76 research outputs found
An Efficient Local Search for Partial Latin Square Extension Problem
A partial Latin square (PLS) is a partial assignment of n symbols to an nxn
grid such that, in each row and in each column, each symbol appears at most
once. The partial Latin square extension problem is an NP-hard problem that
asks for a largest extension of a given PLS. In this paper we propose an
efficient local search for this problem. We focus on the local search such that
the neighborhood is defined by (p,q)-swap, i.e., removing exactly p symbols and
then assigning symbols to at most q empty cells. For p in {1,2,3}, our
neighborhood search algorithm finds an improved solution or concludes that no
such solution exists in O(n^{p+1}) time. We also propose a novel swap
operation, Trellis-swap, which is a generalization of (1,q)-swap and
(2,q)-swap. Our Trellis-neighborhood search algorithm takes O(n^{3.5}) time to
do the same thing. Using these neighborhood search algorithms, we design a
prototype iterated local search algorithm and show its effectiveness in
comparison with state-of-the-art optimization solvers such as IBM ILOG CPLEX
and LocalSolver.Comment: 17 pages, 2 figure
Scheduling Algorithms for Procrastinators
This paper presents scheduling algorithms for procrastinators, where the
speed that a procrastinator executes a job increases as the due date
approaches. We give optimal off-line scheduling policies for linearly
increasing speed functions. We then explain the computational/numerical issues
involved in implementing this policy. We next explore the online setting,
showing that there exist adversaries that force any online scheduling policy to
miss due dates. This impossibility result motivates the problem of minimizing
the maximum interval stretch of any job; the interval stretch of a job is the
job's flow time divided by the job's due date minus release time. We show that
several common scheduling strategies, including the "hit-the-highest-nail"
strategy beloved by procrastinators, have arbitrarily large maximum interval
stretch. Then we give the "thrashing" scheduling policy and show that it is a
\Theta(1) approximation algorithm for the maximum interval stretch.Comment: 12 pages, 3 figure
Optimal ordering rule for a stochastic sequencing model
In this note, necessary and sufficient conditions are derived for the optimality of a sequencing rule for a class of stochastic sequential models. The optimal sequential rule generalizes the deterministic results, given in Refs. 1–2, for situations when some of the parameters of the problem are random variables. Two cases are given to demonstrate the usefulness of the results.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45247/1/10957_2005_Article_BF02275359.pd
On generalized surrogate duality in mixed-integer nonlinear programming
The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global -optimality with spatial branch and bound is a tight, computationally
tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solvers
can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we
exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and
challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the
algorithm’s ability to generate strong dual bounds through extensive computational experiments
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