On the Parameterized Complexity of Reconfiguration Problems
We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration variant of an optimization problem Q takes as input two feasible solutions S and T and determines if there is a sequence of reconfiguration steps, i.e. a reconfiguration sequence, that c...
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Veröffentlicht in: | Algorithmica 2017-05, Vol.78 (1), p.274-297 |
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Sprache: | eng |
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Zusammenfassung: | We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration variant of an optimization problem
Q
takes as input two feasible solutions
S
and
T
and determines if there is a sequence of reconfiguration steps, i.e. a reconfiguration sequence, that can be applied to transform
S
into
T
such that each step results in a feasible solution to
Q
. For most of the results in this paper,
S
and
T
are sets of vertices of a given graph and a reconfiguration step adds or removes a vertex. Our study is motivated by results establishing that for many
NP
-hard problems, the classical complexity of reconfiguration is
PSPACE
-complete. We address the question for several important graph properties under two natural parameterizations:
k
, a bound on the size of solutions, and
ℓ
, a bound on the length of reconfiguration sequences. Our first general result is an algorithmic paradigm, the reconfiguration kernel, used to obtain fixed-parameter tractable algorithms for reconfiguration variants of
Vertex Cover
and, more generally,
Bounded Hitting Set
and
Feedback Vertex Set
, all parameterized by
k
. In contrast, we show that reconfiguring
Unbounded Hitting Set
is
W[2]
-hard when parameterized by
k
+
ℓ
. We also demonstrate the
W[1]
-hardness of reconfiguration variants of a large class of maximization problems parameterized by
k
+
ℓ
, and of their corresponding deletion problems parameterized by
ℓ
; in doing so, we show that there exist problems in
FPT
when parameterized by
k
, but whose reconfiguration variants are
W[1]
-hard when parameterized by
k
+
ℓ
. |
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ISSN: | 0178-4617 1432-0541 |
DOI: | 10.1007/s00453-016-0159-2 |