Algorithms for finding the weight-constrained k longest paths in a tree and the length-constrained k maximum-sum segments of a sequence

Liu, Hsiao-Fei; Chao, Kun-Mao

doi:10.1016/j.tcs.2008.06.052

Cited by 14 publications

(11 citation statements)

References 35 publications

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“…Wu et al (2009Wu et al ( , 1999 improve on this by presenting an optimal algorithm for computing a maximum density path in a tree in time O(n log n) in the presence of both a lower and upper length bounds. They also give an O(n log 2 n) algorithm for finding a heaviest path in a tree in the presence of length constraints (Wu et al 1999), which is improved to O(n log n) by Liu and Chao (2008). Problems involving Steiner constraints have been widely studied in computer science for a long time.…”

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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Section: Problem Maximum Density Steiner Subgraphmentioning

confidence: 99%

The density maximization problem in graphs

et al. 2012

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“…Wu et al [21,20] improve on this by presenting an optimal algorithm for computing a maximum density path in a tree in time O(n log n) in the presence of both a lower and upper length bounds. They also give an O(n log 2 n) algorithm for finding a heaviest path in a tree in the presence of length constraints [21], which is improved to O(n log n) by Liu and Chao [14] .…”

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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“…, a j ) to be j − i + 1, j h=i a h , and ( j h=i a h )/( j − i + 1), respectively. The Maximum-Sum Segment problem, which finds a segment maximizing the sum, is widely formulated in pattern recognition [16,22], image processing [15], biological sequence analysis [1,13,17,20,23,25], and data mining [14,15]. It was first surveyed by Bentley in his "Programming Pearls" column of CACM [6,7] and is lineartime solvable using Kadane's algorithm [6].…”

Section: Introductionmentioning

confidence: 99%

“…Since then, there have been many variants proposed. The k MaximumSum Segments problem [3][4][5]10,12,18,19] is to locate the k segments whose sums are the k largest among all possible sums, and is solvable in O (n + k) time [10,21]. The Range Maximum-Sum Segment Query (RMSQ) problem is to preprocess the input sequence such that any range maximum-sum segment query can be answered quickly, where a range maximum-sum segment query specifies two intervals [i, j] and [k, l] and the goal is to find a segment A(x, y) with maximum sum subject to i x j and k y .…”

Section: Introductionmentioning

confidence: 99%