4 research outputs found
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
Minimum Size Tree-Decompositions
International audienceWe study in this paper the problem of computing a tree-decomposition of a graph with width at most k and minimum number of bags. More precisely, we focus on the following problem: given a fixed , what is the complexity of computing a tree-decomposition of width at most k with minimum number of bags in the class of graphs with treewidth at most k? We prove that the problem is NP-complete for any fixed 4 and polynomial for ; for , we show that it is polynomial in the class of trees and 2-connected outerplanar graphs
Subexponential time algorithms for finding small tree and path decompositions
The Minimum Size Tree Decomposition (MSTD) and Minimum Size Path Decomposition (MSPD) problems ask for a given n-vertex graph G and integer k, what is the minimum number of bags of a tree decomposition (respectively, path decomposition) of width at most k. The problems are known to be NP-complete for each fixed kÂż=Âż4. In this paper we present algorithms that solve both problems for fixed k in 2^O(n/ logn) time and show that they cannot be solved in 2^o(n / logn) time, assuming the Exponential Time Hypothesis
Subexponential time algorithms for finding small tree and path decompositions
The Minimum Size Tree Decomposition (MSTD) and Minimum Size Path Decomposition (MSPD) problems ask for a given nvertex graph G and integer k, what is the minimum number of bags of a tree decomposition (respectively, path decomposition) of width at most k. The problems are known to be NP-complete for each fixed k â„ 4. In this paper we present algorithms that solve both problems for fixed k in 2O(n/ log n) time and show that they cannot be solved in 2o(n/ log n) time, assuming the Exponential Time Hypothesis