114 research outputs found
Convex Matroid Optimization
We consider a problem of optimizing convex functionals over matroid bases. It
is richly expressive and captures certain quadratic assignment and clustering
problems. While generally NP-hard, we show it is polynomial time solvable when
a suitable parameter is restricted
Two graph isomorphism polytopes
The convex hull of certain tensors was considered
recently in connection with graph isomorphism. We consider the convex hull
of the diagonals among these tensors. We show: 1. The polytope
is a face of . 2. Deciding if a graph has a subgraph
isomorphic to reduces to optimization over . 3. Optimization over
reduces to optimization over . In particular, this implies
that the subgraph isomorphism problem reduces to optimization over
Nowhere-Zero Flow Polynomials
In this article we introduce the flow polynomial of a digraph and use it to
study nowhere-zero flows from a commutative algebraic perspective. Using
Hilbert's Nullstellensatz, we establish a relation between nowhere-zero flows
and dual flows. For planar graphs this gives a relation between nowhere-zero
flows and flows of their planar duals. It also yields an appealing proof that
every bridgeless triangulated graph has a nowhere-zero four-flow
Convex Integer Optimization by Constantly Many Linear Counterparts
In this article we study convex integer maximization problems with composite
objective functions of the form , where is a convex function on
and is a matrix with small or binary entries, over
finite sets of integer points presented by an oracle or by
linear inequalities.
Continuing the line of research advanced by Uri Rothblum and his colleagues
on edge-directions, we introduce here the notion of {\em edge complexity} of
, and use it to establish polynomial and constant upper bounds on the number
of vertices of the projection \conv(WS) and on the number of linear
optimization counterparts needed to solve the above convex problem.
Two typical consequences are the following. First, for any , there is a
constant such that the maximum number of vertices of the projection of
any matroid by any binary matrix is
regardless of and ; and the convex matroid problem reduces to
greedily solvable linear counterparts. In particular, . Second, for any
, there is a constant such that the maximum number of
vertices of the projection of any three-index
transportation polytope for any by any binary
matrix is ; and the convex three-index transportation problem
reduces to linear counterparts solvable in polynomial time
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