2,452 research outputs found
A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface
Given a graph cellularly embedded on a surface of genus , a
cut graph is a subgraph of such that cutting along yields a
topological disk. We provide a fixed parameter tractable approximation scheme
for the problem of computing the shortest cut graph, that is, for any
, we show how to compute a approximation of
the shortest cut graph in time .
Our techniques first rely on the computation of a spanner for the problem
using the technique of brick decompositions, to reduce the problem to the case
of bounded tree-width. Then, to solve the bounded tree-width case, we introduce
a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which
may be of independent interest
Polyhedral characteristics of balanced and unbalanced bipartite subgraph problems
We study the polyhedral properties of three problems of constructing an
optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the
first problem we consider a balanced biclique with the same number of vertices
in both parts and arbitrary edge weights. In the other two problems we are
dealing with unbalanced subgraphs of maximum and minimum weight with
nonnegative edges. All three problems are established to be NP-hard. We study
the polytopes and the cone decompositions of these problems and their
1-skeletons. We describe the adjacency criterion in 1-skeleton of the polytope
of the balanced complete bipartite subgraph problem. The clique number of
1-skeleton is estimated from below by a superpolynomial function. For both
unbalanced biclique problems we establish the superpolynomial lower bounds on
the clique numbers of the graphs of nonnegative cone decompositions. These
values characterize the time complexity in a broad class of algorithms based on
linear comparisons
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
The generalized minimum spanning tree polytope and related polytopes
The Generalized Minimum Spanning Tree problem denoted by GMST is a variant of the classical Minimum Spanning Tree problem in which nodes are partitioned into clusters and the problem calls for a minimum cost tree spanning at least one node from each cluster. A different version of the problem, called E-GMST arises when exactly one node from each cluster has to be visited. Both GMST problem and E-GMST problem are NP-hard problems. In this paper, we model GMST problem and E-GMST problem as integer linear programs and study the facial structure of the corresponding polytopes
Average case polyhedral complexity of the maximum stable set problem
We study the minimum number of constraints needed to formulate random
instances of the maximum stable set problem via linear programs (LPs), in two
distinct models. In the uniform model, the constraints of the LP are not
allowed to depend on the input graph, which should be encoded solely in the
objective function. There we prove a lower bound with
probability at least for every LP that is exact for a randomly
selected set of instances; each graph on at most n vertices being selected
independently with probability . In the
non-uniform model, the constraints of the LP may depend on the input graph, but
we allow weights on the vertices. The input graph is sampled according to the
G(n, p) model. There we obtain upper and lower bounds holding with high
probability for various ranges of p. We obtain a super-polynomial lower bound
all the way from to . Our upper bound is close to this as there is only an essentially quadratic
gap in the exponent, which currently also exists in the worst-case model.
Finally, we state a conjecture that would close this gap, both in the
average-case and worst-case models
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