383 research outputs found
Polynomial-Time Algorithms for the Longest Induced Path and Induced Disjoint Paths Problems on Graphs of Bounded Mim-Width
We give the first polynomial-time algorithms on graphs of bounded maximum induced matching width (mim-width) for problems that are not locally checkable. In particular, we give n^O(w)-time algorithms on graphs of mim-width at most w, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and H-Induced Topological Minor for fixed H. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Per- mutation and Circular Permutation graphs, Convex graphs, k-Trapezoid, Circular k-Trapezoid, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k
Mim-Width II. The Feedback Vertex Set Problem
Under embargo until: 2020-07-18We give a first polynomial-time algorithm for (WEIGHTED) FEEDBACK VERTEX SET on graphs of bounded maximum induced matching width (mim-width). Explicitly, given a branch decomposition of mim-width w, we give an nO(w)-time algorithm that solves FEEDBACK VERTEX SET. This provides a unified polynomial-time algorithm for many well-known classes, such as INTERVAL graphs, PERMUTATION graphs, and LEAF POWER graphs (given a leaf root), and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mim-width, such as CIRCULAR PERMUTATION and CIRCULAR k-TRAPEZOID graphs (given a circular k-trapezoid model) for fixed k. We complement our result by showing that FEEDBACK VERTEX SET is W[1]-hard when parameterized by w and the hardness holds even when a linear branch decomposition of mim-width w is given.acceptedVersio
Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
A bipartite graph is convex if the vertices in can be
linearly ordered such that for each vertex , the neighbors of are
consecutive in the ordering of . An induced matching of is a
matching such that no edge of connects endpoints of two different edges of
. We show that in a convex bipartite graph with vertices and
weighted edges, an induced matching of maximum total weight can be computed in
time. An unweighted convex bipartite graph has a representation of
size that records for each vertex the first and last neighbor
in the ordering of . Given such a compact representation, we compute an
induced matching of maximum cardinality in time.
In convex bipartite graphs, maximum-cardinality induced matchings are dual to
minimum chain covers. A chain cover is a covering of the edge set by chain
subgraphs, that is, subgraphs that do not contain induced matchings of more
than one edge. Given a compact representation, we compute a representation of a
minimum chain cover in time. If no compact representation is given, the
cover can be computed in time.
All of our algorithms achieve optimal running time for the respective problem
and model. Previous algorithms considered only the unweighted case, and the
best algorithm for computing a maximum-cardinality induced matching or a
minimum chain cover in a convex bipartite graph had a running time of
A Unified Polynomial-Time Algorithm for Feedback Vertex Set on Graphs of Bounded Mim-Width
We give a first polynomial-time algorithm for (Weighted) Feedback Vertex Set on graphs of bounded maximum induced matching width (mim-width). Explicitly, given a branch decomposition of mim-width w, we give an n^{O(w)}-time algorithm that solves Feedback Vertex Set. This provides a unified algorithm for many well-known classes, such as Interval graphs and Permutation graphs, and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mim-width, such as Circular Permutation and Circular k-Trapezoid graphs for fixed k. In all these classes the decomposition is computable in polynomial time, as shown by Belmonte and Vatshelle [Theor. Comput. Sci. 2013].
We show that powers of graphs of tree-width w-1 or path-width w and powers of graphs of clique-width w have mim-width at most w. These results extensively provide new classes of bounded mim-width. We prove a slight strengthening of the first statement which implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most 1. Given a tree decomposition of width w-1, a path decomposition of width w, or a clique-width w-expression of a graph G, one can for any value of k find a mim-width decomposition of its k-power in polynomial time, and apply our algorithm to solve Feedback Vertex Set on the k-power in time n^{O(w)}.
In contrast to Feedback Vertex Set, we show that Hamiltonian Cycle is NP-complete even on graphs of linear mim-width 1, which further hints at the expressive power of the mim-width parameter
On the approximability of the maximum induced matching problem
In this paper we consider the approximability of the maximum induced matching problem (MIM). We give an approximation algorithm with asymptotic performance ratio <i>d</i>-1 for MIM in <i>d</i>-regular graphs, for each <i>d</i>≥3. We also prove that MIM is APX-complete in <i>d</i>-regular graphs, for each <i>d</i>≥3
On the asymptotics of dimers on tori
We study asymptotics of the dimer model on large toric graphs. Let be a weighted -periodic planar graph, and let
be a large-index sublattice of . For bipartite we
show that the dimer partition function on the quotient
has the asymptotic expansion , where is the area of ,
is the free energy density in the bulk, and is a finite-size
correction term depending only on the conformal shape of the domain together
with some parity-type information. Assuming a conjectural condition on the zero
locus of the dimer characteristic polynomial, we show that an analogous
expansion holds for non-bipartite. The functional form of the
finite-size correction differs between the two classes, but is universal within
each class. Our calculations yield new information concerning the distribution
of the number of loops winding around the torus in the associated double-dimer
models.Comment: 48 pages, 18 figure
Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes
We investigate how to make the surface of a convex polyhedron (a polytope) by
folding up a polygon and gluing its perimeter shut, and the reverse process of
cutting open a polytope and unfolding it to a polygon. We explore basic
enumeration questions in both directions: Given a polygon, how many foldings
are there? Given a polytope, how many unfoldings are there to simple polygons?
Throughout we give special attention to convex polygons, and to regular
polygons. We show that every convex polygon folds to an infinite number of
distinct polytopes, but that their number of combinatorially distinct gluings
is polynomial. There are, however, simple polygons with an exponential number
of distinct gluings.
In the reverse direction, we show that there are polytopes with an
exponential number of distinct cuttings that lead to simple unfoldings. We
establish necessary conditions for a polytope to have convex unfoldings,
implying, for example, that among the Platonic solids, only the tetrahedron has
a convex unfolding. We provide an inventory of the polytopes that may unfold to
regular polygons, showing that, for n>6, there is essentially only one class of
such polytopes.Comment: 54 pages, 33 figure
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