Abstract
We consider the following problem: Given k independent edges in G. Is there a polynomial time algorithm to decide whether or not G has a cycle through all of these edges ? If the answer is yes, detect such a cycle in polynomial time.
This problem can be viewed as an algorithmic aspect of the conjecture of Lovász [22] and Woodall [34]. For fixed k, it follows from the seminal result of Robertson and Seymour [29] that there is a polynomial time algorithm to decide this problem. But, the proof of its correctness requires the full power of machinery from the graph minor series of papers, which consist of more than 20 papers and > 500 pages. In addition, the hidden constant is an extremely rapidly growing function of k. Even k = 3, the algorithm is not practical at all.
Our main result is to give a better algorithm for the problem in the following sense.
-
1
Even when k is a non-trivially super-constant number (up to O((loglogn)1/10)), there is a polynomial time algorithm for the above problem (So the hidden constant is not too large).
-
1
The time complexity is O(n 2), which improves Robertson and Seymour’s algorithm whose time complexity is O(n 3).
Our algorithm has several appealing features. Although our approach makes use of several ideas underlying the Robertson and Seymour’s algorithm, our new algorithmic components allow us to give a self-contained proof within 10 pages, which is much shorter and simpler than Robertson and Seymour’s. In addition, if an input is a planar graph or a bounded genus graph, we can get a better bound for the hidden constant. More precisely, for the planar case, when k is a non-trivially super-constant number up to k ≤ O((logn/(loglogn))1/4), there is a polynomial time algorithm, and for the bounded genus case, when k is a non-trivially super-constant number up to k ≤ O((log(n/g)/(loglog(n/g)))1/4), there is a polynomial time algorithm, where g is the Euler genus.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math. 23, 11–24 (1989)
Berge, C.: Graphs and Hypergraphs. North-Holland, Amsterdam (1973)
Bodlaender, H.L.: A linear-time algorithm for finding tree-decomposition of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)
Bondy, J.A., Lovász, L.: Cycles thourhg specified vertices of a graph. Combinatorica 1, 117–140 (1981)
Demaine, E.D., Fomin, F., Hajiaghayi, M., Thilikos, D.: Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM 52, 1–29 (2005)
Diestel, R., Gorbunov, K.Y., Jensen, T.R., Thomassen, C.: Highly connected sets and the excluded grid theorem. J. Combin. Theory Ser. B 75, 61–73 (1999)
Dirac, G.A.: In abstrakten Graphen vorhandene vollständige 4-Graphen und ihre unterteilungen. Math. Nachr. 22, 61–85 (1960)
Erdős, P.L., Győri, E.: Any four independent edges of a 4-connected graph are contained in a circuit. Acta Math. Hungar. 46, 311–313 (1985)
Gabow, H.: Finding paths and cycles of superpolylogarithmic lentgh. In: STOC 2004, Chicago, Illinois, USA, pp. 407–416 (2004)
Häggkvist, R., Thomassen, C.: Circuits through specified edges. Discrete Math. 41, 29–34 (1982)
Holton, D.A., McKay, B.D., Plummer, M.D., Thomassen, C.: A nine point theorem for 3-connected graphs. Combinatorica 2, 53–62 (1982)
Ibaraki, T., Nagamichi, H.: A linear time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica 7, 583–596 (1992)
Johnson, D.: The Many Faces of Polynomial Time. J. Algorithms 8, 285–303 (1987)
Kawarabayashi, K.: One or two disjoint ciruits cover independent edges, Loväsz-Woodall Conjecture. J. Combin. Theory Ser. B. 84, 1–44 (2002)
Kawarabayashi, K.: Two circuits through independent edges (manuscript, 1999)
Kawarabayashi, K.: An extremal problem for two circuits through independent edges (manuscript, 1999)
Kawarabayashi, K.: Proof of Lovász-Woodall Conjecture (in preparation)
Kawarabayashi, K.: Cycles through prescribed vertex set in N-connected graphs. J. Combin. Theory Ser. B 90, 315–323 (2004)
Kleinberg, J.: Decision algorithms for unsplittable flows and the half-disjoint paths problem. In: Proc. 30th ACM Symposium on Theory of Computing, pp. 530–539 (1998)
LaPaugh, A.S., Rivest, R.L.: The subgraph homomorphism problem. J. Comput. Sys. Sci. 20, 133–149 (1980)
Lomonosov, M.V.: Cycles through prescribed elements in a graph. In: Korte, Lovász, Prőmel, Schrijver (eds.) Paths, Flows, and VLSI Layout, pp. 215–234. Springer, Berlin (1990)
Lovász, L.: Problem 5. Period. Math. Hungar, 82 (1974)
Lovász, L.: Exercise 6.67. In: Combinatorial Problems and Exercises. North-Holland, Amsterdam (1979)
Perkovic, L., Reed, B.: An improved algorithm for finding tree decompositions of small width. International Journal on the Foundations of Computing Science 11, 81–85 (2000)
Reed, B.: Tree width and tangles: a new connectivity measure and some applications. In: Surveys in Combinatorics, London. London Math. Soc. Lecture Note Ser., vol. 241, pp. 87–162. Cambridge Univ.Press, Cambridge (1997)
Reed, B.: Rooted Routing in the Plane. Discrete Applied Mathematics 57, 213–227 (1995)
Reed, B., Robertson, N., Schrijver, A., Seymour, P.D.: Finding disjoint trees in planar graphs in linear time. In: Graph structure theory (Seattle, WA, 1991) Contemp. Math., pp. 295–301. Amer. Math. Soc., Providence (1993)
Robertson, N., Seymour, P.D.: Graph minors. V. Excluding a planar graph. J. Combin. Theory Ser. B 41, 92–114 (1986)
Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Combin. Theory Ser. B 63, 65–110 (1995)
Robertson, N., Seymour, P.D.: Graph minors. XVI. Excluding a non-planar graph. J. Combin. Theory Ser. B 89, 43–76 (2003)
Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a planar graph. J. Combin. Theory Ser. B 62, 323–348 (1994)
Sanders, D.P.: On circuits through five edges. Discrete Math. 159, 199–215 (1996)
Thomassen, C.: Note on circuits containing specified edges. J. Combin. Theory Ser. B. 22, 279–280 (1977)
Woodall, D.R.: Circuits containing specified edges. J. Combin. Theory Ser. B. 22, 274–278 (1977)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kawarabayashi, Ki. (2008). An Improved Algorithm for Finding Cycles Through Elements. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2008. Lecture Notes in Computer Science, vol 5035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68891-4_26
Download citation
DOI: https://doi.org/10.1007/978-3-540-68891-4_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-68886-0
Online ISBN: 978-3-540-68891-4
eBook Packages: Computer ScienceComputer Science (R0)