113 research outputs found
Extended formulations for a class of polyhedra with bimodular cographic constraint matrices
We are motivated by integer linear programs (ILPs) defined by constraint
matrices with bounded determinants. Such matrices generalize the notion of
totally-unimodular matrices. When the determinants are bounded by , the
matrix is called bimodular. Artmann et al. give a polynomial-time algorithm for
solving any ILP defined by a bimodular constraint matrix. Complementing this
result, Conforti et al. give a compact extended formulation for a particular
class of bimodular-constrained ILPs, namely those that model the stable set
polytope of a graph with odd cycle packing number . We demonstrate that
their compact extended formulation can be modified to hold for polyhedra such
that (1) the constraint matrix is bimodular, (2) the row-matroid generated by
the constraint matrix is cographic and (3) the right-hand side is a linear
combination of the columns of the constraint matrix. This generalizes the
important special case from Conforti et al. concerning 4-connected graphs with
odd cycle transversal number at least four. Moreover, our results yield compact
extended formulations for a new class of polyhedra
Sensitivity Conjecture and Log-rank Conjecture for functions with small alternating numbers
The Sensitivity Conjecture and the Log-rank Conjecture are among the most
important and challenging problems in concrete complexity. Incidentally, the
Sensitivity Conjecture is known to hold for monotone functions, and so is the
Log-rank Conjecture for and with monotone
functions , where and are bit-wise AND and XOR,
respectively. In this paper, we extend these results to functions which
alternate values for a relatively small number of times on any monotone path
from to . These deepen our understandings of the two conjectures,
and contribute to the recent line of research on functions with small
alternating numbers
Convex Analysis and Optimization with Submodular Functions: a Tutorial
Set-functions appear in many areas of computer science and applied
mathematics, such as machine learning, computer vision, operations research or
electrical networks. Among these set-functions, submodular functions play an
important role, similar to convex functions on vector spaces. In this tutorial,
the theory of submodular functions is presented, in a self-contained way, with
all results shown from first principles. A good knowledge of convex analysis is
assumed
Algorithms to Approximate Column-Sparse Packing Problems
Column-sparse packing problems arise in several contexts in both
deterministic and stochastic discrete optimization. We present two unifying
ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain
improved approximation algorithms for some well-known families of such
problems. As three main examples, we attain the integrality gap, up to
lower-order terms, for known LP relaxations for k-column sparse packing integer
programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set
packing (Bansal et al., Algorithmica, 2012), and go "half the remaining
distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and
Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM
Transactions of Algorithm
Convex Relaxation for Combinatorial Penalties
In this paper, we propose an unifying view of several recently proposed
structured sparsity-inducing norms. We consider the situation of a model
simultaneously (a) penalized by a set- function de ned on the support of the
unknown parameter vector which represents prior knowledge on supports, and (b)
regularized in Lp-norm. We show that the natural combinatorial optimization
problems obtained may be relaxed into convex optimization problems and
introduce a notion, the lower combinatorial envelope of a set-function, that
characterizes the tightness of our relaxations. We moreover establish links
with norms based on latent representations including the latent group Lasso and
block-coding, and with norms obtained from submodular functions.Comment: 35 pag
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