73 research outputs found
Breaking symmetries to rescue Sum of Squares in the case of makespan scheduling
The Sum of Squares (\sos{}) hierarchy gives an automatized technique to
create a family of increasingly tight convex relaxations for binary programs.
There are several problems for which a constant number of rounds of this
hierarchy give integrality gaps matching the best known approximation
algorithms. For many other problems, however, ad-hoc techniques give better
approximation ratios than \sos{} in the worst case, as shown by corresponding
lower bound instances. Notably, in many cases these instances are invariant
under the action of a large permutation group. This yields the question how
symmetries in a formulation degrade the performance of the relaxation obtained
by the \sos{} hierarchy. In this paper, we study this for the case of the
minimum makespan problem on identical machines. Our first result is to show
that rounds of \sos{} applied over the \emph{configuration linear
program} yields an integrality gap of at least , where is the
number of jobs. Our result is based on tools from representation theory of
symmetric groups. Then, we consider the weaker \emph{assignment linear program}
and add a well chosen set of symmetry breaking inequalities that removes a
subset of the machine permutation symmetries. We show that applying
rounds of the SA hierarchy to this stronger
linear program reduces the integrality gap to , which yields a
linear programming based polynomial time approximation scheme. Our results
suggest that for this classical problem, symmetries were the main barrier
preventing the \sos{}/ SA hierarchies to give relaxations of polynomial
complexity with an integrality gap of~. We leave as an open
question whether this phenomenon occurs for other symmetric problems
Quasi-PTAS for Scheduling with Precedences using LP Hierarchies
A central problem in scheduling is to schedule n unit size jobs with precedence constraints on m identical machines so as to minimize the makespan. For m=3, it is not even known if the problem is NP-hard and this is one of the last open problems from the book of Garey and Johnson.
We show that for fixed m and epsilon, {polylog}(n) rounds of Sherali-Adams hierarchy applied to a natural LP of the problem provides a (1+epsilon)-approximation algorithm running in quasi-polynomial time. This improves over the recent result of Levey and Rothvoss, who used r=(log n)^{O(log log n)} rounds of Sherali-Adams in order to get a (1+epsilon)-approximation algorithm with a running time of n^O(r)
Sherali-Adams gaps, flow-cover inequalities and generalized configurations for capacity-constrained Facility Location
Metric facility location is a well-studied problem for which linear
programming methods have been used with great success in deriving approximation
algorithms. The capacity-constrained generalizations, such as capacitated
facility location (CFL) and lower-bounded facility location (LBFL), have proved
notorious as far as LP-based approximation is concerned: while there are
local-search-based constant-factor approximations, there is no known linear
relaxation with constant integrality gap. According to Williamson and Shmoys
devising a relaxation-based approximation for \cfl\ is among the top 10 open
problems in approximation algorithms.
This paper advances significantly the state-of-the-art on the effectiveness
of linear programming for capacity-constrained facility location through a host
of impossibility results for both CFL and LBFL. We show that the relaxations
obtained from the natural LP at levels of the Sherali-Adams
hierarchy have an unbounded gap, partially answering an open question of
\cite{LiS13, AnBS13}. Here, denotes the number of facilities in the
instance. Building on the ideas for this result, we prove that the standard CFL
relaxation enriched with the generalized flow-cover valid inequalities
\cite{AardalPW95} has also an unbounded gap. This disproves a long-standing
conjecture of \cite{LeviSS12}. We finally introduce the family of proper
relaxations which generalizes to its logical extreme the classic star
relaxation and captures general configuration-style LPs. We characterize the
behavior of proper relaxations for CFL and LBFL through a sharp threshold
phenomenon.Comment: arXiv admin note: substantial text overlap with arXiv:1305.599
Scheduling with Outliers
In classical scheduling problems, we are given jobs and machines, and have to
schedule all the jobs to minimize some objective function. What if each job has
a specified profit, and we are no longer required to process all jobs -- we can
schedule any subset of jobs whose total profit is at least a (hard) target
profit requirement, while still approximately minimizing the objective
function?
