1,184 research outputs found
Reconfiguration on sparse graphs
A vertex-subset graph problem Q defines which subsets of the vertices of an
input graph are feasible solutions. A reconfiguration variant of a
vertex-subset problem asks, given two feasible solutions S and T of size k,
whether it is possible to transform S into T by a sequence of vertex additions
and deletions such that each intermediate set is also a feasible solution of
size bounded by k. We study reconfiguration variants of two classical
vertex-subset problems, namely Independent Set and Dominating Set. We denote
the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete
on graphs of bounded bandwidth and W[1]-hard parameterized by k on general
graphs. We show that ISR is fixed-parameter tractable parameterized by k when
the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we
answer positively an open question concerning the parameterized complexity of
the problem on graphs of bounded treewidth. Moreover, our techniques generalize
recent results showing that ISR is fixed-parameter tractable on planar graphs
and graphs of bounded degree. For DSR, we show the problem fixed-parameter
tractable parameterized by k when the input graph does not contain large
bicliques, a class of graphs which includes graphs of bounded degeneracy and
nowhere-dense graphs
Complexity of Token Swapping and its Variants
In the Token Swapping problem we are given a graph with a token placed on
each vertex. Each token has exactly one destination vertex, and we try to move
all the tokens to their destinations, using the minimum number of swaps, i.e.,
operations of exchanging the tokens on two adjacent vertices. As the main
result of this paper, we show that Token Swapping is -hard parameterized
by the length of a shortest sequence of swaps. In fact, we prove that, for
any computable function , it cannot be solved in time where is the number of vertices of the input graph, unless the ETH
fails. This lower bound almost matches the trivial -time algorithm.
We also consider two generalizations of the Token Swapping, namely Colored
Token Swapping (where the tokens have different colors and tokens of the same
color are indistinguishable), and Subset Token Swapping (where each token has a
set of possible destinations). To complement the hardness result, we prove that
even the most general variant, Subset Token Swapping, is FPT in nowhere-dense
graph classes.
Finally, we consider the complexities of all three problems in very
restricted classes of graphs: graphs of bounded treewidth and diameter, stars,
cliques, and paths, trying to identify the borderlines between polynomial and
NP-hard cases.Comment: 23 pages, 7 Figure
On the Parameterized Complexity of Reconfiguration of Connected Dominating Sets
In a reconfiguration version of a decision problem ? the input is an instance of ? and two feasible solutions S and T. The objective is to determine whether there exists a step-by-step transformation between S and T such that all intermediate steps also constitute feasible solutions. In this work, we study the parameterized complexity of the Connected Dominating Set Reconfiguration problem (CDS-R). It was shown in previous work that the Dominating Set Reconfiguration problem (DS-R) parameterized by k, the maximum allowed size of a dominating set in a reconfiguration sequence, is fixed-parameter tractable on all graphs that exclude a biclique K_{d,d} as a subgraph, for some constant d ? 1. We show that the additional connectivity constraint makes the problem much harder, namely, that CDS-R is W[1]-hard parameterized by k+?, the maximum allowed size of a dominating set plus the length of the reconfiguration sequence, already on 5-degenerate graphs. On the positive side, we show that CDS-R parameterized by k is fixed-parameter tractable, and in fact admits a polynomial kernel on planar graphs
Complexity of token swapping and its variants
AbstractIn the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is W[1]-hard parameterized by the length k of a shortest sequence of swaps. In fact, we prove that, for any computable function f, it cannot be solved in time f(k)no(k/logk) where n is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial nO(k)-time algorithm. We also consider two generalizations of the Token Swapping, namely Colored Token Swapping (where the tokens have colors and tokens of the same color are indistinguishable), and Subset Token Swapping (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, Subset Token Swapping, is FPT in nowhere-dense graph classes. Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases
Complexity of token swapping and its variants
AbstractIn the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is W[1]-hard parameterized by the length k of a shortest sequence of swaps. In fact, we prove that, for any computable function f, it cannot be solved in time f(k)no(k/logk) where n is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial nO(k)-time algorithm. We also consider two generalizations of the Token Swapping, namely Colored Token Swapping (where the tokens have colors and tokens of the same color are indistinguishable), and Subset Token Swapping (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, Subset Token Swapping, is FPT in nowhere-dense graph classes. Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases
Galactic Token Sliding
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