Fixed-Parameter Algorithms for Maximum-Profit Facility Location Under Matroid Constraints

We study the parameterized and classical complexity of two problems that are concerned with induced paths on three vertices, called P 3 s, in undirected graphs G = (V, E). In Strong Triadic Closure we aim to label the edges in E as strong and weak such that at most k edges are weak and G contains no induced P 3 with two strong edges. In Cluster Deletion we aim to destroy all induced P 3 s by a minimum number of edge deletions. We first show that Strong Triadic Closure admits a 4k-vertex kernel. Then, we study parameterization by ℓ := |E| − k and show that both problems are fixed-parameter tractable and unlikely to admit a polynomial kernel with respect to ℓ. Finally, we give a dichotomy of the classical complexity of both problems on H-free graphs for all H of order at most four. IntroductionWe study two related graph problems arising in social network analysis and data clustering. Assume we are given a social network where vertices represent agents and edges represent relationships between these agents, and want to predict which of the relationships are important. In online social networks for example, one could aim to distinguish between close friends and spurious relationships. Sintos and Tsaparas [28] proposed to use the notion of strong triadic closure for this problem. This notion goes back to Granovetter's sociological work [10]. Informally, it is the assumption that if one agent has strong relationships with two other agents, then these two other agents should have at least a weak relationship. The aim in the computational problem is then to label a maximum number of edges of the given social network as strong while fulfilling this requirement. Formally, we are looking for an STC-labeling defined as follows. ⋆ A preliminary version of this work appeared inKnown Results. STC is NP-hard [28], even when restricted to graphs with maximum degree four [16] or to split graphs [17]. In contrast, STC is solvable in polynomial time when the input graph is bipartite [28], subcubic [16], a proper interval graph [17], or a cograph, that is, a graph with no induced P 4 [16]. STC can be solved in O(1.28 k + nm) time and admits a polynomial kernel when parameterized by k. These two results follow from a parameter-preserving reduction to Vertex Cover, which asks if it is possible to delete at most k vertices of a given graph such that the remaining graph does not contain any edge. This parameter-preserving reduction [28] computes the so-called Gallai graph [18,29] of the input graph and directly gives the above-mentioned running time bound. The existence of a kernel for parameter k is implied by two facts: First, Vertex Cover admits a polynomial kernel for the number k of vertices to delete [6,7]. Second, Vertex Cover is in NP and STC is NP-hard. Hence, the Vertex Cover instance of size poly(k) which we obtain by first reducing from STC to Vertex Cover and then applying the kernelization can be transformed into an equivalent STC instance by a polynomial-time reduction. The STC instance then has size poly(k).

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“…, X t } is internally connected via strong links [25]. This problem has been shown to be a special case of a facility location problem [30].…”

Section: Strong Triadic Closure (Stc)mentioning

confidence: 99%

On the Relation of Strong Triadic Closure and Cluster Deletion

Grüttemeier

Komusiewicz

2019

Algorithmica

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“…For example, in the k-center problem, the goal is minimizing the maximum distance of each client from its nearest facility center [7]. The facility location problem has received a good deal of attention in the parameterized perspective [1,6,20,21].…”

Section: Related Workmentioning

confidence: 99%

On the Parameterized Complexity of the Expected Coverage Problem

Fomin

Ramamoorthi

2022

Theory Comput Syst

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The Maximum Covering Location Problem (MCLP) is a well-studied problem in the field of operations research. Given a network with positive or negative demands on the nodes, a positive integer k, the MCLP seeks to find k potential facility centers in the network such that the neighborhood coverage is maximized. We study the variant of MCLP where edges of the network are subject to random failures due to some disruptive events. One of the popular models capturing the unreliable nature of the facility location is the linear reliability ordering (LRO) model. In this model, with every edge e of the network, we associate its survival probability 0 ≤ pe ≤ 1, or equivalently, its failure probability 1 − pe. The failure correlation in LRO is the following: If an edge e fails then every edge $e^{\prime }$ e ′ with $p_{e^{\prime }} \leq p_{e}$ p e ′ ≤ p e surely fails. The task is to identify the positions of k facilities that maximize the expected coverage. We refer to this problem as Expected Coverage problem. We study the Expected Coverage problem from the parameterized complexity perspective and obtain the following results. 1. For the parameter pathwidth, we show that the Expected Coverage problem is W[1]-hard. We find this result a bit surprising, because the variant of the problem with non-negative demands is fixed-parameter tractable (FPT) parameterized by the treewidth of the input graph. 2. We complement the lower bound by the proof that Expected Coverage is FPT being parameterized by the treewidth and the maximum vertex degree. We give an algorithm that solves the problem in time $ 2^{{\mathcal {O}}({\textbf {tw}} \log {\varDelta })} n^{{\mathcal {O}}(1)}$ 2 O ( tw log Δ ) n O ( 1 ) , where tw is the treewidth, Δ is the maximum vertex degree, and n the number of vertices of the input graph. In particular, since Δ ≤ n, it means the problem is solvable in time $ n^{{\mathcal {O}}({\textbf {tw}})} $ n O ( tw ) , that is, is in XP parameterized by treewidth.

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“…In another line of work, van Bevern et al [52] considered a matroid constrained variant of facility location. Their approach can be shown to yield an FPT algorithm for weighted coverage functions subject to ℓ matroid constraints.…”

Section: Related Workmentioning

confidence: 99%

FPT-Algorithms for the \ell-Matchoid Problem with a Coverage Objective

Huang

Ward²

2020

Preprint

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We design a fixed-parameter deterministic algorithm for computing a maximum weight feasible set under a ℓ-matchoid of rank k, parameterized by ℓ and k. Unlike previous work that presumes linear representativity of matroids, we consider the general oracle model. Our result, combined with the lower bounds of Lovasz [37], and Jensen and Korte [28], demonstrates a separation between the ℓ-matchoid and the matroid ℓ-parity problems in the setting of fixedparameter tractability.Our algorithms are obtained by means of kernelization: we construct a small representative set which contains an optimal solution. Such a set gives us much flexibility in adapting to other settings, allowing us to optimize not only a linear function, but also several important submodular functions. It also helps to transform our algorithms into streaming algorithms.In the streaming setting, we show that we can find a feasible solution of value z and the number of elements to be stored in memory depends only on z and ℓ but totally independent of n. This shows that it is possible to circumvent the recent space lower bound of Feldman et al [20], by parameterizing the solution value. This result, combined with existing lower bounds, also provides a new separation between the space and time complexity of maximizing an arbitrary submodular function and a coverage function in the value oracle model.

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Fixed-Parameter Algorithms for Maximum-Profit Facility Location Under Matroid Constraints

Cited by 13 publications

References 40 publications

On the Relation of Strong Triadic Closure and Cluster Deletion

On the Relation of Strong Triadic Closure and Cluster Deletion

On the Parameterized Complexity of the Expected Coverage Problem

FPT-Algorithms for the \ell-Matchoid Problem with a Coverage Objective

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