Abstract
For a graph \(G=(V,E)\) the minimum line-distortion problem asks for the minimum k such that there is a mapping f of the vertices into points of the line such that for each pair of vertices x, y the distance on the line \(|f(x) - f(y)|\) can be bounded by the term \(d_G(x, y)\le |f(x)-f(y)|\le k \, d_G(x, y)\), where \(d_G(x, y)\) is the distance in the graph. The minimum bandwidth problem minimizes the term \(\max _{uv\in E}|f(u)-f(v)|\), where f is a mapping of the vertices of G into the integers \(\{1, \ldots , n\}\). We investigate the minimum line-distortion and the minimum bandwidth problems on unweighted graphs and their relations with the minimum length of a Robertson–Seymour’s path-decomposition. The length of a path-decomposition of a graph is the largest diameter of a bag in the decomposition. The path-length of a graph is the minimum length over all its path-decompositions. In particular, we show:
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there is a simple polynomial time algorithm that embeds an arbitrary unweighted input graph G into the line with distortion \(\mathcal{O}(k^2)\), where k is the minimum line-distortion of G;
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if a graph G can be embedded into the line with distortion k, then G admits a Robertson–Seymour’s path-decomposition with bags of diameter at most k in G;
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for every class of graphs with path-length bounded by a constant, there exist an efficient constant-factor approximation algorithm for the minimum line-distortion problem and an efficient constant-factor approximation algorithm for the minimum bandwidth problem;
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there is an efficient 2-approximation algorithm for computing the path-length of an arbitrary graph;
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AT-free graphs and some intersection families of graphs have path-length at most 2;
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for AT-free graphs, there exist a linear time 8-approximation algorithm for the minimum line-distortion problem and a linear time 4-approximation algorithm for the minimum bandwidth problem.
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References
Assman, S.F., Peck, G.W., Syslo, M.M., Zak, J.: The bandwidth of caterpillars with hairs of length 1 and 2. SIAM J. Algebraic Discrete Methods 2, 387–392 (1981)
Blache, G., Karpinski, M., Wirtgen, J.: On approximation intractability of the bandwidth problem, Technical report TR98-014. University of Bonn (1997)
Bădoiu, M., Chuzhoy, J., Indyk, P., Sidiropoulos, A.: Low-distortion embeddings of general metrics into the line, In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC 2005). Baltimore, MD, USA, ACM, pp. 225–233, 22–24 May 2005
Bǎdoiu, M., Dhamdhere, K., Gupta, A., Rabinovich, Y., Raecke, H., Ravi, R., Sidiropoulos, A.: Approximation algorithms for low-distortion embeddings into low-dimensional spaces, In: Proceedings of the ACM/SIAM Symposium on Discrete Algorithms, (2005)
Brandstädt, A., Le, V.B., Spinrad, J.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (1999)
Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)
Corneil, D.G., Olariu, S., Stewart, L.: Asteroidal triple-free graphs. SIAM J. Discrete Math. 10, 399–430 (1997)
Corneil, D.G., Olariu, S., Stewart, L.: Linear time algorithms for dominating pairs in asteroidal triple-free graphs. SIAM J. Comput. 28, 292–302 (1997)
Diestel, R.: Graph Theory (Graduate Texts in Mathematics), 2nd edn. Springer, Berlin (2000)
Dourisboure, Y., Gavoille, C.: Tree-decompositions with bags of small diameter. Discrete Math. 307, 208–229 (2007)
Dragan, F.F., Köhler, E.: An Approximation Algorithm for the tree \(t\)-spanner problem on unweighted graphs via generalized chordal graphs, approximation, randomization, and combinatorial optimization. algorithms and techniques. In: Proceedings of the 14th International Workshop, APPROX 2011, and 15th International Workshop, RANDOM 2011, Princeton, NJ, USA, 17–19 Aug 2011. Lecture Notes in Computer Science 6845, Springer, pp. 171–183; Algorithmica 69 (2014), 884–905
Dragan, F.F., Köhler, E., Leitert, A.: Line-distortion, Bandwidth and path-length of a graph. In Proceedings of 14th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2014), July 2-4 2014. Copenhagen, Denmark, Lecture Notes in Computer Science 8503, pp. 146–257 (2014)
Dubey, Ch., Feige, U., Unger, W.: Hardness results for approximating the bandwidth. J. Comput. Syst. Sci. 77, 62–90 (2011)
