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.

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

2019 ◽  
Author(s):  
Siddharth Gupta ◽  
Guy Sa'ar ◽  
Meirav Zehavi
Keyword(s):  
Motion Planning ◽  
Parameterized Complexity ◽  
Initial Position ◽  
Polynomial Kernel ◽  
Planning Problem ◽  
Color Coding ◽  
Grid Graphs ◽  
Input Size ◽  
Motion Problems ◽  
General Graphs

We study a motion-planning problem inspired by the game Snake that models scenarios like the transportation of linked wagons towed by a locomotor to the movement of a group of agents that travel in an ``ant-like'' fashion. Given a ``snake-like'' robot with initial and final positions in an environment modeled by a graph, our goal is to decide whether the robot can reach the final position from the initial position without intersecting itself. Already on grid graphs, this problem is PSPACE-complete [Biasi and Ophelders, 2018]. Nevertheless, we prove that even on general graphs, it is solvable in time k^{O(k)}|I|^{O(1)} where k is the size of the robot, and |I| is the input size. Towards this, we give a novel application of color-coding to sparsify the configuration graph of the problem. We also show that the problem is unlikely to have 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 problems has been~largely~neglected, thus our work is pioneering in this regard.

Perfect domination and small cycles

2017 ◽  
Vol 09 (03) ◽  
pp. 1750030 ◽  
Author(s):  
Minghui Jiang ◽  
Yong Zhang
Keyword(s):  
Parameterized Complexity ◽  
Fixed Parameter Tractable ◽  
Fixed Parameter ◽  
Domination In Graphs

We study the parameterized complexity of several problems related to perfect domination in graphs with or without small cycles. When parameterized by the solution size, these problems are W-hard in graphs with girth at most four, but are fixed-parameter tractable in graphs with girth at least five.

scholarly journals Novel Structural Parameters for Acyclic Planning Using Tree Embeddings

2018 ◽  
Author(s):  
Christer Bäckström ◽  
Peter Jonsson ◽  
Sebastian Ordyniak
Keyword(s):  
Computer Science ◽  
Bounded Domain ◽  
Parameterized Complexity ◽  
Structural Parameters ◽  
Domain Size ◽  
Fixed Parameter Tractable ◽  
Bounded Treewidth ◽  
Fixed Parameter ◽  
Causal Graphs ◽  
Computational Properties

We introduce two novel structural parameters for acyclic planning (planning restricted to instances with acyclic causal graphs): up-depth and down-depth. We show that cost-optimal acyclic planning restricted to instances with bounded domain size and bounded up- or down-depth can be solved in polynomial time. For example, many of the tractable subclasses based on polytrees are covered by our result. We analyze the parameterized complexity of planning with bounded up- and down-depth: in a certain sense, down-depth has better computational properties than up-depth. Finally, we show that computing up- and down-depth are fixed-parameter tractable problems, just as many other structural parameters that are used in computer science. We view our results as a natural step towards understanding the complexity of acyclic planning with bounded treewidth and other parameters.

scholarly journals A Parameterized Perspective on Protecting Elections

2019 ◽  
Author(s):  
Palash Dey ◽  
Neeldhara Misra ◽  
Swaprava Nath ◽  
Garima Shakya
Keyword(s):  
Parameterized Complexity ◽  
Greedy Algorithms ◽  
Optimal Defense ◽  
Fixed Parameter Tractable ◽  
Scoring Rule ◽  
Voting Rule ◽  
Fixed Parameter ◽  
Voter Groups

We study the parameterized complexity of the optimal defense and optimal attack problems in voting. In both the problems, the input is a set of voter groups (every voter group is a set of votes) and two integers k_a and k_d corresponding to respectively the number of voter groups the attacker can attack and the number of voter groups the defender can defend. A voter group gets removed from the election if it is attacked but not defended. In the optimal defense problem, we want to know if it is possible for the defender to commit to a strategy of defending at most k_d voter groups such that, no matter which k_a voter groups the attacker attacks, the out-come of the election does not change. In the optimal attack problem, we want to know if it is possible for the attacker to commit to a strategy of attacking k_a voter groups such that, no matter which k_d voter groups the defender defends, the outcome of the election is always different from the original (without any attack) one. We show that both the optimal defense problem and the optimal attack problem are computationally intractable for every scoring rule and the Condorcet voting rule even when we have only3candidates. We also show that the optimal defense problem for every scoring rule and the Condorcet voting rule is W[2]-hard for both the parameters k_a and k_d, while it admits a fixed parameter tractable algorithm parameterized by the combined parameter (ka, kd). The optimal attack problem for every scoring rule and the Condorcet voting rule turns out to be much harder – it is W[1]-hard even for the combined parameter (ka, kd). We propose two greedy algorithms for the OPTIMAL DEFENSE problem and empirically show that they perform effectively on reasonable voting profiles.

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.

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 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 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 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).

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