A Multivariate Complexity Analysis of Lobbying in Multiple Referenda

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.

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Maximum Exact Satisfiability: NP-completeness Proofs and Exact Algorithms

BRICS Report Series ◽

10.7146/brics.v11i19.21844 ◽

2004 ◽

Vol 11 (19) ◽

Cited By ~ 3

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.

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How to Approximate Ontology-Mediated Queries

10.24963/kr.2021/31 ◽

2021 ◽

Author(s):

Anneke Haga ◽

Carsten Lutz ◽

Leif Sabellek ◽

Frank Wolter

Keyword(s):

Computational Complexity ◽

Linear Time ◽

Description Logics ◽

Data Complexity ◽

Fixed Parameter Tractable ◽

Ontology Language ◽

Fixed Parameter ◽

Almost All

We introduce and study several notions of approximation for ontology-mediated queries based on the description logics ALC and ALCI. Our approximations are of two kinds: we may (1) replace the ontology with one formulated in a tractable ontology language such as ELI or certain TGDs and (2) replace the database with one from a tractable class such as the class of databases whose treewidth is bounded by a constant. We determine the computational complexity and the relative completeness of the resulting approximations. (Almost) all of them reduce the data complexity from coNP-complete to PTime, in some cases even to fixed-parameter tractable and to linear time. While approximations of kind (1) also reduce the combined complexity, this tends to not be the case for approximations of kind (2). In some cases, the combined complexity even increases.

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

ACM Transactions on Privacy and Security ◽

10.1145/3450768 ◽

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.

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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 ◽

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.

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Learning Bayesian Networks Under Sparsity Constraints: A Parameterized Complexity Analysis

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

10.24963/ijcai.2020/586 ◽

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.

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