Feige, U.: Approximating the bandwidth via volume respecting embedding. J. Comput. Syst. Sci. 60, 510–539 (2000)
Feige, U., Talwar, K.: Approximating the bandwidth of caterpillars. In: Proceedings of 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2005) and 9th International Workshop on Randomization and Computation (RANDOM 2005), Berkeley, CA, USA, Lecture Notes in Computer Science 3624, 2005, pp 62–73, 22–24 Aug 2005
Fellows, M.R., Fomin, F.V., Lokshtanov, D., Losievskaja, E., Rosamond, F.A., Saurabh, S.: Distortion is fixed parameter tractable. ACM Trans. Comput. Theory 5, 16.1–16.20 (2013)
Fomin, F.V., Lokshtanov, D., Saurabh, S.: An exact algorithm for minimum distortion embedding. Theor. Comput. Sci. 412, 3530–3536 (2011)
Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)
Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and interval graphs. Can. J. Math. 16, 539–548 (1964)
Golovach, P.A., Heggernes, P., Kratsch, D., Lokshtanov, D., Meister, D., Saurabh, S.: Bandwidth on AT-free graphs. Theor. Comput. Sci. 412, 7001–7008 (2011)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)
Golumbic, M.C., Rotem, D., Urrutia, J.: Comparability graphs and intersection graphs. Discrete Math. 43, 37–46 (1983)
Gupta, A.: Improved bandwidth approximation for trees and chordal graphs. J. Algorithms 40, 24–36 (2001)
Heggernes, P., Kratsch, D., Meister, D.: Bandwidth of bipartite permutation graphs in polynomial time. J. Discrete Algorithms 7, 533–544 (2009)
Heggernes, P., Meister, D.: Hardness and approximation of minimum distortion embeddings. Inf. Process. Lett. 110, 312–316 (2010)
Heggernes, P., Meister, D., Proskurowski, A.: Computing minimum distortion embeddings into a path of bipartite permutation graphs and threshold graphs. Theor. Comput. Sci. 412, 1275–1297 (2011)
Indyk, P.: Algorithmic applications of low-distortion geometric embeddings. In: Proceedings of 42nd IEEE Symposium on Foundations of Computer Science (FOCS 2001). IEEE, pp. 10–35
Indyk, P., Matousek, J.: Low-distortion embeddings of finite metric spaces. In: Goodman J.E., O’Rourke J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., pp. 177–196. CRC press, New York (2004)
Kleitman, D.J., Vohra, R.V.: Computing the bandwidth of interval graphs. SIAM J. Discrete Math. 3, 373–375 (1990)
Kloks, T., Kratsch, D., Müller, H.: Approximating the bandwidth for asteroidal triple-free graphs. J. Algorithms 32, 41–57 (1999)
Kratsch, D., Spinrad, J.: Between \({\cal O}(n m)\), In: Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms (SODA 2003). ACM, pp. 709–716
Kratsch, D., Stewart, L.: Approximating bandwidth by mixing layouts of interval graphs. SIAM J. Discrete Math. 15, 435–449 (2002)
Lokshtanov, D.: On the complexity of computing treelength. Discrete Appl. Math. 158(7), 820–827 (2010)
Ma, T.H., Spinrad, J.P.: On the two-chain subgraph cover and related problems. J. Algorithms 17, 251–268 (1994)
McConnell, R.M., Spinrad, J.P.: Linear-time transitive orientation. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, Louisiana 5–7, 19–25 (1997)
Monien, B.: The bandwidth-minimization problem for caterpillars with hair length 3 is NP-complete. SIAM J. Algebraic Discrete Methods 7, 505–512 (1986)
Olariu, S.: An optimal greedy heuristic to color interval graphs. Inf. Process. Lett. 37, 65–80 (1991)
Proskurowski, A., Telle, J.A.: Classes of graphs with restricted interval models. Discrete Math. Theor. Comput. Sci. 3, 167–176 (1999)
Räcke, H.: Lecture notes at http://ttic.uchicago.edu/~harry/teaching/pdf/lecture15
Robertson, N., Seymour, P.: Graph minors. I. Excluding a forest. J. Comb. Theory Ser. B 35, 39–61 (1983)
Shrestha, A.M.S., Tayu, S., Ueno, S.: Bandwidth of convex bipartite graphs and related graphs. Inf. Process. Lett. 112, 411–417 (2012)
Sprague, A.P.: An \({\cal O}(n \log n)\) algorithm for bandwidth of interval graphs. SIAM J. Discrete Math. 7, 213–220 (1994)
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000)
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We are very grateful to anonymous referees for many useful suggestions.
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Results of this paper were partially presented at the SWAT 2014 conference [12].
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Dragan, F.F., Köhler, E. & Leitert, A. Line-Distortion, Bandwidth and Path-Length of a Graph. Algorithmica 77, 686–713 (2017). https://doi.org/10.1007/s00453-015-0094-7
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DOI: https://doi.org/10.1007/s00453-015-0094-7