Abstract
We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of maximum independent sets; (3) a polynomial-time algorithm for counting the number of independent sets of a fixed size. With similar ideas, we show that enumeration (namely, listing) of the independent sets, the maximum independent sets, and the independent sets of a fixed size in a chordal graph can be done in constant amortized time per output. On the other hand, we prove that the following problems for a chordal graph are # P-complete: (1) counting the number of maximal independent sets; (2) counting the number of minimum maximal independent sets. With similar ideas, we also show that finding a minimum weighted maximal independent set in a chordal graph is NP-hard, and even hard to approximate.
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References
Beeri, C., Fagin, R., Maier, D., Yannakakis, M.: On the Desirability of Acyclic Database Schemes. Journal of the ACM 30, 479–513 (1983)
Blair, J.R.S., Peyton, B.: An Introduction to Chordal Graphs and Clique Trees. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds.) Graph Theory and Sparse Matrix Computation. IMA, vol. 56, pp. 1–29. Springer, Heidelberg (1993)
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM, Philadelphia (1999)
Buneman, P.: A Characterization of Rigid Circuit Graphs. Discrete Mathematics 9, 205–212 (1974)
Chandran, L.S.: A Linear Time Algorithm for Enumerating All the Minimum and Minimal Separators of a Chordal Graph. In: Wang, J. (ed.) COCOON 2001. LNCS, vol. 2108, pp. 308–317. Springer, Heidelberg (2001)
Chandran, L.S., Ibarra, L., Ruskey, F., Sawada, J.: Generating and Characterizing the Perfect Elimination Orderings of a Chordal Graph. Theoretical Computer Science 307, 303–317 (2003)
Eppstein, D.: All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs. In: Proc. 16th Ann. ACM-SIAM Symp. on Discrete Algorithms, pp. 451–459. ACM, New York (2005)
Farber, M.: Independent Domination in Chordal Graphs. Operations Research Letters 1(4), 134–138 (1982)
Flum, J., Grohe, M.: The Parameterized Complexity of Counting Problems. SIAM Journal on Computing 33(4), 892–922 (2004)
Fulkerson, D.R., Gross, O.A.: Incidence Matrices and Interval Graphs. Pacific J. Math. 15, 835–855 (1965)
Gavril, F.: Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph. SIAM Journal on Computing 1(2), 180–187 (1972)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, 2nd edn. Annals of Discrete Mathematics, vol. 57. Elsevier, Amsterdam (2004)
Klein, P.N.: Efficient Parallel Algorithms for Chordal Graphs. SIAM Journal on Computing 25(4), 797–827 (1996)
Leung, J.Y.-T.: Fast Algorithms for Generating All Maximal Independent Sets of Interval, Circular-Arc and Chordal Graphs. Journal of Algorithms 5, 22–35 (1984)
Provan, J.S., Ball, M.O.: The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected. SIAM Journal on Computing 12, 777–788 (1983)
Spinrad, J.P.: Efficient Graph Representations. American Mathematical Society, Providence (2003)
Vadhan, S.P.: The Complexity of Counting in Sparse, Regular, and Planar Graphs. SIAM Journal on Computing 31(2), 398–427 (2001)
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Okamoto, Y., Uno, T., Uehara, R. (2005). Linear-Time Counting Algorithms for Independent Sets in Chordal Graphs. In: Kratsch, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2005. Lecture Notes in Computer Science, vol 3787. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11604686_38
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DOI: https://doi.org/10.1007/11604686_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-31000-6
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