Abstract
Given a directed acyclic graph with positive edge-weights, two vertices s and t, and a threshold-weight L, we present a fully-polynomial time approximation-scheme for the problem of counting the s-t paths of length at most L. We extend the algorithm for the case of two (or more) instances of the same problem. That is, given two graphs that have the same vertices and edges and differ only in edge-weights, and given two threshold-weights L 1 and L 2, we show how to approximately count the s-t paths that have length at most L 1 in the first graph and length not much larger than L 2 in the second graph. We believe that our algorithms should find application in counting approximate solutions of related optimization problems, where finding an (optimum) solution can be reduced to the computation of a shortest path in a purpose-built auxiliary graph.
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References
Broder, A.Z., Mayr, E.W.: Counting minimum weight spanning trees. J. Algorithms 24, 171–176 (1997)
Buhmann, J.M., Mihalák, M., Šrámek, R., Widmayer, P.: Robust optimization in the presence of uncertainty. In: Proc. 4th Conference on Innovations in Theoretical Computer Sciencei (ITCS), pp. 505–514. ACM, New York (2013)
Burge, C., Karlin, S.: Prediction of complete gene structures in human genomic DNA. Journal of Molecular Biology 268(1), 78–94 (1997)
Chen, T., Kao, M.Y., Tepel, M., Rush, J., Church, G.M.: A dynamic programming approach to de novo peptide sequencing via tandem mass spectrometry. Journal of Computational Biology 8(3), 325–337 (2001)
Durbin, R., Eddy, S.R., Krogh, A., Mitchison, G.: Biological sequence analysis: Probabilistic models of proteins and nucleic acids. Cambridge university press (1998)
Dyer, M., Frieze, A., Kannan, R., Kapoor, A., Perkovic, L., Vazirani, U.: A mildly exponential time algorithm for approximating the number of solutions to a multidimensional knapsack problem. Combinatorics, Probability and Computing 2(3), 271–284 (1993)
Gopalan, P., Klivans, A., Meka, R., Štefankovič, D., Vempala, S., Vigoda, E.: An FPTAS for # knapsack and related counting problems. In: Proc. 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 817–826 (2011)
Jerrum, M.: Two-dimensional monomer-dimer systems are computationally intractable. Journal of Statistical Physics 48(1-2), 121–134 (1987)
Karp, R.M.: Reducibility among combinatorial problems. Springer (1972)
Kreher, D.L., Stinson, D.R.: Combinatorial Algorithms: Generation, Enumeration, and Search (1998)
Lu, B., Chen, T.: A suboptimal algorithm for de novo peptide sequencing via tandem mass spectrometry. Journal of Computational Biology 10(1), 1–12 (2003)
Mihalák, M., Šrámek, R., Widmayer, P.: Counting approximately-shortest paths in directed acyclic graphs. arXiv preprint arXiv:1304.6707 (2013)
Naor, D., Brutlag, D.: On suboptimal alignments of biological sequences. In: Apostolico, A., Crochemore, M., Galil, Z., Manber, U. (eds.) CPM 1993. LNCS, vol. 684, pp. 179–196. Springer, Heidelberg (1993)
Štefankovič, D., Vempala, S., Vigoda, E.: A deterministic polynomial-time approximation scheme for counting knapsack solutions. SIAM Journal on Computing 41(2), 356–366 (2012)
Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8(2), 189–201 (1979)
Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979)
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Mihalák, M., Šrámek, R., Widmayer, P. (2014). Counting Approximately-Shortest Paths in Directed Acyclic Graphs. In: Kaklamanis, C., Pruhs, K. (eds) Approximation and Online Algorithms. WAOA 2013. Lecture Notes in Computer Science, vol 8447. Springer, Cham. https://doi.org/10.1007/978-3-319-08001-7_14
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DOI: https://doi.org/10.1007/978-3-319-08001-7_14
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