Abstract
A knot in a directed graph G is a strongly connected subgraph Q of G with size at least two, such that no vertex in V(Q) is an in-neighbor of a vertex in \(V(G)\setminus V(Q)\). Knots are a very important graph structure in the networked computation field, because they characterize deadlock occurrences into a classical distributed computation model, the so-called OR-model. Given a directed graph G and a positive integer k, in this paper we present a parameterized complexity analysis of the Knot-Free Vertex Deletion (KFVD) problem, which consists of determining whether G has a subset \(S \subseteq V(G)\) of size at most k such that \(G[V\setminus S]\) contains no knot. KFVD is a graph problem with natural applications in deadlock resolution, and it is closely related to Directed Feedback Vertex Set. It is known that KFVD is NP-complete on planar graphs with bounded degree, but it is polynomial time solvable on subcubic graphs. In this paper we prove that: KFVD is W[1]-hard when parameterized by the size of the solution; it can be solved in \(2^{k\log \varphi }n^{O(1)}\) time, but assuming SETH it cannot be solved in \((2-\epsilon )^{k\log \varphi }n^{O(1)}\) time, where \(\varphi \) is the size of the largest strongly connected subgraph of G; it can be solved in \(2^{\phi }n^{O(1)}\) time, but assuming ETH it cannot be solved in \(2^{o(\phi )}n^{O(1)}\) time, where \(\phi \) is the number of vertices with out-degree at most k; unless \(PH = \varSigma _p^3\), KFVD does not admit polynomial kernel even when \(\varphi =2\) and k is the parameter.
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Barbosa, V.C.: The combinatorics of resource sharing. In: Corrêa, R., Dutra, I., Fiallos, M., Gomes, F. (eds.) Models for Parallel and Distributed Computation. APOP, vol. 67, pp. 27–52. Springer, Boston (2002). https://doi.org/10.1007/978-1-4757-3609-0_2
Barbosa, V.C., Benevides, M.R.: A graph-theoretic characterization of AND-OR deadlocks. Technical report COPPE-ES-472/98, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil (1998)
Barbosa, V.C., Carneiro, A.D.A., Protti, F., Souza, U.S.: Deadlock models in distributed computation: foundations, design, and computational complexity. In: Proceedings of the 31st ACM/SIGAPP Symposium on Applied Computing, pp. 538–541 (2016)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications, vol. 290. Macmilan, London (1976)
Carneiro, A.D.A., Protti, F., Souza, U.S.: Deletion graph problems based on deadlock resolution. In: Cao, Y., Chen, J. (eds.) COCOON 2017. LNCS, vol. 10392, pp. 75–86. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-62389-4_7
Chahar, P., Dalal, S.: Deadlock resolution techniques: an overview. Int. J. Sci. Res. Publ. 3(7), 1–5 (2013)
Chandy, K.M., Lamport, L.: Distributed snapshots: determining global states of distributed systems. ACM Trans. Comput. Syst. 3, 63–75 (1985)
Chen, J., Liu, Y., Lu, S., O’sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM (JACM) 55(5), 21 (2008)
Chen, J., Meng, J.: On parameterized intractability: hardness and completeness. Comput. J. 51(1), 39–59 (2007)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2009)
Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3
de Mendívil, J.G., Fariña, F., Garitagotia, J.R., Alastruey, C.F., Bernabeu-Auban, J.M.: A distributed deadlock resolution algorithm for the and model. IEEE Trans. Parallel Distrib. Syst. 10(5), 433–447 (1999)
Dom, M., Lokshtanov, D., Saurabh, S.: Kernelization lower bounds through colors and IDs. ACM Trans. Algorithms (TALG) 11(2), 13 (2014)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science, p. 87. Springer, New York (1999). https://doi.org/10.1007/978-1-4612-0515-9
Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-29953-X
Gary, M.R., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness (1979)
Holt, R.C.: Some deadlock properties of computer systems. ACM Comput. Surv. (CSUR) 4(3), 179–196 (1972)
Impagliazzo, R., Paturi, R.: Complexity of k-SAT. In: 1999 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity, pp. 237–240. IEEE (1999)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? In: 1998 Proceedings 39th Annual Symposium on Foundations of Computer Science, pp. 653–662. IEEE (1998)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations. IRSS, pp. 85–103. Springer, Boston (1972). https://doi.org/10.1007/978-1-4684-2001-2_9
Leung, E.K., Lai, J.Y.-T.: On minimum cost recovery from system deadlock. IEEE Trans. Comput. 9(C–28), 671–677 (1979)
Lokshtanov, D., Marx, D., Saurabh, S., et al.: Lower bounds based on the exponential time hypothesis. Bull. EATCS 3(105), 41–72 (2013)
Niedermeier, R.: Invitation to fixed-parameter algorithms (2006)
Terekhov, I., Camp, T.: Time efficient deadlock resolution algorithms. Inf. Process. Lett. 69(3), 149–154 (1999)
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Carneiro, A.D.A., Protti, F., Souza, U.S. (2018). Fine-Grained Parameterized Complexity Analysis of Knot-Free Vertex Deletion – A Deadlock Resolution Graph Problem. In: Wang, L., Zhu, D. (eds) Computing and Combinatorics. COCOON 2018. Lecture Notes in Computer Science(), vol 10976. Springer, Cham. https://doi.org/10.1007/978-3-319-94776-1_8
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