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39
A Threshold of ln n for Approximating Set Cover
 JOURNAL OF THE ACM
, 1998
"... Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhar ..."
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Cited by 778 (5 self)
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Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhard. We prove that (1 \Gamma o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1 \Gamma o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max kcover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .
On the Approximability of Minimizing Nonzero Variables Or Unsatisfied Relations in Linear Systems
, 1997
"... We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning, operations research and pattern recognition. In the first class (Min ULR) one wishes, given a possibly infeasib ..."
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Cited by 115 (3 self)
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We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning, operations research and pattern recognition. In the first class (Min ULR) one wishes, given a possibly infeasible system of linear relations, to find a solution that violates as few relations as possible while satisfying all the others. In the second class (Min RVLS) the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible. For both Min ULR and Min RVLS the four basic types of relational operators =, , ? and 6= are considered. While Min RVLS with equations was known to be NPhard in [27], we established in [2, 5] that Min ULR with equalities and inequalities are NPhard even when restricted to homogeneous systems with bipolar coefficients. The latter problems have been shown hard to approximate in [8]. In this paper we determine strong bou...
The bchromatic number of a graph
 Discrete Applied Math
, 1999
"... The achromatic number ψ(G) of a graph G = (V, E) is the maximum k such that V has a partition V1, V2,..., Vk into independent sets, the union of no pair of which is independent. Here we show that ψ(G) can be viewed as the maximum over all minimal elements of a partial order defined on the set of all ..."
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Cited by 37 (0 self)
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The achromatic number ψ(G) of a graph G = (V, E) is the maximum k such that V has a partition V1, V2,..., Vk into independent sets, the union of no pair of which is independent. Here we show that ψ(G) can be viewed as the maximum over all minimal elements of a partial order defined on the set of all colourings of G. We introduce a natural refinement of this partial order, giving rise to a new parameter, which we call the bchromatic number, ϕ(G), of G. We prove that determining ϕ(G) is NPhard for general graphs, but polynomialtime solvable for trees.
Independent Sets With Domination Constraints
, 1999
"... A #independent set S in a graph is parameterized by a set # of nonnegative integers that constrains how the independent set S can dominate the remaining vertices (#v ## S : N(v) # S # #.) For all values of #, we classify as either NPcomplete or polynomialtime solvable the problems of ..."
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Cited by 23 (8 self)
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A #independent set S in a graph is parameterized by a set # of nonnegative integers that constrains how the independent set S can dominate the remaining vertices (#v ## S : N(v) # S # #.) For all values of #, we classify as either NPcomplete or polynomialtime solvable the problems of deciding if a given graph has a #independent set. We complement this with approximation algorithms and inapproximability results, for all the corresponding optimization problems. These approximation results extend also to several related independence problems. In particular, we obtain a # m approximation of the Set Packing problem, where m is the number of base elements, as well as a # n approximation of the maximum independent set in power graphs G t , for t even. 1
Complexity Issues in Discrete Hopfield Networks
, 1994
"... We survey some aspects of the computational complexity theory of discretetime and discretestate Hopfield networks. The emphasis is on topics that are not adequately covered by the existing survey literature, most significantly: 1. the known upper and lower bounds for the convergence times of Hopfi ..."
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Cited by 19 (4 self)
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We survey some aspects of the computational complexity theory of discretetime and discretestate Hopfield networks. The emphasis is on topics that are not adequately covered by the existing survey literature, most significantly: 1. the known upper and lower bounds for the convergence times of Hopfield nets (here we consider mainly worstcase results); 2. the power of Hopfield nets as general computing devices (as opposed to their applications to associative memory and optimization); 3. the complexity of the synthesis ("learning") and analysis problems related to Hopfield nets as associative memories. Draft chapter for the forthcoming book The Computational and Learning Complexity of Neural Networks: Advanced Topics (ed. Ian Parberry).
Minimum independent dominating sets of random cubic graphs. Random Structures and Algorithms
, 2002
"... We present a heuristic for finding a small independent dominating set, D, of cubic graphs. We analyse the performance of this heuristic, which is a random greedy algorithm, on random cubic graphs using differential equations and obtain an upper bound on the expected size of D. A corresponding lower ..."
