Theoretical Computer Science
Note on Max Lin-2 above Average
Information Processing Letters
Note on maximal bisection above tight lower bound
Information Processing Letters
Betweenness parameterized above tight lower bound
Journal of Computer and System Sciences
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
A probabilistic approach to problems parameterized above or below tight bounds
Journal of Computer and System Sciences
Polynomial kernels for proper interval completion and related problems
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
Journal of Computer and System Sciences
Subexponential parameterized algorithm for minimum fill-in
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Slightly superexponential parameterized problems
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Polynomial time and parameterized approximation algorithms for boxicity
IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
Split Vertex Deletion meets Vertex Cover: New fixed-parameter and exact exponential-time algorithms
Information Processing Letters
A faster FPT algorithm for Bipartite Contraction
Information Processing Letters
Polynomial kernels for Proper Interval Completion and related problems
Information and Computation
| Hi-index | 0.01 |
We present an algorithm with runtime $O(k^{2k}n^3m)$ for the following NP-complete problem [M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco, 1979, problem GT35]: Given an arbitrary graph $G$ on $n$ vertices and $m$ edges, can we obtain an interval graph by adding at most $k$ new edges to $G$? This resolves the long-standing open question [H. Kaplan, R. Shamir, and R. E. Tarjan, SIAM J. Comput., 28 (1999), pp. 1906-1922; R. G. Downey and M. R. Fellows, Parameterized Complexity, Springer-Verlag, New York, 1999; M. Serna and D. Thilikos, Bull. Eur. Assoc. Theory Comput. Sci. EATCS, 86 (2005), pp. 41-65; G. Gutin, S. Szeider, and A. Yeo, in Proceedings IWPEC 2006, Lecture Notes in Comput. Sci. 4169, Springer-Verlag, Berlin, 2006, pp. 60-71], first posed by Kaplan, Shamir, and Tarjan, of whether this problem was fixed parameter tractable. The problem has applications in profile minimization for sparse matrix computations [J. A. George and J. W. H. Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981; R. E. Tarjan, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, 1976, pp. 3-22], and our results show tractability for the case of a small number $k$ of zero elements in the envelope. Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph and combines this with a greedy algorithm when graphs of a certain structure are reached by the search.