Abstract
For a function \(f:2^{V} \to \mathbb {Z}_{+}\) on a finite set V with f(∅)=f(V)=0, a digraph D=(V,A) is called f-connected if it satisfies the f-cut condition, that is, δ D (X)≥f(X) for any X⊆V, where δ D (X) is the number of arcs from X to V∖X. We show that, for any crossing supermodular function f, the f-connectivity can be tested with a constant number of queries in the general digraph model with average degree bound. As immediate corollaries, we obtain constant-time testers for k-edge-connectivity, rooted-(k,l)-edge-connectivity, and the property of having k arc-disjoint arborescences. We also give a corresponding result for the undirected case.
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References
Benczúr, A., Frank, A.: Covering symmetric supermodular functions by graphs. Math. Program. 84(3), 483–503 (1999)
Edmonds, J.: Edge-disjoint branchings. Comb. Algorithms 9, 91–96 (1973)
Frank, A.: Connectivity augmentation problems in network design. In: Birge, J., Murty, K. (eds.) Mathematical Programming: State of the Art, vol. 1994, pp. 34–63. The University of Michigan Press, Michigan (1994)
Frank, A.: Connections in Combinatorial Optimization. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2011)
Frank, A., Király, T.: A survey on covering supermodular functions. In: Cook, W., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, pp. 87–126. Springer, Berlin (2009)
Goldreich, O., Ron, D.: Property testing in bounded degree graphs. Algorithmica 32(2), 302–343 (2002)
Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45(4), 653–750 (1998)
Ito, H., Tanigawa, S., Yoshida, Y.: Constant-time algorithms for sparsity matroids. In: Proc. 39th International Colloquium on Automata, Languages and Programming (ICALP), pp. 498–509 (2012)
Kaufman, T., Krivelevich, M., Ron, D.: Tight bounds for testing bipartiteness in general graphs. SIAM J. Comput. 33(6), 1441–1483 (2004)
Orenstein, Y.: Property testing in directed graphs. Master’s thesis, Tel-Aviv University (2010)
Parnas, M., Ron, D.: Testing the diameter of graphs. Random Struct. Algorithms 20(2), 165–183 (2002)
Yoshida, Y., Ito, H.: Testing k-edge-connectivity of digraphs. J. Syst. Sci. Complex. 23(1), 91–101 (2010)
Yoshida, Y., Ito, H.: Property testing on k-vertex-connectivity of graphs. Algorithmica 62(3), 701–712 (2012)
Acknowledgements
We thank anonymous referees for helpful comments and simpler proofs of Lemmas 1 and 2. Shin-ichi Tanigawa is supported by JSPS Grant-in-Aid for Scientific Research (B). Yuichi Yoshida is supported by JSPS Grant-in-Aid for Research Activity Start-up (24800082), MEXT Grant-in-Aid for Scientific Research on Innovative Areas (24106001), and JST, ERATO, Kawarabayashi Large Graph Project.
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Tanigawa, SI., Yoshida, Y. Testing the Supermodular-Cut Condition. Algorithmica 71, 1065–1075 (2015). https://doi.org/10.1007/s00453-013-9842-8
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DOI: https://doi.org/10.1007/s00453-013-9842-8