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Linear Time Approximation Algorithms for Degree Constrained Subgraph Problems

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Research Trends in Combinatorial Optimization

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Many real-world problems require graphs of such large size that polynomial time algorithms are too costly as soon as their runtime is superlinear. Examples include problems in VLSI-design or problems in bioinformatics. For such problems the question arises: What is the best solution that can be obtained in linear time? We survey linear time approximation algorithms for some classical problems from combinatorial optimization, e.g. matchings and branchings.

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References

  • Althaus, E., Mehlhorn, K.: Maximum network flow with floating point arithmetic. Inf. Process. Lett. 66(3), 109ā€“113 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Avis, D.: Two greedy heuristics for the weighted matching problem. Congr. Numer. XXI, 65ā€“76 (1978). Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing (1978)

    MathSciNet  Google Scholar 

  • Bast, H., Mehlhorn, K., SchƤfer, G., Tamaki, H.: Matching algorithms are fast in sparse random graphs. Theory Comput. Syst. 39(1), 3ā€“14 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Berge, C.: Two theorems in graph theory. Proc. Natl. Acad. Sci. U.S.A. 43(9), 842ā€“844 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  • Blum, N.: A new approach to maximum matching in general graphs. In: Proc. of 17th ICALP (1990). Lecture Notes in Computer Science, vol. 443, pp. 586ā€“597. Springer, Berlin (1990)

    Google Scholar 

  • Bock, F.: An algorithm to construct a minimum directed spanning tree in a directed network. In: Avi Itzhak, B. (ed.) Developments in Operations Research, Proceedings of the Third Annual Israel Conference on Operations Research, July 1969, vol. 1, pp. 29ā€“44. Gordon and Breach, New York (1971). Paper 1-2

    Google Scholar 

  • Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305ā€“1317 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Chu, Y.-J., Liu, T.-H.: On the shortest arborescence of a directed graph. Sci. Sin. 14(10), 1396ā€“1400 (1965)

    MATH  Google Scholar 

  • Cook, W., Rohe, A.: Computing minimum-weight perfect matchings. INFORMS J. Comput. 11(2), 138ā€“148 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Drake, D.E., Hougardy, S.: Linear time local improvements for weighted matchings in graphs. In: Jansen, K. et al. (eds.) International Workshop on Experimental and Efficient Algorithms (WEA) 2003. Lecture Notes in Computer Science, vol. 2647, pp. 107ā€“119. Springer, Berlin (2003a)

    Chapter  Google Scholar 

  • Drake, D.E., Hougardy, S.: A simple approximation algorithm for the weighted matching problem. Inf. Process. Lett. 85(4), 211ā€“213 (2003b)

    Article  MathSciNet  Google Scholar 

  • Drake Vinkemeier, D.E., Hougardy, S.: A linear-time approximation algorithm for weighted matchings in graphs. ACM Trans. Algorithms 1(1), 107ā€“122 (2005)

    Article  MathSciNet  Google Scholar 

  • Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17(3), 449ā€“467 (1965)

    MATH  MathSciNet  Google Scholar 

  • Edmonds, J.: Optimum branchings. J. Res. Natl. Bur. Stand., B Math. Math. Phys. 71(4), 233ā€“240 (1967)

    MATH  MathSciNet  Google Scholar 

  • Feder, T., Motwani, R.: Clique partitions, graph compression and speeding-up algorithms. J. Comput. Syst. Sci. 51(2), 261ā€“272 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Fischer, T., Goldberg, A.V., Haglin, D.J., Plotkin, S.: Approximating matchings in parallel. Inf. Process. Lett. 46(3), 115ā€“118 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Can. J. Math. 8, 399ā€“404 (1956)

    MATH  MathSciNet  Google Scholar 

  • Frank, A.: On Kuhnā€™s Hungarian methodā€”a tribute from Hungary. Nav. Res. Logist. 52(1), 2ā€“5 (2005)

    Article  MATH  Google Scholar 

  • Fremuth-Paeger, C., Jungnickel, D.: Balanced network flows. VIII. A revised theory of phase-ordered algorithms and the \(O(\sqrt{n}m\log (n^{2}/m)/\log n)\) bound for the nonbipartite cardinality matching problem. Networks 41(3), 137ā€“142 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Gabow, H.N.: An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems. In: STOC ā€™83: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pp. 448ā€“456. ACM Press, New York (1983)

    Chapter  Google Scholar 

  • Gabow, H.N.: Data structures for weighted matching and nearest common ancestors with linking. In: SODA ā€™90: Proceedings of the First Annual ACMā€“SIAM Symposium on Discrete Algorithms, pp. 434ā€“443. SIAM, Philadelphia (1990)

    Google Scholar 

  • Gabow, H.N., Tarjan, R.E.: Algorithms for two bottleneck optimization problems. J. Algorithms 9(3), 411ā€“417 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  • Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for general graph-matching problems. J. Assoc. Comput. Mach. 38(4), 815ā€“853 (1991)