We refer to this class of problems as scheduling with outliers. This model
was initiated by Charikar and Khuller (SODA'06) on the minimum max-response
time in broadcast scheduling. We consider three other well-studied scheduling
objectives: the generalized assignment problem, average weighted completion
time, and average flow time, and provide LP-based approximation algorithms for
them. For the minimum average flow time problem on identical machines, we give
a logarithmic approximation algorithm for the case of unit profits based on
rounding an LP relaxation; we also show a matching integrality gap. For the
average weighted completion time problem on unrelated machines, we give a
constant factor approximation. The algorithm is based on randomized rounding of
the time-indexed LP relaxation strengthened by the knapsack-cover inequalities.
For the generalized assignment problem with outliers, we give a simple
reduction to GAP without outliers to obtain an algorithm whose makespan is
within 3 times the optimum makespan, and whose cost is at most (1 + \epsilon)
times the optimal cost.Comment: 23 pages, 3 figure
An exact extended formulation for the unrelated parallel machine total weighted completion time problem
The plethora of research on NP-hard parallel machine scheduling problems is focused on heuristics due to the theoretically and practically challenging nature of these problems. Only a handful of exact approaches are available
in the literature, and most of these suffer from scalability issues. Moreover, the majority of the papers on the subject are restricted to the identical parallel machine scheduling environment. In this context, the main contribution of this work is to recognize and prove that a particular preemptive relaxation for the problem of minimizing the total weighted completion time (TWCT) on a set of unrelated parallel machines naturally admits a non-preemptive optimal solution and gives rise to an exact mixed integer linear programming formulation of the problem. Furthermore, we exploit the structural properties of TWCT and attain a very fast and scalable exact Benders decomposition-based algorithm for solving this formulation. Computationally, our approach holds great promise and may even be embedded into iterative algorithms for more complex shop scheduling problems as instances with up to 1000 jobs and 8 machines are solved to optimality within a few seconds
Algorithmic and game-theoretic perspectives on scheduling
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2008.Includes bibliographical references (p. 103-110).(cont.) Second, for almost all 0-1 bipartite instances, we give a lower bound on the integrality gap of various linear programming relaxations of this problem. Finally, we show that for almost all 0-1 bipartite instances, all feasible schedules are arbitrarily close to optimal. Finally, we consider the problem of minimizing the sum of weighted completion times in a concurrent open shop environment. We present some interesting properties of various linear programming relaxations for this problem, and give a combinatorial primal-dual 2-approximation algorithm.In this thesis, we study three problems related to various algorithmic and game-theoretic aspects of scheduling. First, we apply ideas from cooperative game theory to study situations in which a set of agents faces super modular costs. These situations appear in a variety of scheduling contexts, as well as in some settings related to facility location and network design. Although cooperation is unlikely when costs are super modular, in some situations, the failure to cooperate may give rise to negative externalities. We study the least core value of a cooperative game -- the minimum penalty we need to charge a coalition for acting independently that ensures the existence of an efficient and stable cost allocation -- as a means of encouraging cooperation. We show that computing the least core value of supermodular cost cooperative games is strongly NP-hard, and design an approximation framework for this problem that in the end, yields a (3 + [epsilon])-approximation algorithm. We also apply our approximation framework to obtain better results for two special cases of supermodular cost cooperative games that arise from scheduling and matroid optimization. Second, we focus on the classic precedence- constrained single-machine scheduling problem with the weighted sum of completion times objective. We focus on so-called 0-1 bipartite instances of this problem, a deceptively simple class of instances that has virtually the same approximability behavior as arbitrary instances. In the hope of improving our understanding of these instances, we use models from random graph theory to look at these instances with a probabilistic lens. First, we show that for almost all 0-1 bipartite instances, the decomposition technique of Sidney (1975) does not yield a non-trivial decomposition.by Nelson A. Uhan.Ph.D
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