We consider a variant of the matroid median problem introduced by Krishnaswamy et al. [SODA 2011]: an uncapacitated discrete facility location problem where the task is to decide which facilities to open and which clients to serve for maximum profit so that the facilities form an independent set in given facility-side matroids and the clients form an independent set in given client-side matroids. We show that the problem is fixed-parameter tractable parameterized by the number of matroids and the minimum rank among the client-side matroids. To this end, we derive fixed-parameter algorithms for computing representative families for matroid intersections and maximum-weight set packings with multiple matroid constraints. To illustrate the modeling capabilities of the new problem, we use it to obtain algorithms for a problem in social network analysis. We complement our tractability results by lower bounds.Keywords: matroid set packing · matroid parity · matroid median · representative families · social network analysis · strong triadic closure 4 The State Register of Waste Disposal Facilities of the Russian Federation lists 18 dumps for municipal solid waste in Moscow region-the largest city of Europe.
Given an undirected graph with edge weights and a subset R of its edges, the Rural Postman Problem (RPP) is to find a closed walk of minimum total weight containing all edges of R. We prove that RPP is WK[1]-complete parameterized by the number and cost d of edges traversed additionally to the required ones. Thus, in particular, RPP instances cannot be polynomial-time compressed to instances of size polynomial in d unless the polynomial-time hierarchy collapses. In contrast, denoting by b ≤ 2d the number of vertices incident to an odd number of edges of R and by c ≤ d the number of connected components formed by the edges in R, we show how to reduce any RPP instance I to an RPP instance I with 2b + O(c/ε) vertices in O(n 3 ) time so that any α-approximate solution for I gives an α(1 + ε)-approximate solution for I, for any α ≥ 1 and ε > 0. That is, we provide a polynomial-size approximate kernelization scheme (PSAKS). We experimentally evaluate it on wide-spread benchmark data sets as well as on two real snow plowing instances from Berlin. On instances with few connected components, the number of vertices and required edges is reduced to about 50 % at a 1 % solution quality loss. We also make first steps towards a PSAKS for the parameter c. Problem 1.1 (Rural Postman Problem, RPP).Input: An undirected graph G = (V, E) with n vertices, edge weights ω : E → N ∪ {0}, and a multiset R of required edges of G. Task: Find a closed walk W * in G containing each edge of R and minimizing the total weight ω(W * ) of the edges on W * .We call any closed walk containing each edge of R an RPP tour. We will also consider the decision variant k-RPP, where one additionally gets a non-negative integer k ∈ N in the input and the task is to decide whether there is an RPP tour W of cost ω(W) ≤ k. RPP has direct applications in snow plowing, street sweeping, meter reading [14,22], vehicle depot location [29], drilling, and plotting [28,31]. The undirected version occurs especially in rural areas, where service vehicles can operate in both directions even on one-way roads [19]. Moreover, RPP is a special case of the Capacitated Arc Routing Problem (CARP) [30] and used in all "route first, cluster second" algorithms for CARP [1,10,49], which are notably the only ones with proven approximation guarantees [6,36,50]. Improved approximations for RPP automatically lead to better approximations for CARP.There is a folklore polynomial-time 3/2-approximation for RPP based on the Christofides-Serdyukov algorithm for the metric Traveling Salesman Problem [11,46] (we refer to Eiselt et al. [22] or van Bevern et al. [7] for a detailed algorithm description). We aim for (1 + ε)-approximations for all ε > 0. Unfortunately, containing the metric Traveling Salesman Problem as a special case, RPP is APX-hard [37]. Thus, finding such approximations typically requires exponential time, we present data reduction rules for this task. Their effectivity depends on the desired approximation factor. Graph theory. We generally consider multigraphs G = (V, E) wit...
Given a graph G = (V, E), two vertices s, t ∈ V, and two integers k, , the SHORT SECLUDED PATH problem is to find a simple s-t-path with at most k vertices and neighbors. We study the parameterized complexity of the problem with respect to four structural graph parameters: the vertex cover number, treewidth, feedback vertex number, and feedback edge number. In particular, we completely settle the question of the existence of problem kernels with size polynomial in these parameters and their combinations with k and . We also obtain a 2 O(tw) ⋅ 2 ⋅ n-time algorithm for n-vertex graphs of treewidth tw, which yields subexponential-time algorithms in several graph classes. ]. This version contains full proof details, new kernelization results with respect to the feedback vertex number as parameter, and the algorithm for graphs of bounded treewidth has been generalized to a more general problem variant and accelerated. Networks. 2020;75:34-63. wileyonlinelibrary.com/journal/net © 2019 Wiley Periodicals, Inc. 34 VAN BEVERN ET AL. 35 TABLE 1 Overview of our results par. Positive results Negative results vc Size vc O(r) -kernel in K r,r -subgraph-free graphs (Theorem 3.8) No polynomial kernel and WK[1]-hard w.r.t. vc (Theorem 3.1) fes Size poly(fes)-kernel (Theorem 5.15) fvs O(fvs ⋅ (k + ) 2 )-vertex kernel (Theorem 5.4) No kernel with size poly(fvs + ) (Theorem 5.20) tw 2 O(tw) ⋅ 2 ⋅ n-time algorithm (Theorem 4.2) No kernel with size poly(tw + k + ) even in planar graphs with const. Δ (Theorem 4.14)Herein, n, tw, vc, fes, fvs, and Δ denote the number of vertices, treewidth, vertex cover number, feedback edge number, feedback vertex number, and maximum degree of the input graph, respectively. FIGURE 1Overview on the existence of polynomial kernelization. Gray: no polynomial-size kernel unless coNP ⊆ NP/poly. White: polynomial-size kernel exists. An arrow from parameter p to p ′ means that the value of p can be upper-bounded by a polynomial in p ′ [25]. Thus, hardness results for p ′ also hold for p and polynomial-size kernels for p also hold for p ′ call a problem fixed-parameter tractable if it can be solved in f (k) ⋅ n O(1) time on inputs of length n and some function f depending only on some parameter k. In contrast to an algorithm that merely runs in polynomial time for fixed k, say, in O(n k ) time, which is intractable even for small values of k, fixed-parameter algorithms can solve NP-hard problems quickly if k is small.Provably effective polynomial-time data reduction. Parameterized complexity theory also provides a framework for data reduction with performance guarantees-problem kernelization [16,21,27,47].Kernelization allows for provably effective polynomial-time data reduction. Note that a result of the form "our polynomial-time data reduction algorithm reduces the input size by at least one bit, preserving optimality of solutions" is impossible for NP-hard problems unless P = NP. In contrast, a kernelization algorithm reduces a problem instance into an equivalent one (the problem kernel) whose size depends o...
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