Abstract
It is well known that in a bipartite (and more generally in a König) graph, the size of the minimum vertex cover is equal to the size of the maximum matching. We first address the question whether (and if not when) this property still holds in a König graph if we insist on forcing one of the two vertices of some of the matching edges in the vertex cover solution. We characterize such graphs using the classical notions of augmenting paths and flowers used in Edmonds’ matching algorithm. Using this characterization, we develop an O *(9k) algorithm for the question of whether a general graph has a vertex cover of size at most m + k where m is the size of the maximum matching. Our algorithm for this well studied Above Guarantee Vertex Cover problem uses the technique of iterative compression and the notion of important separators, and improves the runtime of the previous best algorithm that took O *(15k) time. As a consequence of this result we get that well known problems like Almost 2 SAT (deleting at most k clauses to get a satisfying 2 SAT formula) and König Vertex Deletion (deleting at most k vertices to get a König graph) also have an algorithm with O *(9k) running time, improving on the previous bound of O *(15k).
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References
Chen, J., Kanj, I.A., Xia, G.: Improved upper bounds for vertex cover. Theor. Comput. Sci. 411(40-42), 3736–3756 (2010)
Chen, J., Liu, Y., Lu, S.: An improved parameterized algorithm for the minimum node multiway cut problem. Algorithmica 55(1), 1–13 (2009)
Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM 55(5) (2008)
Demaine, E., Gutin, G., Marx, D., Stege, U.: Open problems from dagstuhl seminar 07281 (2007)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)
Edmonds, J.: Paths, trees, and flowers. Canad. J. Math. 17, 449–467 (1965)
Flum, J., Grohe, M.: Parameterized Complexity Theory. In: Texts in Theoretical Computer Science. Springer, Berlin (2006)
Khot, S., Raman, V.: Parameterized complexity of finding subgraphs with hereditary properties. Theor. Comput. Sci. 289(2), 997–1008 (2002)
Lovász, L., Plummer, M.D.: Matching Theory. North-Holland, Amsterdam (1986)
Mahajan, M., Raman, V.: Parameterizing above guaranteed values: Maxsat and maxcut. J. Algorithms 31(2), 335–354 (1999)
Marx, D.: Parameterized graph separation problems. Theoret. Comput. Sci. 351(3), 394–406 (2006)
Marx, D., Razgon, I.: Constant ratio fixed-parameter approximation of the edge multicut problem. Inf. Process. Lett. 109(20), 1161–1166 (2009)
Marx, D., Razgon, I.: Fixed-parameter tractability of multicut parameterized by the size of the cutset. In: STOC, pp. 469–478 (2011)
Mishra, S., Raman, V., Saurabh, S., Sikdar, S., Subramanian, C.R.: The complexity of könig subgraph problems and above-guarantee vertex cover. Algorithmica 58 (2010)
Razgon, I., O’Sullivan, B.: Almost 2-sat is fixed-parameter tractable. J. Comput. Syst. Sci. 75(8), 435–450 (2009)
Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)
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Raman, V., Ramanujan, M.S., Saurabh, S. (2011). Paths, Flowers and Vertex Cover. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_33
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DOI: https://doi.org/10.1007/978-3-642-23719-5_33
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