Parameterized Counting of Partially Injective Homomorphisms

Linear Combinations ◽

Counting Problem ◽

Large Target ◽

Graph Homomorphisms ◽

Join Queries ◽

Fixed Parameter ◽

Complexity Classification ◽

Injective Homomorphisms

AbstractWe study the parameterized complexity of the problem of counting graph homomorphisms with given partial injectivity constraints, i.e., inequalities between pairs of vertices, which subsumes counting of graph homomorphisms, subgraph counting and, more generally, counting of answers to equi-join queries with inequalities. Our main result presents an exhaustive complexity classification for the problem in fixed-parameter tractable and $$\#\mathsf {W[1]}$$ # W [ 1 ] -complete cases. The proof relies on the framework of linear combinations of homomorphisms as independently discovered by Chen and Mengel (PODS 16) and by Curticapean, Dell and Marx in the recent breakthrough result regarding the exact complexity of the subgraph counting problem (STOC 17). Moreover, we invoke Rota’s NBC-Theorem to obtain an explicit criterion for fixed-parameter tractability based on treewidth. The abstract classification theorem is then applied to the problem of counting locally injective graph homomorphisms from small pattern graphs to large target graphs. As a consequence, we are able to fully classify its parameterized complexity depending on the class of allowed pattern graphs.

Perfect domination and small cycles

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830917500306 ◽

2017 ◽

Vol 09 (03) ◽

pp. 1750030 ◽

Cited By ~ 1

Author(s):

Minghui Jiang ◽

Yong Zhang

Keyword(s):

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.

Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence ◽

Novel Structural Parameters for Acyclic Planning Using Tree Embeddings

10.24963/ijcai.2018/647 ◽

2018 ◽

Author(s):

Christer Bäckström ◽

Peter Jonsson ◽

Sebastian Ordyniak

Keyword(s):

Computer Science ◽

Bounded Domain ◽

Structural Parameters ◽

Domain Size ◽

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.

A Compendium of Parameterized Problems at Higher Levels of the Polynomial Hierarchy

Algorithms ◽

10.3390/a12090188 ◽

2019 ◽

Vol 12 (9) ◽

pp. 188 ◽

Cited By ~ 1

Author(s):

Ronald de Haan ◽

Stefan Szeider

Keyword(s):

Polynomial Hierarchy ◽

Fixed Parameter ◽

Complexity Classification

We present a list of parameterized problems together with a complexity classification of whether they allow a fixed-parameter tractable reduction to SAT or not. These problems are parameterized versions of problems whose complexity lies at the second level of the Polynomial Hierarchy or higher.

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence ◽

A Parameterized Perspective on Protecting Elections

10.24963/ijcai.2019/34 ◽

2019 ◽

Cited By ~ 3

Author(s):

Palash Dey ◽

Neeldhara Misra ◽

Swaprava Nath ◽

Garima Shakya

Keyword(s):

Greedy Algorithms ◽

Optimal Defense ◽

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.

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

10.5753/ctd.2015.9996 ◽

2020 ◽

Author(s):

Uéverton Souza ◽

Fábio Protti ◽

Maise Da Silva ◽

Dieter Rautenbach

Keyword(s):

Polynomial Time ◽

Complexity Analysis ◽

Np Hard ◽

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.

The Parameterized Complexity of Motion Planning for Snake-Like Robots

Journal of Artificial Intelligence Research ◽

10.1613/jair.1.11864 ◽

2020 ◽

Vol 69 ◽

pp. 191-229

Author(s):

Siddharth Gupta ◽

Guy Sa'ar ◽

Meirav Zehavi

Keyword(s):

Motion Planning ◽

Video Game ◽

Initial Position ◽

Polynomial Kernel ◽

Final Position ◽

Grid Graphs ◽

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.

Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence ◽

Learning Bayesian Networks Under Sparsity Constraints: A Parameterized Complexity Analysis

10.24963/ijcai.2020/586 ◽

2020 ◽

Author(s):

Niels Grüttemeier ◽

Christian Komusiewicz

Keyword(s):

Maximum Degree ◽

Complexity Analysis ◽

Structural Constraints ◽

Sparse Graph ◽

Optimal Network ◽

Fixed Parameter ◽

Graph Class ◽

Sparsity Constraints

We study the problem of learning the structure of an optimal Bayesian network when additional structural constraints are posed on the network or on its moralized graph. More precisely, we consider the constraint that the moralized graph can be transformed to a graph from a sparse graph class Π by at most k vertex deletions. We show that for Π being the graphs with maximum degree 1, an optimal network can be computed in polynomial time when k is constant, extending previous work that gave an algorithm with such a running time for Π being the class of edgeless graphs [Korhonen & Parviainen, NIPS 2015]. We then show that further extensions or improvements are presumably impossible. For example, we show that when Π is the set of graphs in which each component has size at most three, then learning an optimal network is NP-hard even if k=0. Finally, we show that learning an optimal network with at most k edges in the moralized graph presumably is not fixed-parameter tractable with respect to k and that, in contrast, computing an optimal network with at most k arcs can be computed is fixed-parameter tractable in k.