48 research outputs found
Exact Algorithm for Graph Homomorphism and Locally Injective Graph Homomorphism
For graphs and , a homomorphism from to is a function , which maps vertices adjacent in to adjacent vertices
of . A homomorphism is locally injective if no two vertices with a common
neighbor are mapped to a single vertex in . Many cases of graph homomorphism
and locally injective graph homomorphism are NP-complete, so there is little
hope to design polynomial-time algorithms for them. In this paper we present an
algorithm for graph homomorphism and locally injective homomorphism working in
time , where is the bandwidth of the
complement of
Tight Euler tours in uniform hypergraphs - computational aspects
By a tight tour in a -uniform hypergraph we mean any sequence of its
vertices such that for all the set
is an edge of (where operations on
indices are computed modulo ) and the sets for are
pairwise different. A tight tour in is a tight Euler tour if it contains
all edges of . We prove that the problem of deciding if a given -uniform
hypergraph has a tight Euler tour is NP-complete, and that it cannot be solved
in time (where is the number of edges in the input hypergraph),
unless the ETH fails. We also present an exact exponential algorithm for the
problem, whose time complexity matches this lower bound, and the space
complexity is polynomial. In fact, this algorithm solves a more general problem
of computing the number of tight Euler tours in a given uniform hypergraph
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
A polynomial bound on the number of minimal separators and potential maximal cliques in -free graphs of bounded clique number
In this note we show a polynomial bound on the number of minimal separators
and potential maximal cliques in -free graphs of bounded clique number