Fixed-Parameter Tractable Algorithm and Polynomial Kernel for Max-Cut Above Spanning Tree

2019 ◽  
Vol 64 (1) ◽  
pp. 62-100
Author(s):  
Jayakrishnan Madathil ◽  
Saket Saurabh ◽  
Meirav Zehavi
Keyword(s):  
Spanning Tree ◽  
Polynomial Kernel ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter

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.

scholarly journals Towards a Polynomial Kernel for Directed Feedback Vertex Set

Algorithmica ◽  
2020 ◽  
Author(s):  
Benjamin Bergougnoux ◽  
Eduard Eiben ◽  
Robert Ganian ◽  
Sebastian Ordyniak ◽  
M. S. Ramanujan
Keyword(s):  
Undirected Graph ◽  
Polynomial Kernel ◽  
Directed Cycle ◽  
Feedback Vertex Set ◽  
Open Problems ◽  
Fixed Parameter Tractable ◽  
Parameterized Algorithmics ◽  
Fixed Parameter ◽  
Vertex Set ◽  
Directed Feedback Vertex Set

Abstract In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph D and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen et al. (J ACM 55(5):177–186, 2008); since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics. Since this problem has remained open in spite of the best efforts of a number of prominent researchers and pioneers in the field, a natural step forward is to study the kernelization complexity of DFVS parameterized by a natural larger parameter. In this paper, we study DFVS parameterized by the feedback vertex set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus.

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.

scholarly journals The Parameterized Complexity of Motion Planning for Snake-Like Robots

2020 ◽  
Vol 69 ◽  
pp. 191-229
Author(s):  
Siddharth Gupta ◽  
Guy Sa'ar ◽  
Meirav Zehavi
Keyword(s):  
Motion Planning ◽  
Video Game ◽  
Parameterized Complexity ◽  
Initial Position ◽  
Polynomial Kernel ◽  
Final Position ◽  
Grid Graphs ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
General Graphs

We study the parameterized complexity of a variant of the classic video game Snake that models real-world problems of motion planning. Given a snake-like robot with an initial position and a final position in an environment (modeled by a graph), our objective is to determine whether the robot can reach the final position from the initial position without intersecting itself. Naturally, this problem models a wide-variety of scenarios, ranging from the transportation of linked wagons towed by a locomotor at an airport or a supermarket to the movement of a group of agents that travel in an “ant-like” fashion and the construction of trains in amusement parks. Unfortunately, already on grid graphs, this problem is PSPACE-complete. Nevertheless, we prove that even on general graphs, the problem is solvable in FPT time with respect to the size of the snake. In particular, this shows that the problem is fixed-parameter tractable (FPT). Towards this, we show how to employ color-coding to sparsify the configuration graph of the problem to reduce its size significantly. We believe that our approach will find other applications in motion planning. Additionally, we show that the problem is unlikely to admit a polynomial kernel even on grid graphs, but it admits a treewidth-reduction procedure. To the best of our knowledge, the study of the parameterized complexity of motion planning problems (where the intermediate configurations of the motion are of importance) has so far been largely overlooked. Thus, our work is pioneering in this regard.

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).

A general method to speed up fixed-parameter-tractable algorithms

2000 ◽  
Vol 73 (3-4) ◽  
pp. 125-129 ◽  
Author(s):  
Rolf Niedermeier ◽  
Peter Rossmanith
Keyword(s):  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
Speed Up ◽  
General Method

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
Sign in / Sign up

Export Citation Format

Share Document