Abstract
It is known that every multigraph with an even number of edges has an even orientation (i.e., all indegrees are even). We study parity constrained graph orientations under additional constraints. We consider two types of constraints for a multigraph G = (V,E): (1) an exact conflict constraint is an edge set C ⊆ E and a vertex v ∈ V such that C should not equal the set of incoming edges at v; (2) a subset conflict constraint is an edge set C ⊆ E and a vertex v ∈ V such that C should not be a subset of incoming edges at v. We show that it is NP-complete to decide whether G has an even orientation with exact or subset conflicts, for all conflict sets of size two or higher. We present efficient algorithms for computing parity constrained orientations with disjoint exact or subset conflict pairs.
Research was supported by NSF Grants #CCF-0830734 and #CBET-0941538. Work by Tóth was also supported by NSERC grant RGPIN 35586.
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References
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall (February 1993)
Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge Univ. Press (2009)
Cannon, S., Ishaque, M., Tóth, C.D.: Conflict-free graph orientations with parity and degree constraints, arXiv:1203.3256 (2012) (manuscript)
Darmann, A., Pferschy, U., Schauer, J., Woeginger, G.J.: Paths, trees, and matchings under disjunctive constraints. Discrete Appl. Math. 159(16), 1726–1735 (2011)
Felsner, S., Fusy, É., Noy, M.: Asymptotic enumeration of orientations. Discrete Math. Theor. Comp. Sci. 12(2), 249–262 (2010)
Felsner, S., Zickfeld, F.: On the number of planar orientations with prescribed degrees. Electron. J. Comb. 15(1), article R77 (2008)
Frank, A.: On the orientaiton of graphs. J. Combin. Theor. B 28, 251–261 (1980)
Frank, A., Gyárfás, A.: How to orient the edges of a graph. Coll. Math. Soc. J. Bolyai 18, 353–364 (1976)
Frank, A., Jordán, T., Szigeti, Z.: An orientation theorem with parity conditions. Discrete Appl. Math. 115, 37–47 (2001)
Frank, A., Király, Z.: Graph orientations with edge-connection and parity constraints. Combinatorica 22, 47–70 (2002)
Frank, A., Tardos, É., Sebő, A.: Covering directed and odd cuts. Math Prog. Stud. 22, 99–112 (1984)
Hakimi, S.L.: On the degrees of the vertices of a directed graph. J. Franklin Inst. 279, 280–308 (1965)
Khanna, S., Naor, J., Shepherd, F.B.: Directed network design with orientation constraints. SIAM J. Discre. Math. 19, 245–257 (2005)
Lovász, L., Plummer, M.D.: Matching Theory. AMS Chelsea (2009)
Szabó, T., Welzl, E.: Unique sink orientations of cubes. In: Proc. 42nd FOCS, pp. 547–555. IEEE (2001)
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Cannon, S., Ishaque, M., Tóth, C.D. (2012). Conflict-Free Graph Orientations with Parity Constraints. In: Kranakis, E., Krizanc, D., Luccio, F. (eds) Fun with Algorithms. FUN 2012. Lecture Notes in Computer Science, vol 7288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30347-0_9
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