A linear time algorithm for finding tree-decompositions of small treewidth
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
Fixed-parameter tractability and completeness II: on completeness for W[1]
Theoretical Computer Science
Journal of the ACM (JACM)
Spanning Trees---Short or Small
SIAM Journal on Discrete Mathematics
Bicriteria network design problems
Journal of Algorithms
An efficient algorithm for the length-constrained heaviest path problem on a tree
Information Processing Letters
Subgraph isomorphism in planar graphs and related problems
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Data structures for weighted matching and nearest common ancestors with linking
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Improved Steiner tree approximation in graphs
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Journal of Computer and System Sciences - Computational biology 2002
The non-approximability of bicriteria network design problems
Journal of Discrete Algorithms
An Optimal Algorithm for the Maximum-Density Segment Problem
SIAM Journal on Computing
Journal of Computer and System Sciences
Finding a maximum-density path in a tree under the weight and length constraints
Information Processing Letters
Fast Algorithms for the Density Finding Problem
Algorithmica
An optimal algorithm for the maximum-density path in a tree
Information Processing Letters
Finding a weight-constrained maximum-density subtree in a tree
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
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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 驴驴 and edge lengths ℓ e 驴驴 for e驴E we define the density of a pattern subgraph H=(V驴,E驴)⊆G as the ratio 驴(H)=驴 e驴E驴 w e /驴 e驴E驴 ℓ 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 biconstrained 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.