26,416 research outputs found
The linearization problem of a binary quadratic problem and its applications
We provide several applications of the linearization problem of a binary
quadratic problem. We propose a new lower bounding strategy, called the
linearization-based scheme, that is based on a simple certificate for a
quadratic function to be non-negative on the feasible set. Each
linearization-based bound requires a set of linearizable matrices as an input.
We prove that the Generalized Gilmore-Lawler bounding scheme for binary
quadratic problems provides linearization-based bounds. Moreover, we show that
the bound obtained from the first level reformulation linearization technique
is also a type of linearization-based bound, which enables us to provide a
comparison among mentioned bounds. However, the strongest linearization-based
bound is the one that uses the full characterization of the set of linearizable
matrices. Finally, we present a polynomial-time algorithm for the linearization
problem of the quadratic shortest path problem on directed acyclic graphs. Our
algorithm gives a complete characterization of the set of linearizable matrices
for the quadratic shortest path problem
Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives
A well-studied nonlinear extension of the minimum-cost flow problem is to
minimize the objective over feasible flows ,
where on every arc of the network, is a convex function. We give
a strongly polynomial algorithm for the case when all 's are convex
quadratic functions, settling an open problem raised e.g. by Hochbaum [1994].
We also give strongly polynomial algorithms for computing market equilibria in
Fisher markets with linear utilities and with spending constraint utilities,
that can be formulated in this framework (see Shmyrev [2009], Devanur et al.
[2011]). For the latter class this resolves an open question raised by Vazirani
[2010]. The running time is for quadratic costs,
for Fisher's markets with linear utilities and
for spending constraint utilities.
All these algorithms are presented in a common framework that addresses the
general problem setting. Whereas it is impossible to give a strongly polynomial
algorithm for the general problem even in an approximate sense (see Hochbaum
[1994]), we show that assuming the existence of certain black-box oracles, one
can give an algorithm using a strongly polynomial number of arithmetic
operations and oracle calls only. The particular algorithms can be derived by
implementing these oracles in the respective settings
Service Chain (SC) Mapping with Multiple SC Instances in a Wide Area Network
Network Function Virtualization (NFV) aims to simplify deployment of network
services by running Virtual Network Functions (VNFs) on commercial
off-the-shelf servers. Service deployment involves placement of VNFs and
in-sequence routing of traffic flows through VNFs comprising a Service Chain
(SC). The joint VNF placement and traffic routing is usually referred as SC
mapping. In a Wide Area Network (WAN), a situation may arise where several
traffic flows, generated by many distributed node pairs, require the same SC,
one single instance (or occurrence) of that SC might not be enough. SC mapping
with multiple SC instances for the same SC turns out to be a very complex
problem, since the sequential traversal of VNFs has to be maintained while
accounting for traffic flows in various directions. Our study is the first to
deal with SC mapping with multiple SC instances to minimize network resource
consumption. Exact mathematical modeling of this problem results in a quadratic
formulation. We propose a two-phase column-generation-based model and solution
in order to get results over large network topologies within reasonable
computational times. Using such an approach, we observe that an appropriate
choice of only a small set of SC instances can lead to solution very close to
the minimum bandwidth consumption
The Metric Nearness Problem
Metric nearness refers to the problem of optimally restoring metric properties to
distance measurements that happen to be nonmetric due to measurement errors or otherwise. Metric
data can be important in various settings, for example, in clustering, classification, metric-based
indexing, query processing, and graph theoretic approximation algorithms. This paper formulates
and solves the metric nearness problem: Given a set of pairwise dissimilarities, find a ānearestā set
of distances that satisfy the properties of a metricāprincipally the triangle inequality. For solving
this problem, the paper develops efficient triangle fixing algorithms that are based on an iterative
projection method. An intriguing aspect of the metric nearness problem is that a special case turns
out to be equivalent to the all pairs shortest paths problem. The paper exploits this equivalence and
develops a new algorithm for the latter problem using a primal-dual method. Applications to graph
clustering are provided as an illustration. We include experiments that demonstrate the computational
superiority of triangle fixing over general purpose convex programming software. Finally, we
conclude by suggesting various useful extensions and generalizations to metric nearness
- ā¦