    MATH  MathSciNet  Google Scholar 

  • Gabow, H.N., Galil, Z., Spencer, T., Tarjan, R.E.: Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica 6(2), 109ā€“122 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • Goldberg, A.V., Karzanov, A.V.: Maximum skew-symmetric flows and matchings. Math. Program. 100(3), 537ā€“568 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Harvey, N.J.A.: Algebraic structures and algorithms for matching and matroid problems. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCSā€™06), pp. 531ā€“542. IEEE Computer Society, Washington (2006)

    Google Scholar 

  • Hassin, R., Lahav (Haddad), S.: Maximizing the number of unused colors in the vertex coloring problem. Inf. Process. Lett. 52(2), 87ā€“90 (1994)

    Article  MATH  Google Scholar 

  • Hopcroft, J.E., Karp, R.M.: An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225ā€“231 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  • Jenkyns, T.A.: The efficacy of the greedy algorithm. Congr. Numer. 17, 341ā€“350 (1976)

    MathSciNet  Google Scholar 

  • Karypis, G., Kumar, V.: Multilevel k-way partitioning scheme for irregular graphs. J. Parallel Distrib. Comput. 48(1), 96ā€“129 (1998)

    Article  MathSciNet  Google Scholar 

  • Korte, B., Hausmann, D.: An analysis of the greedy heuristic for independence systems. Ann. Discrete Math. 2, 65ā€“74 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  • Kuhn, H.W.: The Hungarian method for the assignment problem. Nav. Res. Logist. Q. 2, 83ā€“97 (1955)

    Article  MathSciNet  Google Scholar 

  • Mehlhorn, K., SchƤfer, G.: Implementation of O(nmlogā€‰n) weighted matchings in general graphs: the power of data structures. ACM J. Exp. Algorithmics 7, 4 (2002)

    Article  Google Scholar 

  • Mestre, J.: Greedy in approximation algorithms. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. Lecture Notes in Computer Science, vol. 4168, pp. 528ā€“539. Springer, Berlin (2006)

    Chapter  Google Scholar 

  • Micali, S., Vazirani, V.V.: An \(O(\sqrt {|v|}\cdot |E|)\) algorithm for finding maximum matching in general graphs. In: Proc. of 21st Annual Symposium on Foundations of Computer Science (21st FOCS, Syracuse, New York, 1980), pp. 17ā€“27 (1980)

    Google Scholar 

  • Motwani, R.: Average-case analysis of algorithms for matchings and related problems. J. Assoc. Comput. Mach. 41(6), 1329ā€“1356 (1994)

    MATH  MathSciNet  Google Scholar 

  • Mucha, M., Sankowski, P.: Maximum matchings via Gaussian elimination. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCSā€™04), pp. 248ā€“255. IEEE Computer Society, Washington (2004)

    Chapter  Google Scholar 

  • Petersen, J.: Die Theorie der regulƤren Graphs. Acta Math. 15(1), 193ā€“220 (1891)

    Article  MathSciNet  Google Scholar 

  • Pettie, S., Sanders, P.: A simpler linear time 2/3āˆ’Īµ approximation for maximum weight matching. Inf. Process. Lett. 91(6), 271ā€“276 (2004)

    Article  MathSciNet  Google Scholar 

  • Preis, R.: Linear time \(\frac{1}{2}\) -approximation algorithm for maximum weighted matching in general graphs. In: Meinel, C., Tison, S. (eds.) Symposium on Theoretical Aspects in Computer Science (STACS). Lecture Notes in Computer Science, vol. 1563, pp. 259ā€“269. Springer, Berlin (1999)

    Google Scholar 

  • Shiloach, Y.: Another look at the degree constrained subgraph problem. Inf. Process. Lett. 12(2), 89ā€“92 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  • Vazirani, V.V.: A theory of alternating paths and blossoms for proving correctness of the \(O(\sqrt{V}E)\) general graph maximum matching algorithm. Combinatorica 14(1), 71ā€“109 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  • Ziegler, V.: Approximating optimum branchings in linear time. Technical report, Humboldt-UniversitƤt zu Berlin, Institut fĆ¼r Informatik (2008)

    Google Scholar 

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Authors and Affiliations

  1. Research Institute for Discrete Mathematics, University of Bonn, LennƩstr. 2, 53113, Bonn, Germany

    Stefan Hougardy

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  1. Stefan Hougardy
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Editors and Affiliations

  1. Industrial and Systems Engineering, Georgia Tech, 765 Ferst Drive, Atlanta, Georgia, 30332-0205, USA

    William Cook

  2. Institute of Mathematics, Eƶtvƶs LorƔnd University, PƔzmƔny PƩter sƩtƔny 1/C, H-1117, Budapest, Hungary

    LĆ”szlĆ³ LovĆ”sz

  3. Research Institute for Discrete Mathematics, University of Bonn, LennƩstr. 2, 53113, Bonn, Germany

    Jens Vygen

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Hougardy, S. (2009). Linear Time Approximation Algorithms for Degree Constrained Subgraph Problems. In: Cook, W., LovĆ”sz, L., Vygen, J. (eds) Research Trends in Combinatorial Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76796-1_9

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