scholarly journals Learning Bayesian Networks Under Sparsity Constraints: A Parameterized Complexity Analysis

2020 ◽  
Author(s):  
Niels Grüttemeier ◽  
Christian Komusiewicz
Keyword(s):  
Parameterized Complexity ◽  
Maximum Degree ◽  
Complexity Analysis ◽  
Structural Constraints ◽  
Sparse Graph ◽  
Fixed Parameter Tractable ◽  
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.

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.

Chapter 17. Fixed-Parameter Tractability

2021 ◽  
Author(s):  
Marko Samer ◽  
Stefan Szeider
Keyword(s):  
Polynomial Time ◽  
Parameterized Complexity ◽  
Complexity Analysis ◽  
Fixed Parameter Tractability ◽  
Graph Representations ◽  
Fine Grained ◽  
Input Size ◽  
Backdoor Sets ◽  
Fixed Parameter ◽  
Problem Instances

Parameterized complexity is a new theoretical framework that considers, in addition to the overall input size, the effects on computational complexity of a secondary measurement, the parameter. This two-dimensional viewpoint allows a fine-grained complexity analysis that takes structural properties of problem instances into account. The central notion is “fixed-parameter tractability” which refers to solvability in polynomial time for each fixed value of the parameter such that the order of the polynomial time bound is independent of the parameter. This chapter presents main concepts and recent results on the parameterized complexity of the satisfiability problem and it outlines fundamental algorithmic ideas that arise in this context. Among the parameters considered are the size of backdoor sets with respect to various tractable base classes and the treewidth of graph representations of satisfiability instances.

scholarly journals Fixed-Parameter Tractable Distances to Sparse Graph Classes

Algorithmica ◽  
2016 ◽  
Vol 79 (1) ◽  
pp. 139-158 ◽  
Author(s):  
Jannis Bulian ◽  
Anuj Dawar
Keyword(s):  
Sparse Graph ◽  
Fixed Parameter Tractable ◽  
Graph Classes ◽  
Fixed Parameter

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 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 Towards Better Understanding of User Authorization Query Problem via Multi-variable Complexity Analysis

2021 ◽  
Vol 24 (3) ◽  
pp. 1-22
Author(s):  
Jason Crampton ◽  
Gregory Z. Gutin ◽  
Diptapriyo Majumdar
Keyword(s):  
Complexity Analysis ◽  
State Of The Art ◽  
Role Based Access Control ◽  
Fixed Parameter Tractable ◽  
Multiple Parameters ◽  
The Past ◽  
Number Of Factors ◽  
Fpt Algorithm ◽  
Fixed Parameter ◽  
Role Based

User authorization queries in the context of role-based access control have attracted considerable interest in the past 15 years. Such queries are used to determine whether it is possible to allocate a set of roles to a user that enables the user to complete a task, in the sense that all the permissions required to complete the task are assigned to the roles in that set. Answering such a query, in general, must take into account a number of factors, including, but not limited to, the roles to which the user is assigned and constraints on the sets of roles that can be activated. Answering such a query is known to be NP-hard. The presence of multiple parameters and the need to find efficient and exact solutions to the problem suggest that a multi-variate approach will enable us to better understand the complexity of the user authorization query problem (UAQ). In this article, we establish a number of complexity results for UAQ. Specifically, we show the problem remains hard even when quite restrictive conditions are imposed on the structure of the problem. Our fixed-parameter tractable (FPT) results show that we have to use either a parameter with potentially quite large values or quite a restricted version of UAQ. Moreover, our second FPT algorithm is complex and requires sophisticated, state-of-the-art techniques. In short, our results show that it is unlikely that all variants of UAQ that arise in practice can be solved reasonably quickly in general.

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.

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