Affiliation:
1. University of California, San Diego
2. Indian Institute of Technology, Madras
3. Institute of Mathematical Sciences, Chennai
Abstract
We investigate the parameterized complexity of
Vertex Cover
parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an
O
*(2.618
k
) algorithm for the problem. Here,
k
is the excess of the vertex cover size over the LP optimum, and we write
O
*(
f
(
k
)) for a time complexity of the form
O
(
f
(
k
)
n
O
(1)
). We proceed to show that a more sophisticated branching algorithm achieves a running time of
O
*(2.3146
k
).
Following this, using previously known as well as new reductions, we give
O
*(2.3146
k
) algorithms for the parameterized versions of
Above Guarantee Vertex Cover
,
Odd Cycle Transversal
,
Split Vertex Deletion,
and
Almost 2-SAT
, and
O
*(1.5214
k
) algorithms for
König Vertex Deletion
and
Vertex Cover
parameterized by the size of the smallest odd cycle transversal and König vertex deletion set. These algorithms significantly improve the best known bounds for these problems. The most notable improvement among these is the new bound for
Odd Cycle Transversal
—this is the first algorithm that improves on the dependence on
k
of the seminal
O
*(3
k
) algorithm of Reed, Smith, and Vetta. Finally, using our algorithm, we obtain a kernel for the standard parameterization of
Vertex Cover
with at most 2
k
−
c
log
k
vertices. Our kernel is simpler than previously known kernels achieving the same size bound.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)
Cited by
114 articles.
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