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 ℓ e ∈ℕ for e∈E we define the density of a pattern subgraph H=(V′,E′)⊆G as the ratio ϱ(H)=∑ e∈E′ w e /∑ e∈E′ 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-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.
This is a preview of subscription content,to check access.
Access this article
Alon N, Yuster R, Zwick U (1995) Color-coding. J ACM 42(4):844–856
Bálint V (2003) The non-approximability of bicriteria network design problems. J Discrete Algorithms 1:339–355
Bodlaender HL (1993) A linear time algorithm for finding tree-decompositions of small treewidth. In: STOC’93: Proceedings of the 25th annual ACM symposium on theory of computing. ACM, New York, pp 226–234
Chandrasekaran R (1977) Minimal ratio spanning trees. Networks 7(4):335–342
Chinchuluun A, Pardalos P (2007) A survey of recent developments in multiobjective optimization. Ann Oper Res 154:29–50
Chung KM, Lu HI (2005) An optimal algorithm for the maximum-density segment problem. SIAM J Comput 34(2):373–387
Downey RG, Fellows MR (1995) Fixed-parameter tractability and completeness II: On completeness for W. Theor Comput Sci 141(1–2):109–131
Dreyfus S, Wagner R (1971) The Steiner problem in graphs. Networks 1(3):195–207
Eppstein D (1995) Subgraph isomorphism in planar graphs and related problems. In: Proc 6th ann ACM-SIAM sympos disc alg SIAM, Philadelphia, pp 632–640
Gabow HN (1990) Data structures for weighted matching and nearest common ancestors with linking. In: Proceedings of the first annual ACM-SIAM symposium on discrete algorithms, SODA’90. Society for Industrial and Applied Mathematics, Philadelphia, pp 434–443
Garey MR, Johnson DS (1979) Computers and intractability. A guide to the theory of NP-completeness. Freeman, New York
Goldwasser MH, Kao MY, Lu HI (2005) Linear-time algorithms for computing maximum-density sequence segments with bioinformatics applications. J Comput Syst Sci 70(2):128–144
Hsieh SY, Cheng CS (2008) Finding a maximum-density path in a tree under the weight and length constraints. Inf Process Lett 105(5):202–205
Hsieh SY, Chou TY (2005) Finding a weight-constrained maximum-density subtree in a tree. In: Algorithms and computation. LNCS, vol 3827. Springer, Berlin, pp 944–953
Inman RB (1966) A denaturation map of the lambda phage DNA molecule determined by electron microscopy. J Mol Biol 18(3):464–476
Karger D, Motwani R, Ramkumar G (1997) On approximating the longest path in a graph. Algorithmica 18:82–98
Kloks T (1994) Treewidth, Computations and approximations. LNCS. Springer, Berlin
Lau HC, Ngo TH, Nguyen BN (2006) Finding a length-constrained maximum-sum or maximum-density subtree and its application to logistics. Discrete Optim 3(4):385–391
Lee DT, Lin TC, Lu HI (2009) Fast algorithms for the density finding problem. Algorithmica 53(3):298–313
Lin YL, Jiang T, Chao KM (2002) Efficient algorithms for locating the length-constrained heaviest segments with applications to biomolecular sequence analysis. J Comput Syst Sci 65(3):570–586
Liu HF, Chao KM (2008) Algorithms for finding the weight-constrained k longest paths in a tree and the length-constrained k maximum-sum segments of a sequence. Theor Comput Sci 407(1–3):349–358
Lokshtanov D (2009) New methods in parameterized algorithms and complexity. PhD thesis, University of Bergen Norway
Macaya G, Thiery JP, Bernardi G (1976) An approach to the organization of eukaryotic genomes at a macromolecular level. J Mol Biol 108(1):237–254
Marathe MV, Ravi R, Sundaram R, Ravi SS, Rosenkrantz DJ, Hunt HB (1998) Bicriteria network design problems. J Algorithms 28(1):142–171
McCreight EM (1985) Priority search trees. SIAM J Comput 14(2):257–276
Overmars MH, van Leeuwen J (1981) Maintenance of configurations in the plane. J Comput Syst Sci 23(2):166–204
Ravi R, Sundaram R, Marathe MV, Rosenkrantz DJ, Ravi SS (1996) Spanning trees—short or small. SIAM J Discrete Math 9:178–200
Robertson N, Seymour PD (1984) Graph minors. iii. Planar tree-width. J Comb Theory, Ser B 36(1):49–64
Robertson N, Seymour PD (1995) Graph minors. XIII. The disjoint paths problem. J Comb Theory, Ser B 63(1):65–110
Robins G, Zelikovsky A (2000) Improved steiner tree approximation in graphs. In: Proceedings of the eleventh annual ACM–SIAM symposium on discrete algorithms, SODA’00. Society for Industrial and Applied Mathematics, Philadelphia, pp 770–779
Schuurman P, Woeginger G (2011) Approximation schemes—a tutorial. URL www.win.tue.nl/~gwoegi/papers/ptas.pdf. Preliminary version of a chapter in the book Lectures on Scheduling, to appear
Wu BY (2009) An optimal algorithm for the maximum-density path in a tree. Inf Process Lett 109(17):975–979
Wu BY, Chao KM, Tang CY (1999) An efficient algorithm for the length-constrained heaviest path problem on a tree. Inf Process Lett 69(2):63–67
Supported by NSC-DFG Projects NSC98-2221-E-001-007-MY3 and WA 654/18.
Also supported by the Institute of Information Science, Academia Sinica, Taiwan.
About this article
Cite this article
Kao, MJ., Katz, B., Krug, M. et al. The density maximization problem in graphs. J Comb Optim 26, 723–754 (2013). https://doi.org/10.1007/s10878-012-9465-z