119 research outputs found
The Graph Motif problem parameterized by the structure of the input graph
The Graph Motif problem was introduced in 2006 in the context of biological
networks. It consists of deciding whether or not a multiset of colors occurs in
a connected subgraph of a vertex-colored graph. Graph Motif has been mostly
analyzed from the standpoint of parameterized complexity. The main parameters
which came into consideration were the size of the multiset and the number of
colors. Though, in the many applications of Graph Motif, the input graph
originates from real-life and has structure. Motivated by this prosaic
observation, we systematically study its complexity relatively to graph
structural parameters. For a wide range of parameters, we give new or improved
FPT algorithms, or show that the problem remains intractable. For the FPT
cases, we also give some kernelization lower bounds as well as some ETH-based
lower bounds on the worst case running time. Interestingly, we establish that
Graph Motif is W[1]-hard (while in W[P]) for parameter max leaf number, which
is, to the best of our knowledge, the first problem to behave this way.Comment: 24 pages, accepted in DAM, conference version in IPEC 201
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
Flip Distance to a Non-crossing Perfect Matching
A perfect straight-line matching on a finite set of points in the
plane is a set of segments such that each point in is an endpoint of
exactly one segment. is non-crossing if no two segments in cross each
other. Given a perfect straight-line matching with at least one crossing,
we can remove this crossing by a flip operation. The flip operation removes two
crossing segments on a point set and adds two non-crossing segments to
attain a new perfect matching . It is well known that after a finite number
of flips, a non-crossing matching is attained and no further flip is possible.
However, prior to this work, no non-trivial upper bound on the number of flips
was known. If (resp.~) is the maximum length of the longest
(resp.~shortest) sequence of flips starting from any matching of size , we
show that and (resp.~ and
)
Parameterized Hardness of Art Gallery Problems
Given a simple polygon P on n vertices, two points x,y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum set S such that every point in P is visible from a point in S.
The Vertex Guard Art Gallery problem asks for such a set S subset of the vertices of P. A point in the set S is referred to as a guard. For both variants, we rule out a f(k)*n^{o(k/log k)} algorithm, for any computable function f, where k := |S| is the number of guards, unless the Exponential Time Hypothesis fails. These lower bounds almost match the n^{O(k)} algorithms that exist for both problems
An Approximation Algorithm for the Art Gallery Problem
Given a simple polygon on vertices, two points in
are said to be visible to each other if the line segment between
and is contained in . The Point Guard Art Gallery problem
asks for a minimum set such that every point in is visible
from a point in . The set is referred to as guards. Assuming integer
coordinates and a specific general position assumption, we present the first
-approximation algorithm for the point guard problem for
simple polygons. This algorithm combines ideas of a paper of Efrat and
Har-Peled [Inf. Process. Lett. 2006] and Deshpande et. al. [WADS 2007]. We also
point out a mistake in the latter.Comment: 25 pages, 4 pages proof ideas, many figure
Metric Dimension Parameterized By Treewidth
A resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The METRIC DIMENSION problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. METRIC DIMENSION has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of METRIC DIMENSION with respect to treewidth. We provide a first answer to the question. We show that METRIC DIMENSION parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving METRIC DIMENSION in time f(pw)no(pw) on n-vertex graphs of constant degree, with pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter tl+Δ, where tl is the tree-length and Δ the maximum-degree of the input graph.publishedVersio
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