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QoS Constrained Optimal Sink and Relay Placement in Planned Wireless Sensor Networks
"... Abstract — We are given a set of sensors at given locations, a set of potential locations for placing base stations (BSs, or sinks), and another set of potential locations for placing wireless relay nodes. There is a cost for placing a BS and a cost for placing a relay. The problem we consider is to ..."
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Abstract — We are given a set of sensors at given locations, a set of potential locations for placing base stations (BSs, or sinks), and another set of potential locations for placing wireless relay nodes. There is a cost for placing a BS and a cost for placing a relay. The problem we consider is to select a set of BS locations, a set of relay locations, and an association of sensor nodes with the selected BS locations, so that the number of hops in the path from each sensor to its BS is bounded by hmax, and among all such feasible networks, the cost of the selected network is the minimum. The hop count bound suffices to ensure a certain probability of the data being delivered to the BS within a given maximum delay under a light traffic model. We observe that the problem is NPHard, and is hard to even approximate within a constant factor. For this problem, we propose a polynomial time approximation algorithm (SmartSelect) based on a relay
APPROXIMATION ALGORITHMS FOR SUBMODULAR OPTIMIZATION AND GRAPH PROBLEMS
, 2013
"... In this thesis, we consider combinatorial optimization problems involving submodular functions and graphs. The problems we study are NPhard and therefore, assuming that P 6 = NP, there do not exist polynomialtime algorithms that always output an optimal solution. In order to cope with the intracta ..."
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In this thesis, we consider combinatorial optimization problems involving submodular functions and graphs. The problems we study are NPhard and therefore, assuming that P 6 = NP, there do not exist polynomialtime algorithms that always output an optimal solution. In order to cope with the intractability of these problems, we focus on algorithms that construct approximate solutions: An approximation algorithm is a polynomialtime algorithm that, for any instance of the problem, it outputs a solution whose value is within a multiplicative factor ρ of the value of the optimal solution for the instance. The quantity ρ is the approximation ratio of the algorithm and we aim to achieve the smallest ratio possible. Our focus in this thesis is on designing approximation algorithms for several combinatorial optimization problems. In the first part of this thesis, we study a class of constrained submodular minimization problems. We introduce a model that captures allocation problems with submodular costs and we give a generic approach for designing approximation algorithms for problems in this model. Our model captures several problems of interest, such as nonmetric facility location, multiway cut problems in graphs and hypergraphs, uniform metric labeling and its generalization
Halfintegrality, LPbranching and FPT Algorithms
, 2014
"... A recent trend in parameterized algorithms is the application of polytope tools (specifically, LPbranching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). Though the list of work in this direction is short, the results are already interesting, yielding significant speedup ..."
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A recent trend in parameterized algorithms is the application of polytope tools (specifically, LPbranching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). Though the list of work in this direction is short, the results are already interesting, yielding significant speedups for a range of important problems. However, the existing approaches require the underlying polytope to have very restrictive properties, including halfintegrality and NemhauserTrotterstyle persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994). Taking a slightly different approach, we view halfintegrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from {0, 1}V to {0, 1/2, 1}V such that the new problem admits a polynomialtime exact solution. Using tools from CSP (in particular Thapper and Živný, 2012) to study the existence of such relaxations, we are able to provide a much broader class of halfintegral polytopes with the required properties. Our results unify and significantly extend the previously known cases. In addition to the new insight into problems with halfintegral relaxations, our results yield a range of new and improved FPT algo
How to Cut a Graph into Many Pieces
, 2011
"... In this paper we consider the problem of finding a graph separator of a given size that decomposes the graph into the maximum number of connected components. We present the picture of the computational complexity and the approximability of this problem for several natural classes of graphs. We fi ..."
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In this paper we consider the problem of finding a graph separator of a given size that decomposes the graph into the maximum number of connected components. We present the picture of the computational complexity and the approximability of this problem for several natural classes of graphs. We first provide an overview of the hardness of approximation of this problem, which stems mainly from its close relation to the INDEPENDENT SET and to the MAXIMUM CLIQUE problem. Next, we show that the problem is solvable in polynomial time for interval graphs and graphs of bounded treewidth. We also show that MAXINUM COMPONENTS is fixedparameter tractable on planar graphs with the size of the separator as the parameter. Our main contribution is the derivation of an efficient polynomialtime approximation scheme for the problem on planar graphs.
Restricted vertex multicut on permutation graphs
"... Abstract Given an undirected graph and pairs of terminals the Restricted Vertex Multicut problem asks for a minimum set of nonterminal vertices whose removal disconnects each pair of terminals. The problem is known to be NPcomplete for trees and polynomialtime solvable for interval graphs. In thi ..."
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Abstract Given an undirected graph and pairs of terminals the Restricted Vertex Multicut problem asks for a minimum set of nonterminal vertices whose removal disconnects each pair of terminals. The problem is known to be NPcomplete for trees and polynomialtime solvable for interval graphs. In this paper we give a polynomialtime algorithm for the problem on permutation graphs. Furthermore we show that the problem remains NPcomplete on split graphs whereas it becomes polynomialtime solvable for the class of cobipartite graphs.
Prophylactic Vaccination . . .
, 2008
"... Motivated by preventative vaccination in a graph against the worstcase outbreak of an infectious disease, we propose new important graph cut problems. In the most basic problem MinMax Component Size, we are given a capacity bound, and the goal is to remove a set of nodes (or edges) whose capacity ..."
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Motivated by preventative vaccination in a graph against the worstcase outbreak of an infectious disease, we propose new important graph cut problems. In the most basic problem MinMax Component Size, we are given a capacity bound, and the goal is to remove a set of nodes (or edges) whose capacity does not exceed the bound, and minimizes the size of the largest resulting connected component. In generalizations of this problem, we consider the objectives of minimizing the size of the k largest components for a fixed k, and the maximum number of special “terminal ” nodes inside any component. Under the assumption that each edge of a network will deterministically transmit a disease from either endpoint to the other, these problems naturally model the goal of targeting limited vaccinations of nodes (or edges) in such a way that the number of infected nodes in a worstcase outbreak is minimized. We present (O(1), O(log n)) bicriteria approximation algorithms for the MinMax Component Size problem and for the generalization to k infected components. Our algorithms are based on LP rounding and region growing techniques. If instead, a bound k on the number of terminals inside any component is given, and the goal is to minimize the total capacity of nodes (or edges) removed, we improve the approximation guarantee to an (O(1), O(log k)) bicriteria result. Finally, if k is a constant, and the edgecut version is considered, we show how to obtain a combinatorial singlecriterion approximation not violating the bound on the component size, and approximating the capacity of edges removed to within O(log k).
Approximation Algorithms for kHurdle Problems
"... Abstract. The polynomialtime solvable khurdle problem is a natural generalization of the classical st minimum cut problem where we must select a minimumcost subset S of the edges of a graph such that ..."
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Abstract. The polynomialtime solvable khurdle problem is a natural generalization of the classical st minimum cut problem where we must select a minimumcost subset S of the edges of a graph such that