Maximum weight independent sets and cliques in intersection graphs of filaments
Information Processing Letters
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Measure and conquer: a simple O(20.288n) independent set algorithm
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
A 2O (k)poly(n) algorithm for the parameterized Convex Recoloring problem
Information Processing Letters
Improved Approximation Algorithm for Convex Recoloring of Trees
Theory of Computing Systems
Convex recolorings of strings and trees: Definitions, hardness results and algorithms
Journal of Computer and System Sciences
Speeding up dynamic programming for some NP-hard graph recoloring problems
TAMC'08 Proceedings of the 5th international conference on Theory and applications of models of computation
Connected coloring completion for general graphs: algorithms and complexity
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Quadratic kernelization for convex recoloring of trees
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
The complexity of minimum convex coloring
Discrete Applied Mathematics
Discrete Applied Mathematics
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We take a new look at the convex path recoloring (CPR ), convex tree recoloring (CTR ), and convex leaf recoloring (CLR ) problems through the eyes of the independent set problem. This connection allows us to give a complete characterization of the complexity of all these problems in terms of the number of occurrences of each color in the input instance, and consequently, to present simpler NP-hardness proofs for them than those given earlier. For example, we show that the CLR problem on instances in which the number of leaves of each color is at most 3, is solvable in polynomial time, by reducing it to the independent set problem on chordal graphs, and becomes NP-complete on instances in which the number of leaves of each color is at most 4. This connection also allows us to develop improved exact algorithms for the problems under consideration. For instance, we show that the CPR problem on instances in which the number of vertices of each color is at most 2, denoted 2-CPR , proved to be NP-complete in the current paper, is solvable in time 2 n /4 n O (1) (n is the number of vertices on the path) by reducing it after 2 n /4 enumerations to the weighted independent set problem on interval graphs, which is solvable in polynomial time. Then, using an exponential-time reduction from CPR to 2-CPR , we show that CPR is solvable in time 24n /9 n O (1). We also present exact algorithms for CTR and CLR running in time 20.454n n O (1) and 2 n /3 n O (1), respectively.