Abstract
We define a notion called leftmost separator of size at most k. A leftmost separator of size k is a minimal separator S that separates two given sets of vertices X and Y such that we “cannot move S more towards X” such that |S| remains smaller than the threshold. One of the incentives is that by using leftmost separators we can improve the time complexity of treewidth approximation. Treewidth approximation is a problem which is known to have a linear time FPT algorithm in terms of input size, and only single exponential in terms of the parameter, treewidth. It is not known whether this result can be improved theoretically. However, the coefficient of the parameter k (the treewidth) in the exponent is large. Hence, our goal is to decrease the coefficient of k in the exponent, in order to achieve a more practical algorithm. Hereby, we trade a linear-time algorithm for an \(\mathcal {O}(n \log n)\)-time algorithm. The previous known \(\mathcal {O}(f(k) n \log n)\)-time algorithms have dependences of \(2^{24k}k!\), \(2^{8.766k}k^2\) (a better analysis shows that it is \(2^{7.671k}k^2\)), and higher. In this paper, we present an algorithm for treewidth approximation which runs in time \(\mathcal {O}(2^{6.755k}\ n \log n)\),
Furthermore, we count the number of leftmost separators and give a tight upper bound for them. We show that the number of leftmost separators of size \(\le k\) is at most \(C_{k-1}\) (Catalan number). Then, we present an algorithm which outputs all leftmost separators in time \(\mathcal {O}(\frac{4^k}{\sqrt{k}}n)\).
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Notes
- 1.
Please note that the full version of this result with all the proofs and the figures can be found on the ArXiv version [2]. We have omitted some details here due to space constraints.
- 2.
Notice that all leftmost separators are minimal per definition.
- 3.
\(C_n\) is the nth Catalan number.
- 4.
We drop the term “minimal” because it is the default for the separators throughout this paper unless mentioned otherwise.
- 5.
any path from a vertex in X to a vertex in \(S_{out}\).
- 6.
note that by Claim 3 \(G_x = G[V_{X, S} \cup S]\).
- 7.
Notice the difference between the restricted Dyck Language and the Dyck language itself. In every non-trivial prefix of a Dyck word, the number of “[” is \(\ge \) the number of “]”, whereas for the restricted version it should be strictly greater.
References
Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Algebraic Discrete Methods 8(2), 277–284 (1987)
Belbasi, M., Fürer, M.: Finding all leftmost separators of size \(\le k\). arXiv preprint arXiv:2111.02614 (2021)
Belbasi, M., Fürer, M.: An improvement of Reed’s treewidth approximation. In: Uehara, R., Hong, S.-H., Nandy, S.C. (eds.) WALCOM 2021. LNCS, vol. 12635, pp. 166–181. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-68211-8_14
Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: A \(\text{ c}^\text{ kn }\) 5-approximation algorithm for treewidth. SIAM J. Comput. 45(2), 317–378 (2016)
Chitnis, R., Hajiaghayi, M.T., Marx, D.: Fixed-parameter tractability of directed multiway cut parameterized by the size of the cutset. SIAM J. Comput. 42(4), 1674–1696 (2013)
Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)
Flum, J., Grohe, M.: Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series). TTCS, Springer, Heidelberg (2006). https://doi.org/10.1007/3-540-29953-X
Korhonen, T.: A single-exponential time 2-approximation algorithm for treewidth. arXiv e-prints [to appear in FOCS] arXiv:2104.07463 (2021)
Lokshtanov, D., Marx, D.: Clustering with local restrictions. Inf. Comput. 222, 278–292 (2013)
Marx, D.: Parameterized graph separation problems. Theor. Comput. Sci. 351(3), 394–406 (2006)
Marx, D., Razgon, I.: Fixed-parameter tractability of multicut parameterized by the size of the cutset. SIAM J. Comput. 43(2), 355–388 (2014)
Reed, B.A.: Finding approximate separators and computing tree width quickly. In: Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing (STOC), pp. 221–228 (1992)
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Belbasi, M., Fürer, M. (2021). Finding All Leftmost Separators of Size \(\le k\). In: Du, DZ., Du, D., Wu, C., Xu, D. (eds) Combinatorial Optimization and Applications. COCOA 2021. Lecture Notes in Computer Science(), vol 13135. Springer, Cham. https://doi.org/10.1007/978-3-030-92681-6_23
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