Experimental design is a classical area in statistics [21] and has also found new applications in machine learning[2]. In the combinatorial experimental design problem, the aim is to estimate an unknown m-dimensional ...
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ISBN:
(纸本)9781611975031
Experimental design is a classical area in statistics [21] and has also found new applications in machine learning[2]. In the combinatorial experimental design problem, the aim is to estimate an unknown m-dimensional vector x from linear measurements where a Gaussian noise is introduced in each measurement. The goal is to pick k out of the given n experiments so as to make the most accurate estimate of the unknown parameter x. Given a set S of chosen experiments, the most likelihood estimate x' can be obtained by a least squares computation. One of the robust measures of error estimation is the D-optimality criterion [27] which aims to minimize the generalized variance of the estimator. This corresponds to minimizing the volume of the standard confidence ellipsoid for the estimation error x - x'. The problem gives rise to two natural variants depending on whether repetitions of experiments is allowed or not. The latter variant, while being more general, has also found applications in geographical location of sensors [19]. We show a close connection between approximation algorithms for the D-optimal design problem and constructions of approximately m-wise positively correlated distributions. This connection allows us to obtain a 1/e-approximation for the D-optimal design problem with and without repetitions giving the first constant factor approximation for the problem. We then consider the case when the number of experiments chosen is much larger than the dimension m and show one can obtain (1-epsilon)-approximation if k >= 2m/epsilon when repetitions are allowed and if k = O(m/epsilon + 1/epsilon(2) . log 1/epsilon) when no repetitions are allowed improving on previous work.
Facility location is a prominent optimization problem that has inspired a large quantity of both theoretical and practical studies in combinatorial optimization. Although the problem has been investigated under variou...
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We give new approximation algorithms for the submodular joint replenishment problem and the inventory routing problem, using an iterative rounding approach. In both problems, we are given a set of N items and a discre...
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Some of the most fundamental and well-studied graph parameters are the Diameter (the largest shortest paths distance) and Radius (the smallest distance for which a "center" node can reach all other nodes). T...
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The Steiner tree problem is one of the fundamental and classical problems in combinatorial optimization. In this paper we study this problem in the CONGEST ED CLIQUE model of distributed computing and present two dete...
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We describe approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in a hypergraph of rank r. Our main result is a deterministic algorithm to generate a ...
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We describe approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in a hypergraph of rank r. Our main result is a deterministic algorithm to generate a matching which is an O(r)-approximation to the maximum weight matching, running in Õ(r log Δ + log 2 Δ + log* n) rounds. (Here, the Õ() notations hides polyloglog Δ and polylog r factors). This is based on a number of new derandomization techniques extending methods of Ghaffari, Harris & Kuhn (2017). The first main application is to nearly-optimal algorithms for the long-studied problem of maximum-weight graph matching. Specifically, we get a (1+ε) approximation algorithm using Õ(log Δ/ε 3 + polylog(1/ε, log log n)) randomized time and Õ(log 2 Δ/ε 4 + log*n/ε) deterministic time. The second application is a faster algorithm for hypergraph maximal matching, a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of local graph algorithms. This gives an algorithm for (2Δ - 1) -edge-list coloring in Õ(log 2 Δ log n) rounds deterministically or Õ((log log n) 3 ) rounds randomly. Another consequence (with additional optimizations) is an algorithm which generates an edge-orientation with out-degree at most ⌈(1+ε)λ⌉ for a graph of arboricity λ; for fixed ε this runs in Õ(log 6 n) rounds deterministically or Õ(log 3 n ) rounds randomly.
As a dual problem of influence maximization, the seed minimization problem asks for the minimum number of seed nodes to influence a required number η of users in a given social network G. Existing algorithms for seed...
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We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-f...
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Given a social network G, the profit maximization (PM) problem asks for a set of seed nodes to maximize the profit, i.e., revenue of influence spread less the cost of seed selection. The target profit maximization (TP...
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Given an undirected graph with edge costs, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless netwo...
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ISBN:
(纸本)9783030046934;9783030046927
Given an undirected graph with edge costs, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider two network design problems under the power minimization criteria. In both problems we are given a graph G = (V, E) with edge costs and a set T subset of V of terminals. The goal is to find a minimum power edge subset F subset of E such that the graph H = (V, F) satisfies some prescribed requirements. In the MIN-POWER EDGECOVER problem, H should contain an edge incident to every terminal. Using the Iterative Randomized Rounding (IRR) method, we give an algorithm with expected approximation ratio 1.41;the ratio is reduced to 73/60 < 1.217 when T is an independent set in G. In the case of unit costs we also achieve ratio 73/60, and in addition give a simple efficient combinatorial algorithm with ratio 5/4. For all these NP-hard problems the previous best known ratio was 3/2. In the related MINPOWER TERMINAL BACKUP problem, H should contain a path from every t E T to some node in T \ {t}. We obtain ratio 3/2 for this NPhard problem, improving the trivial ratio of 2.
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