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32
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 626 (6 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 69 (4 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 21 (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.
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 18 (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).
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 17 (7 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
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 16 (10 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
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 12 (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...
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...
Improved Approximations of Independent Dominating Set in Bounded Degree Graphs
, 1996
"... We consider the problem of finding an independent dominating set of minimum cardinality in bounded degree and regular graphs. We first give an approximation algorithm for at most cubic graphs, that achieves ratio 2, based on greedy and local search techniques. We then propose an heuristic based on a ..."
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Cited by 8 (0 self)
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We consider the problem of finding an independent dominating set of minimum cardinality in bounded degree and regular graphs. We first give an approximation algorithm for at most cubic graphs, that achieves ratio 2, based on greedy and local search techniques. We then propose an heuristic based on an iterative application of the greedy technique on graphs of lower and lower degree. When the graph is at most cubic, a local search phase is executed in order to improve the performance. Keywords: Minimum Independent Dominating Set, Bounded Degree Graphs, Regular Graphs, Cubic Graphs, Greedy, Local Search. 1 Introduction It is widely known that many NPcomplete graph problems remain NPcomplete even if restricted to bounded degree and regular graphs [3]. On the other hand, variation in which the degree of the graph is bounded by a constant often allows to achieve different results with respect to the approximation properties. Namely, problems that for general graphs cannot be approximated ...
On the Approximability of Removing the Smallest Number of Relations from Linear Systems to Achieve Feasibility
 Department of Mathematics, Swiss Federal Institute of Technology, Lausanne and
, 1995
"... We investigate the computational complexity of the problem which consists, given a system of linear relations, of finding a solution violating as few relations as possible while satisfying all the others. This general combinatorial problem, referred to as Min ULR, is considered for the four basic ty ..."
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Cited by 6 (4 self)
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We investigate the computational complexity of the problem which consists, given a system of linear relations, of finding a solution violating as few relations as possible while satisfying all the others. This general combinatorial problem, referred to as Min ULR, is considered for the four basic types of relational operators =, , ? and 6=. We proved in [3] that Min ULR with =, or ? relations is NPhard even when restricted to homogeneous systems with bipolar coefficients, whereas it is trivial for 6= relations. In this paper we determine strong bounds on the approximability of various intractable variants, including constrained ones where the variables are restricted to take bounded discrete values. The various NPhard versions of Min ULR belong to different approximability classes depending on the type of relations and the additional constraints, but none of them can be approximated within any constant factor unless P=NP. In the process of studying Min ULR we also derive strong boun...