G-2018-08
Algorithms and computational complexity on the Pk-hitting set problem
BibTeX reference
The Pk
-hitting set problem consists in removing a minimum number ψk(G)
of vertices of a given graph G
so that the resulting graph does not contain path Pk
on k
vertices as a subgraph. For instance, the corresponding vertex set for k=2
is a vertex cover.
Cubic graphs received more attention for this problem in the last decade, especially for approximation algorithms.
In the present paper, we prove that the Pk
-hitting set problem is NP-complete in the class of cubic graphs, for any fixed constant
k⩾4
. Then, we design for cubic graphs a polynomial-time algorithm which is
a 1.25
-approximation for the P3
-hitting set problem, better than the best-known (1.57
-approximation algorithm),
and a 2
-approximation for the P4
-hitting set problem. Compared to other approximation algorithms, one of its force is its simplicity.
We conjecture that for any cubic graph G
and any constant k⩾5
, |V(G)|k⩽ψk(G)
.
Moreover, if true, this lower bound is achieved by a family of graphs with arbitrary large value of~ψk
.
As a consequence of Bollobas, Robinson and Wormald's result~\cite{Bollobas,RW92,RW94}, the conjecture is true for almost all cubic graphs. So our algorithm becomes a k2
-approximation for the Pk
-hitting set problem for almost all cubic graphs, which is better than other constant approximation ratios.
Finally, we design, for subcubic graphs, a polynomial-time algorithm which transforms any minimum Pk
-hitting set into another one with the following property: itself and its complement are Pk
-hitting sets.
Besides, there does not exist such algorithm if the graph has at least one vertex with degree d⩾4
.
Published February 2018 , 15 pages
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