Therefore, generating subgraphs play an important role in finding WCW(G). The decision problem whether a subgraph of an input graph is generating is known to be NP-complete. In this article we prove NP- completeness of the problem for graphs without cycles of length 3 and 5, and for bipartite graphs with girth at least 6. On the other hand, we supply polynomial algorithms for recognizing generating subgraphs and finding WCW(G), when the input graph is bipartite without cycles of length 6. We also present a polynomial algorithm which finds WCW(G) when [Formula: see text] does not contain cycles of lengths 3, 4, 5, and 7.

scholarly journals On the geodetic hull number for complementary prisms II

2020 ◽  
Author(s):  
Diane Castonguay ◽  
Erika Morais Martins Coelho ◽  
Hebert Coelho ◽  
Julliano Nascimento
Keyword(s):  
Convex Hull ◽  
Shortest Path ◽  
Disjoint Union ◽  
Perfect Matching ◽  
Convex Set ◽  
Upper Bounds ◽  
The Other ◽  
Lower And Upper Bounds ◽  
Split Graph ◽  
Complementary Prism

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is \textit{convex} if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The \textit{convex hull} $H(S)$ of $S$ is the smallest convex set containing $S$. If $H(S) = V(G)$, then $S$ is a \textit{hull set}. The cardinality $h(G)$ of a minimum hull set of $G$ is the \textit{hull number} of $G$. The \textit{complementary prism} $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. A graph $G$ is \textit{autoconnected} if both $G$ and $\overline{G}$ are connected. Motivated by previous work, we study the hull number for complementary prisms of autoconnected graphs. When $G$ is a split graph, we present lower and upper bounds showing that the hull number is unlimited. In the other case, when $G$ is a non-split graph, it is limited by $3$.

scholarly journals Fine-Grained Complexity of Safety Verification

2020 ◽  
Vol 64 (7) ◽  
pp. 1419-1444
Author(s):  
Peter Chini ◽  
Roland Meyer ◽  
Prakash Saivasan
Keyword(s):  
Lower Bounds ◽  
Maximum Size ◽  
Set Cover ◽  
Polynomial Kernel ◽  
Concurrent Programs ◽  
Safety Verification ◽  
Fixed Parameter Tractable ◽  
Fine Grained ◽  
Fixed Parameter ◽  
Key Techniques

Abstract We study the fine-grained complexity of Leader Contributor Reachability ($${\textsf {LCR}} $$ LCR ) and Bounded-Stage Reachability ($${\textsf {BSR}} $$ BSR ), two variants of the safety verification problem for shared memory concurrent programs. For both problems, the memory is a single variable over a finite data domain. Our contributions are new verification algorithms and lower bounds. The latter are based on the Exponential Time Hypothesis ($${\textsf {ETH}} $$ ETH ), the problem $${\textsf {Set~Cover}} $$ Set Cover , and cross-compositions. $${\textsf {LCR}} $$ LCR is the question whether a designated leader thread can reach an unsafe state when interacting with a certain number of equal contributor threads. We suggest two parameterizations: (1) By the size of the data domain $${\texttt {D}}$$ D and the size of the leader $${\texttt {L}}$$ L , and (2) by the size of the contributors $${\texttt {C}}$$ C . We present algorithms for both cases. The key techniques are compact witnesses and dynamic programming. The algorithms run in $${\mathcal {O}}^*(({\texttt {L}}\cdot ({\texttt {D}}+1))^{{\texttt {L}}\cdot {\texttt {D}}} \cdot {\texttt {D}}^{{\texttt {D}}})$$ O ∗ ( ( L · ( D + 1 ) ) L · D · D D ) and $${\mathcal {O}}^*(2^{{\texttt {C}}})$$ O ∗ ( 2 C ) time, showing that both parameterizations are fixed-parameter tractable. We complement the upper bounds by (matching) lower bounds based on $${\textsf {ETH}} $$ ETH and $${\textsf {Set~Cover}} $$ Set Cover . Moreover, we prove the absence of polynomial kernels. For $${\textsf {BSR}} $$ BSR , we consider programs involving $${\texttt {t}}$$ t different threads. We restrict the analysis to computations where the write permission changes $${\texttt {s}}$$ s times between the threads. $${\textsf {BSR}} $$ BSR asks whether a given configuration is reachable via such an $${\texttt {s}}$$ s -stage computation. When parameterized by $${\texttt {P}}$$ P , the maximum size of a thread, and $${\texttt {t}}$$ t , the interesting observation is that the problem has a large number of difficult instances. Formally, we show that there is no polynomial kernel, no compression algorithm that reduces the size of the data domain $${\texttt {D}}$$ D or the number of stages $${\texttt {s}}$$ s to a polynomial dependence on $${\texttt {P}}$$ P and $${\texttt {t}}$$ t . This indicates that symbolic methods may be harder to find for this problem.

Sign in / Sign up

Export Citation Format

Share Document