Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality
Mathematical Programming: Series A and B
Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
Parameterizing above guaranteed values: MaxSat and MaxCut
Journal of Algorithms
Introduction to algorithms
On efficient fixed-parameter algorithms for weighted vertex cover
Journal of Algorithms
Digraphs: Theory, Algorithms and Applications
Digraphs: Theory, Algorithms and Applications
Two Edge Modification Problems without Polynomial Kernels
Parameterized and Exact Computation
A tighter bound for counting max-weight solutions to 2SAT instances
IWPEC'08 Proceedings of the 3rd international conference on Parameterized and exact computation
Improved upper bounds for vertex cover
Theoretical Computer Science
Preprocessing of min ones problems: a dichotomy
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Paths, flowers and vertex cover
ESA'11 Proceedings of the 19th European conference on Algorithms
A kernel of order 2k-clogk for vertex cover
Information Processing Letters
Fast Algorithms for max independent set
Algorithmica
On multiway cut parameterized above lower bounds
IPEC'11 Proceedings of the 6th international conference on Parameterized and Exact Computation
Kernelization algorithms for d-hitting set problems
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
Representative Sets and Irrelevant Vertices: New Tools for Kernelization
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
Parameterized Complexity
| Hi-index | 5.23 |
The problem of finding a satisfying assignment that minimizes the number of variables that are set to 1 is NP-complete even for a satisfiable 2-SAT formula. We call this problem min ones 2-sat. It generalizes the well-studied problem of finding the smallest vertex cover of a graph, which can be modeled using a 2-SAT formula with no negative literals. The natural parameterized version of the problem asks for a satisfying assignment of weight at most k. In this paper, we present a polynomial-time reduction from min ones 2-sat to vertex cover without increasing the parameter and ensuring that the number of vertices in the reduced instance is equal to the number of variables of the input formula. Consequently, we conclude that this problem also has a simple 2-approximation algorithm and a 2k-clogk-variable kernel subsuming (or, in the case of kernels, improving) the results known earlier. Further, the problem admits algorithms for the parameterized and optimization versions whose runtimes will always match the runtimes of the best-known algorithms for the corresponding versions of vertex cover. Finally we show that the optimum value of the LP relaxation of the min ones 2-sat and that of the corresponding vertex cover are the same. This implies that the (recent) results of vertex cover version parameterized above the optimum value of the LP relaxation of vertex cover carry over to the min ones 2-sat version parameterized above the optimum of the LP relaxation of min ones 2-sat.