The non-approximability of bicriteria network design problems

Bálint, Vojtech

doi:10.1016/s1570-8667(03)00033-9

Cited by 6 publications

(6 citation statements)

References 10 publications

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“…A is a criterion on the weight function c and B is a criterion on the weight function l. For instance, A could be cost and B could be diam (as defined in section 2). is a real number not less than 1 and C is a class of graphs that limits the possible solutions (e.g., trees) [Bálint 2003].…”

Section: Related Workmentioning

confidence: 99%

“…The minimum value of criterion B under a budget on A is defined as B = min{B(S)|S 2 C(G), A(S)  A ⇤ }. Note that indirectly defines a budget [Bálint 2003].…”

Section: Related Workmentioning

confidence: 99%

“…The only difference is that here the budget is set indirectly by instead of directly by R on the MSDP, i.e., for a budget R we should set = R/A ⇤ . An (↵, )-approximation to the (A, B, , C)-problem is a spanning subgraph S 2 C(G) such that A(S)  ↵ A ⇤ and B(S)  B [Bálint 2003].…”

Section: Related Workmentioning

confidence: 99%

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On the Approximability of the Minimum Subgraph Diameter Problem

Dadalto¹,

Usberti²,

Felice³

2018

Anais Do Encontro De Teoria Da Computação (ETC)

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This work addresses the minimum subgraph diameter problem (MSDP) by answering an open question with respect to its approximability. Given a graph with lengths and costs associated to its edges, the MSDP consists in finding a spanning subgraph with total cost limited by a given budget, such that its diameter is minimum. We prove that there is no -approximation algorithm for the MSDP, for any constant , unless P = NP. Our proof is grounded on the non-approximability of the minimum spanning tree diameter problem, proven by Bálint in 2013.

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Section: Related Workmentioning

confidence: 99%

“…The minimum value of criterion B under a budget on A is defined as B = min{B(S)|S 2 C(G), A(S)  A ⇤ }. Note that indirectly defines a budget [Bálint 2003].…”

Section: Related Workmentioning

confidence: 99%

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On the Approximability of the Minimum Subgraph Diameter Problem

Dadalto¹,

Usberti²,

Felice³

2018

Anais Do Encontro De Teoria Da Computação (ETC)

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“…Bálint (2003) proves inapproximability for bi-objective network optimization problems, where the task is to minimize the diameter of a spanning subgraph with respect to a given length on the edges, subject to a limited budget on the total cost of the edges. Marathe et al (1998) study biobjective network design problems with two minimization objectives.…”

Section: Problem Maximum Density Steiner Subgraphmentioning

confidence: 99%

The density maximization problem in graphs

et al. 2012

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We consider a framework for bi-objective network construction problems where one objective is to be maximized while the other is to be minimized. Given a host graph G = (V , E) with edge weights w e ∈ Z and edge lengths e ∈ N for e ∈ E we define the density of a pattern subgraph H = (V , E ) ⊆ G as the ratio (H ) = e∈E w e / e∈E e . We consider the problem of computing a maximum density pattern H under various additional constraints. In doing so, we compute a single Pareto-optimal solution with the best weight per cost ratio subject to additional constraints further narrowing down feasible solutions for the underlying bi-objective network construction problem.First, we consider the problem of computing a maximum density pattern with weight at least W and length at most L in a host G. We call this problem the bi- constrained density maximization problem. This problem can be interpreted in terms of maximizing the return on investment for network construction problems in the presence of a limited budget and a target profit. We consider this problem for different classes of hosts and patterns. We show that it is NP-hard, even if the host has treewidth 2 and the pattern is a path. However, it can be solved in pseudo-polynomial linear time if the host has bounded treewidth and the pattern is a graph from a given minor-closed family of graphs. Finally, we present an FPTAS for a relaxation of the density maximization problem, in which we are allowed to violate the upper bound on the length at the cost of some penalty.Second, we consider the maximum density subgraph problem under structural constraints on the vertex set that is used by the patterns. While a maximum density perfect matching can be computed efficiently in general graphs, the maximum density Steiner-subgraph problem, which requires a subset of the vertices in any feasible solution, is NP-hard and unlikely to admit a constant-factor approximation. When parameterized by the number of vertices of the pattern, this problem is W[1]-hard in general graphs. On the other hand, it is FPT on planar graphs if there is no constraint on the pattern and on general graphs if the pattern is a path.

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“…Bálint [1] proves inapproximability for bi-objective network optimization problems, where the task is to minimize the diameter of a spanning subgraph with respect to a given length on the edges, subject to a limited budget on the total cost of the edges. Marathe et al [16] study bi-objective network design problems with two minimization objectives.…”

Section: Introductionmentioning

confidence: 99%

The Density Maximization Problem in Graphs

Kao

Katz

Krug

et al. 2011

Lecture Notes in Computer Science

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Abstract. We consider a framework for bi-objective network construction problems where one objective is to be maximized while the other is to be minimized. Given a host graph G = (V, E) with edge weights we ∈ Z and edge lengths e ∈ N for e ∈ E we define the density of a pattern subgraph H = (V , E ) ⊆ G as the ratio (H) = e∈E we/ e∈E e. We consider the problem of computing a maximum density pattern H with weight at least W and and length at most L in a host G. We call this problem the bi-constrained density maximization problem. In doing so, we compute a single Pareto-optimal solution with the best weight per cost ratio subject to additional constraints further narrowing down feasible solutions for the underlying bi-objective network construction problem. The problems expressible within this framework include maximization of return on investment for network construction problems in the presence of a limited budget and a target profit. We consider this problem for different classes of hosts and patterns. We show that it is NPhard even if the host has treewidth 2 and the pattern is a path. However, it can be solved in pseudo-polynomial linear time if the host has bounded treewidth and the pattern is a graph from a given minor-closed family of graphs. Finally, we present an FPTAS for a relaxation of the density maximization problem, in which we are allowed to violate the upper bound on the length at the cost of some penalty.

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The non-approximability of bicriteria network design problems

Cited by 6 publications

References 10 publications

On the Approximability of the Minimum Subgraph Diameter Problem

On the Approximability of the Minimum Subgraph Diameter Problem

The density maximization problem in graphs

The Density Maximization Problem in Graphs

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