A graph operation that
contracts edges
is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting
k
edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely, the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this article, we study the
F
-Contraction
problem, where
F
is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph
G
and an integer
k
,
F
-Contraction
asks whether there exists
X
⊆
E(G)
such that
G/X
∈
F
and |
X
| ≤
k
. Here,
G/X
is the graph obtained from
G
by contracting edges in
X
. We obtain the following results for the
F
-
Contraction
problem:
•
Clique Contraction
is known to be
FPT
. However, unless NP⊆ coNP/
poly
, it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme (
PSAKS
). That is, it admits a (1 + ε)-approximate kernel with
O
(
k
f(ε))
vertices for every ε > 0.
•
Split Contraction
is known to be
W[1]-Hard
. We deconstruct this intractability result in two ways. First, we give a (2+ε)-approximate polynomial kernel for
Split Contraction
(which also implies a factor (2+ε)-
FPT
-approximation algorithm for
Split Contraction
). Furthermore, we show that, assuming
Gap-ETH
, there is no (5/4-δ)-
FPT
-approximation algorithm for
Split Contraction
. Here, ε, δ > 0 are fixed constants.
•
Chordal Contraction
is known to be
W[2]-Hard
. We complement this result by observing that the existing
W[2]-hardness
reduction can be adapted to show that, assuming
FPT
≠
W[1]
, there is no
F(k)
-
FPT
-approximation algorithm for
Chordal Contraction
. Here,
F(k)
is an arbitrary function depending on
k
alone.
We say that an algorithm is an
h(k)
-
FPT
-approximation algorithm for the
F
-Contraction
problem, if it runs in
FPT
time, and on any input
(G, k)
such that there exists
X
⊆
E(G)
satisfying
G/X
∈
F
and |
X
| ≤
k
, it outputs an edge set
Y
of size at most
h(k)
ċ
k
for which
G/Y
is in
F
.
On the Parameterized Approximability of Contraction to Classes of Chordal Graphs
