Minimum-Cost \(b\)-Edge Dominating Sets on Trees
We consider the minimum-cost \(b\)-edge dominating set problem. This is a generalization of the edge dominating set problem, but the computational complexity for trees is an astonishing open problem. We make steps toward the resolution of this open problem in the following three directions. (1) We give the first combinatorial polynomial-time algorithm for paths. Prior to our work, the polynomial-time algorithm for paths used linear programming, and it was known that the linear-programming approach could not be extended to trees. Thus, our algorithm would yield an alternative approach to a possible polynomial-time algorithm for trees. (2) We give a fixed-parameter algorithm for trees with the number of leaves as a parameter. Thus, a possible NP-hardness proof for trees should make use of trees with unbounded number of leaves. (3) We give a fully polynomial-time approximation scheme for trees. Prior to our work, the best known approximation factor was two. If the problem is NP-hard, then a possible proof cannot be done via a gap-preserving reduction from any APX-hard problem unless P \(=\) NP.
Unable to display preview. Download preview PDF.
- 4.Dadush, D., Peikert, C., Vempala, S.: Enumerative lattice algorithms in any norm via M-ellipsoid coverings. In: FOCS, pp. 580–589 (2011)Google Scholar
- 5.Dadush, D., Vempala, S.: Deterministic construction of an approximate M-ellipsoid and its applications to derandomizing lattice algorithms. In: SODA, pp. 1445–1456 (2012)Google Scholar
- 7.Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer (2004)Google Scholar
- 9.Parekh, O.: Edge dominating and hypomatchable sets. In: SODA, 287–291 (2002)Google Scholar
- 10.Schrijver, A.: Theory of Linear and Integer Programming. John Wiley & Sons (1986)Google Scholar
- 12.Vazirani, V.V.: Approximation algorithms. Springer (2001)Google Scholar