10 research outputs found
On Box-Perfect Graphs
Let be a graph and let be the clique-vertex incidence matrix
of . It is well known that is perfect iff the system , is totally dual integral (TDI). In 1982,
Cameron and Edmonds proposed to call box-perfect if the system
, is box-totally dual
integral (box-TDI), and posed the problem of characterizing such graphs. In
this paper we prove the Cameron-Edmonds conjecture on box-perfectness of parity
graphs, and identify several other classes of box-perfect graphs. We also
develop a general and powerful method for establishing box-perfectness
A Discrete Convex Min-Max Formula for Box-TDI Polyhedra
A min-max formula is proved for the minimum of an integer-valued separable
discrete convex function where the minimum is taken over the set of integral
elements of a box total dual integral (box-TDI) polyhedron. One variant of the
theorem uses the notion of conjugate function (a fundamental concept in
non-linear optimization) but we also provide another version that avoids
conjugates, and its spirit is conceptually closer to the standard form of
classic min-max theorems in combinatorial optimization. The presented framework
provides a unified background for separable convex minimization over the set of
integral elements of the intersection of two integral base-polyhedra,
submodular flows, L-convex sets, and polyhedra defined by totally unimodular
(TU) matrices. As an unexpected application, we show how a wide class of
inverse combinatorial optimization problems can be covered by this new
framework.Comment: 32 page
Recent Progress on Integrally Convex Functions
Integrally convex functions constitute a fundamental function class in
discrete convex analysis, including M-convex functions, L-convex functions, and
many others. This paper aims at a rather comprehensive survey of recent results
on integrally convex functions with some new technical results. Topics covered
in this paper include characterizations of integral convex sets and functions,
operations on integral convex sets and functions, optimality criteria for
minimization with a proximity-scaling algorithm, integral biconjugacy, and the
discrete Fenchel duality. While the theory of M-convex and L-convex functions
has been built upon fundamental results on matroids and submodular functions,
developing the theory of integrally convex functions requires more general and
basic tools such as the Fourier-Motzkin elimination.Comment: 50 page
Analyzing Massive Graphs in the Semi-streaming Model
Massive graphs arise in a many scenarios, for example,
traffic data analysis in large networks, large scale scientific
experiments, and clustering of large data sets.
The semi-streaming model was proposed for processing massive graphs. In the semi-streaming model, we have a random
accessible memory which is near-linear in the number of vertices.
The input graph (or equivalently, edges in the graph)
is presented as a sequential list of edges (insertion-only model)
or edge insertions and deletions (dynamic model). The list
is read-only but we may make multiple passes over the list.
There has been a few results in the insertion-only model
such as computing distance spanners and approximating
the maximum matching.
In this thesis, we present some algorithms and techniques
for (i) solving more complex problems in the semi-streaming model,
(for example, problems in the dynamic model) and (ii) having
better solutions for the problems which have been studied
(for example, the maximum matching problem). In course of both
of these, we develop new techniques with broad applications and
explore the rich trade-offs between the complexity of models
(insertion-only streams vs. dynamic streams), the number
of passes, space, accuracy, and running time.
1. We initiate the study of dynamic graph streams.
We start with basic problems such as the connectivity
problem and computing the minimum spanning tree.
These problems are
trivial in the insertion-only model. However, they require
non-trivial (and multiple passes for computing the exact minimum
spanning tree) algorithms in the
dynamic model.
2. Second, we present a graph sparsification algorithm in the
semi-streaming model. A graph sparsification
is a sparse graph that approximately preserves
all the cut values of a graph.
Such a graph acts as an oracle for solving cut-related problems,
for example, the minimum cut problem and the multicut problem.
Our algorithm produce a graph sparsification with high probability
in one pass.
3. Third, we use the primal-dual algorithms
to develop the semi-streaming algorithms.
The primal-dual algorithms have been widely accepted
as a framework for solving linear programs
and semidefinite programs faster.
In contrast, we apply the method for reducing space and
number of passes in addition to reducing the running time.
We also present some examples that arise in applications
and show how to apply the techniques:
the multicut problem, the correlation clustering problem,
and the maximum matching problem. As a consequence,
we also develop near-linear time algorithms for the -matching
problems which were not known before
On box totally dual integral polyhedra
Edmonds and Giles introduced the class of box totally dual integral polyhedra as a generalization of submodular flow polyhedra. In this paper a geometric characterization of these polyhedra is given. This geometric result is used to show that each TDI defining system for a box TDI polyhedron is in fact a box TDI system, that the class of box TDI polyhedra is in co-NP and is closed under taking projections and dominants, that the class of box perfect graphs is in co-NP, and a result of Edmonds and Giles which is related to the facets of box TDI polyhdera
Hard problems on box-totally dual integral polyhedra
In this paper, we study the complexity of some fundamental questions regarding box-totally dual integral (box-TDI) polyhedra. First, although box-TDI polyhedra have strong integrality properties, we prove that Integer Programming over box-TDI polyhedra is NP-complete, that is, finding an integer point optimizing a linear function over a box-TDI polyhedron is hard. Second, we complement the result of Ding et al. specialIntscript who proved that deciding whether a given system is box-TDI is co-NP-complete: we prove that recognizing whether a polyhedron is box-TDI is co-NP-complete.To derive these complexity results, we exhibit new classes of totally equimodular matrices - a generalization of totally unimodular matrices - by characterizing the total equimodularity of incidence matrices of graphs