720 research outputs found
Conic Mixed-Binary Sets: Convex Hull Characterizations and Applications
We consider a general conic mixed-binary set where each homogeneous conic
constraint involves an affine function of independent continuous variables and
an epigraph variable associated with a nonnegative function, , of common
binary variables. Sets of this form naturally arise as substructures in a
number of applications including mean-risk optimization, chance-constrained
problems, portfolio optimization, lot-sizing and scheduling, fractional
programming, variants of the best subset selection problem, and
distributionally robust chance-constrained programs. When all of the functions
's are submodular, we give a convex hull description of this set that
relies on characterizing the epigraphs of 's. Our result unifies and
generalizes an existing result in two important directions. First, it considers
\emph{multiple general convex cone} constraints instead of a single
second-order cone type constraint. Second, it takes \emph{arbitrary nonnegative
functions} instead of a specific submodular function obtained from the square
root of an affine function. We close by demonstrating the applicability of our
results in the context of a number of broad problem classes.Comment: 21 page
Overcommitment in Cloud Services -- Bin packing with Chance Constraints
This paper considers a traditional problem of resource allocation, scheduling
jobs on machines. One such recent application is cloud computing, where jobs
arrive in an online fashion with capacity requirements and need to be
immediately scheduled on physical machines in data centers. It is often
observed that the requested capacities are not fully utilized, hence offering
an opportunity to employ an overcommitment policy, i.e., selling resources
beyond capacity. Setting the right overcommitment level can induce a
significant cost reduction for the cloud provider, while only inducing a very
low risk of violating capacity constraints. We introduce and study a model that
quantifies the value of overcommitment by modeling the problem as a bin packing
with chance constraints. We then propose an alternative formulation that
transforms each chance constraint into a submodular function. We show that our
model captures the risk pooling effect and can guide scheduling and
overcommitment decisions. We also develop a family of online algorithms that
are intuitive, easy to implement and provide a constant factor guarantee from
optimal. Finally, we calibrate our model using realistic workload data, and
test our approach in a practical setting. Our analysis and experiments
illustrate the benefit of overcommitment in cloud services, and suggest a cost
reduction of 1.5% to 17% depending on the provider's risk tolerance
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
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