160,303 research outputs found

    Constant Factor Approximation for Capacitated k-Center with Outliers

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    The kk-center problem is a classic facility location problem, where given an edge-weighted graph G=(V,E)G = (V,E) one is to find a subset of kk vertices SS, such that each vertex in VV is "close" to some vertex in SS. The approximation status of this basic problem is well understood, as a simple 2-approximation algorithm is known to be tight. Consequently different extensions were studied. In the capacitated version of the problem each vertex is assigned a capacity, which is a strict upper bound on the number of clients a facility can serve, when located at this vertex. A constant factor approximation for the capacitated kk-center was obtained last year by Cygan, Hajiaghayi and Khuller [FOCS'12], which was recently improved to a 9-approximation by An, Bhaskara and Svensson [arXiv'13]. In a different generalization of the problem some clients (denoted as outliers) may be disregarded. Here we are additionally given an integer pp and the goal is to serve exactly pp clients, which the algorithm is free to choose. In 2001 Charikar et al. [SODA'01] presented a 3-approximation for the kk-center problem with outliers. In this paper we consider a common generalization of the two extensions previously studied separately, i.e. we work with the capacitated kk-center with outliers. We present the first constant factor approximation algorithm with approximation ratio of 25 even for the case of non-uniform hard capacities.Comment: 15 pages, 3 figures, accepted to STACS 201

    Online Matching with Stochastic Rewards: Optimal Competitive Ratio via Path Based Formulation

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    The problem of online matching with stochastic rewards is a generalization of the online bipartite matching problem where each edge has a probability of success. When a match is made it succeeds with the probability of the corresponding edge. Introducing this model, Mehta and Panigrahi (FOCS 2012) focused on the special case of identical edge probabilities. Comparing against a deterministic offline LP, they showed that the Ranking algorithm of Karp et al. (STOC 1990) is 0.534 competitive and proposed a new online algorithm with an improved guarantee of 0.5670.567 for vanishingly small probabilities. For the case of vanishingly small but heterogeneous probabilities Mehta et al. (SODA 2015), gave a 0.534 competitive algorithm against the same LP benchmark. For the more general vertex-weighted version of the problem, to the best of our knowledge, no results being 1/21/2 were previously known even for identical probabilities. We focus on the vertex-weighted version and give two improvements. First, we show that a natural generalization of the Perturbed-Greedy algorithm of Aggarwal et al. (SODA 2011), is (11/e)(1-1/e) competitive when probabilities decompose as a product of two factors, one corresponding to each vertex of the edge. This is the best achievable guarantee as it includes the case of identical probabilities and in particular, the classical online bipartite matching problem. Second, we give a deterministic 0.5960.596 competitive algorithm for the previously well studied case of fully heterogeneous but vanishingly small edge probabilities. A key contribution of our approach is the use of novel path-based analysis. This allows us to compare against the natural benchmarks of adaptive offline algorithms that know the sequence of arrivals and the edge probabilities in advance, but not the outcomes of potential matches.Comment: Preliminary version in EC 202

    Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

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    The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph G=(V,E)G=(V,E) and a set DV×V\mathcal{D}\subseteq V\times V of kk demand pairs. The aim is to compute the cheapest network NGN\subseteq G for which there is an sts\to t path for each (s,t)D(s,t)\in\mathcal{D}. It is known that this problem is notoriously hard as there is no k1/4o(1)k^{1/4-o(1)}-approximation algorithm under Gap-ETH, even when parametrizing the runtime by kk [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter kk. For the bi-DSNPlanar_\text{Planar} problem, the aim is to compute a planar optimum solution NGN\subseteq G in a bidirected graph GG, i.e., for every edge uvuv of GG the reverse edge vuvu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for kk. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSNPlanar_\text{Planar}, unless FPT=W[1]. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network NGN\subseteq G needs to strongly connect a given set of kk terminals. It has been observed before that for SCSS a parameterized 22-approximation exists when parameterized by kk [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for kk no parameterized (2ε)(2-\varepsilon)-approximation algorithm exists under Gap-ETH. Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for kk

    Improved Algorithms for MST and Metric-TSP Interdiction

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    We consider the MST-interdiction problem: given a multigraph G = (V, E), edge weights {w_e >= 0}_{e in E}, interdiction costs {c_e >= 0}_{e in E}, and an interdiction budget B >= 0, the goal is to remove a subset R of edges of total interdiction cost at most B so as to maximize the w-weight of an MST of G-R:=(V,E-R). Our main result is a 4-approximation algorithm for this problem. This improves upon the previous-best 14-approximation [Zenklusen, FOCS 2015]. Notably, our analysis is also significantly simpler and cleaner than the one in [Zenklusen, FOCS 2015]. Whereas Zenklusen uses a greedy algorithm with an involved analysis to extract a good interdiction set from an over-budget set, we utilize a generalization of knapsack called the tree knapsack problem that nicely captures the key combinatorial aspects of this "extraction problem." We prove a simple, yet strong, LP-relative approximation bound for tree knapsack, which leads to our improved guarantees for MST interdiction. Our algorithm and analysis are nearly tight, as we show that one cannot achieve an approximation ratio better than 3 relative to the upper bound used in our analysis (and the one in [Zenklusen, FOCS 2015]). Our guarantee for MST-interdiction yields an 8-approximation for metric-TSP interdiction (improving over the 28-approximation in [Zenklusen, FOCS 2015]). We also show that maximum-spanning-tree interdiction is at least as hard to approximate as the minimization version of densest-k-subgraph

    Improved Heterogeneous Distance Functions

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    Instance-based learning techniques typically handle continuous and linear input values well, but often do not handle nominal input attributes appropriately. The Value Difference Metric (VDM) was designed to find reasonable distance values between nominal attribute values, but it largely ignores continuous attributes, requiring discretization to map continuous values into nominal values. This paper proposes three new heterogeneous distance functions, called the Heterogeneous Value Difference Metric (HVDM), the Interpolated Value Difference Metric (IVDM), and the Windowed Value Difference Metric (WVDM). These new distance functions are designed to handle applications with nominal attributes, continuous attributes, or both. In experiments on 48 applications the new distance metrics achieve higher classification accuracy on average than three previous distance functions on those datasets that have both nominal and continuous attributes.Comment: See http://www.jair.org/ for an online appendix and other files accompanying this articl
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