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Cited by 19 (12 self)
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We present a heuristic for finding a small independent dominating set, D, of cubic graphs. We analyse the performance of this heuristic, which is a random greedy algorithm, on random cubic graphs using differential equations and obtain an upper bound on the expected size of D. A corresponding lower bound is derived by means of a direct expectation argument. We prove that D asymptotically almost surely satisfies 0.2641n ≤ D  ≤ 0.2794n. 1
On the Independent Domination Number of Random Regular Graphs
 COMBINATORICS, PROBABILITY AND COMPUTING
, 2003
"... A dominating set D of a graph G is a subset of V (G) such that for every vertex v ∈ V (G), either in v ∈ D or there exists a vertex u ∈ D that is adjacent to v. We are interested in finding dominating sets of small cardinality. A dominating set I of a graph G is said to be independent if no two vert ..."
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Cited by 13 (4 self)
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A dominating set D of a graph G is a subset of V (G) such that for every vertex v ∈ V (G), either in v ∈ D or there exists a vertex u ∈ D that is adjacent to v. We are interested in finding dominating sets of small cardinality. A dominating set I of a graph G is said to be independent if no two vertices of I are connected by an edge of G. The size of a smallest independent dominating set of a graph G is the independent domination number of G. In this paper we present upper bounds on the independent domination number of random regular graphs. This is achieved by analysing the performance of a randomised greedy algorithm on random regular graphs using differential equations.
Strong lower bounds on the approximability of some NPO PBcomplete maximization problems
, 1995
"... The approximability of several NP maximization problems is investigated and strong lower bounds for the studied problems are proved. For some of the problems the bounds are the best that can be achieved, unless P = NP. For example we investigate the approximability of Max PB 0 \Gamma 1 Programming ..."
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Cited by 10 (2 self)
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The approximability of several NP maximization problems is investigated and strong lower bounds for the studied problems are proved. For some of the problems the bounds are the best that can be achieved, unless P = NP. For example we investigate the approximability of Max PB 0 \Gamma 1 Programming, the problem of finding a binary vector x that satisfies a set of linear relations such that the objective value P c i x i is maximized, where c i are binary numbers. We show that, unless P = NP, Max PB 0 \Gamma 1 Programming is not approximable within the factor n 1\Gamma" for any " ? 0, where n is the number of inequalities, and is not approximable within m 1=2\Gamma" for any " ? 0, where m is the number of variables. Similar hardness results are shown for other problems on binary linear systems, some problems on the satisfiability of boolean formulas and the longest induced cycle problem. 1 Introduction Approximation of NPcomplete optimization problems is a very interesting and...
Approximation Algorithms for Certain Network Improvement Problems
 J. Comb. Optim
, 1998
"... . We study budget constrained network upgrading problems. Such problems aim at nding optimal strategies for improving a network under some cost measure subject to certain budget constraints. Given an edge weighted graph G = (V; E), in the edge based upgrading model, it is assumed that each edge e of ..."
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Cited by 9 (0 self)
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. We study budget constrained network upgrading problems. Such problems aim at nding optimal strategies for improving a network under some cost measure subject to certain budget constraints. Given an edge weighted graph G = (V; E), in the edge based upgrading model, it is assumed that each edge e of the given network also has an associated function ce (t) that species the cost of upgrading the edge by an amount t. A reduction strategy species for each edge e the amount by which the length `(e) is to be reduced. In the node based upgrading model, a node v can be upgraded at an expense of c(v). Such an upgrade reduces the delay of each edge incident on v. For a given budget B, the goal is to nd an improvement strategy such that the total cost of reduction is at most the given budget B and the cost of a subgraph (e.g. minimum spanning tree) under the modied edge lengths is the best over all possible strategies which obey the budget constraint. After providing a brief overview of the...
DisC Diversity: Result Diversification based on Dissimilarity and Coverage
, 2012
"... Recently, result diversification has attracted a lot of attention as a means to improve the quality of results retrieved by user queries. In this paper, we propose a new, intuitive definition of diversity called DisC diversity. A DisC diverse subset of a query result contains objects such that each ..."
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Cited by 9 (2 self)
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Recently, result diversification has attracted a lot of attention as a means to improve the quality of results retrieved by user queries. In this paper, we propose a new, intuitive definition of diversity called DisC diversity. A DisC diverse subset of a query result contains objects such that each object in the result is represented by a similar object in the diverse subset and the objects in the diverse subset are dissimilar to each other. We show that locating a minimum DisC diverse subset is an NPhard problem and provide heuristics for its approximation. We also propose adapting DisC diverse subsets to a different degree of diversification. We call this operation zooming. We present efficient implementations of our algorithms based on the Mtree, a spatial index structure, and experimentally evaluate their performance.