Reconstructing the shape of a tree from observed dissimilarity data
Advances in Applied Mathematics
Graph Theory With Applications
Graph Theory With Applications
SIAM Journal on Discrete Mathematics
Parameterized algorithms for feedback set problems and their duals in tournaments
Theoretical Computer Science - Parameterized and exact computation
The Minimum Feedback Arc Set Problem is NP-Hard for Tournaments
Combinatorics, Probability and Computing
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Hardness of fully dense problems
Information and Computation
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Kernelization: New Upper and Lower Bound Techniques
Parameterized and Exact Computation
New results on optimizing rooted triplets consistency
Discrete Applied Mathematics
Fixed-Parameter Tractability of the Maximum Agreement Supertree Problem
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Fixed-parameter tractability results for feedback set problems in tournaments
CIAC'06 Proceedings of the 6th Italian conference on Algorithms and Complexity
Kernel and fast algorithm for dense triplet inconsistency
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Parameterized Complexity
A quadratic vertex kernel for feedback arc set in bipartite tournaments
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
Kernel and fast algorithm for dense triplet inconsistency
Theoretical Computer Science
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We develop a technique that we call Conflict Packing in the context of kernelization [7]. We illustrate this technique on several well-studied problems: FEEDBACK ARC SET IN TOURNAMENTS, DENSE ROOTED TRIPLET INCONSISTENCY and BETWEENNESS IN TOURNAMENTS. For the former, one is given a tournament T = (V, A) and seeks a set of at most k arcs whose reversal in T results in an acyclic tournament. While a linear vertex-kernel is already known for this problem [6], using the Conflict Packing allows us to find a so-called safe partition, the central tool of the kernelization algorithm in [6], with simpler arguments. Regarding the DENSE ROOTED TRIPLET INCONSISTENCY problem, one is given a set of vertices V and a dense collection R of rooted binary trees over three vertices of V and seeks a rooted tree over V containing all but at most k triplets from R. Using again the Conflict Packing, we prove that the DENSE ROOTED TRIPLET INCONSISTENCY problem admits a linear vertex-kernel. This result improves the best known bound of O(k2) vertices for this problem [16]. Finally, we use this technique to obtain a linear vertex-kernel for BETWEENNESS IN TOURNAMENTS, where one is given a set of vertices V and a dense collection R of betweenness triplets and seeks an ordering containing all but at most k triplets from R. To the best of our knowledge this result constitutes the first polynomial kernel for the problem.