12 research outputs found
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
Energy Efficient Scheduling via Partial Shutdown
Motivated by issues of saving energy in data centers we define a collection
of new problems referred to as "machine activation" problems. The central
framework we introduce considers a collection of machines (unrelated or
related) with each machine having an {\em activation cost} of . There
is also a collection of jobs that need to be performed, and is
the processing time of job on machine . We assume that there is an
activation cost budget of -- we would like to {\em select} a subset of
the machines to activate with total cost and {\em find} a schedule
for the jobs on the machines in minimizing the makespan (or any other
metric).
For the general unrelated machine activation problem, our main results are
that if there is a schedule with makespan and activation cost then we
can obtain a schedule with makespan \makespanconstant T and activation cost
\costconstant A, for any . We also consider assignment costs for
jobs as in the generalized assignment problem, and using our framework, provide
algorithms that minimize the machine activation and the assignment cost
simultaneously. In addition, we present a greedy algorithm which only works for
the basic version and yields a makespan of and an activation cost .
For the uniformly related parallel machine scheduling problem, we develop a
polynomial time approximation scheme that outputs a schedule with the property
that the activation cost of the subset of machines is at most and the
makespan is at most for any
Rejecting Jobs to Minimize Load and Maximum Flow-time
Online algorithms are usually analyzed using the notion of competitive ratio
which compares the solution obtained by the algorithm to that obtained by an
online adversary for the worst possible input sequence. Often this measure
turns out to be too pessimistic, and one popular approach especially for
scheduling problems has been that of "resource augmentation" which was first
proposed by Kalyanasundaram and Pruhs. Although resource augmentation has been
very successful in dealing with a variety of objective functions, there are
problems for which even a (arbitrary) constant speedup cannot lead to a
constant competitive algorithm. In this paper we propose a "rejection model"
which requires no resource augmentation but which permits the online algorithm
to not serve an epsilon-fraction of the requests.
The problems considered in this paper are in the restricted assignment
setting where each job can be assigned only to a subset of machines. For the
load balancing problem where the objective is to minimize the maximum load on
any machine, we give O(\log^2 1/\eps)-competitive algorithm which rejects at
most an \eps-fraction of the jobs. For the problem of minimizing the maximum
weighted flow-time, we give an O(1/\eps^4)-competitive algorithm which can
reject at most an \eps-fraction of the jobs by weight. We also extend this
result to a more general setting where the weights of a job for measuring its
weighted flow-time and its contribution towards total allowed rejection weight
are different. This is useful, for instance, when we consider the objective of
minimizing the maximum stretch. We obtain an O(1/\eps^6)-competitive
algorithm in this case.
Our algorithms are immediate dispatch, though they may not be immediate
reject. All these problems have very strong lower bounds in the speed
augmentation model
Approximation Algorithms for Resource Allocation
This thesis is devoted to designing new techniques and algorithms for combinatorial optimization problems arising in various applications of resource allocation. Resource allocation refers to a class of problems where scarce resources must be distributed among competing agents maintaining certain optimization criteria. Examples include scheduling jobs on one/multiple machines maintaining system performance; assigning advertisements to bidders, or items to people maximizing profit/social fairness; allocating servers or channels satisfying networking requirements etc. Altogether they comprise a wide variety of combinatorial optimization problems. However, a majority of these problems are NP-hard in nature and therefore, the goal herein is to develop approximation algorithms that approximate the optimal solution as best as possible in polynomial time.
The thesis addresses two main directions. First, we develop several new techniques, predominantly, a new linear programming rounding methodology and a constructive aspect of a well-known probabilistic method, the Lov\'{a}sz Local Lemma (LLL). Second, we employ
these techniques to applications of resource allocation obtaining substantial improvements over known results. Our research also spurs new direction of study; we introduce new models for achieving energy efficiency in scheduling and a novel framework for assigning advertisements in cellular networks. Both of these lead to a variety of interesting questions.
Our linear programming rounding methodology is a significant generalization of two major rounding approaches in the theory of approximation algorithms, namely the dependent rounding and the iterative relaxation procedure. Our constructive version of LLL leads to first algorithmic results for many combinatorial problems. In addition, it settles a major open question of obtaining a constant factor approximation algorithm for the Santa Claus problem. The Santa Claus problem is a -hard resource allocation problem that received much attention in the last several years. Through out this thesis, we study a number of applications related to scheduling jobs on unrelated parallel machines, such as provisionally shutting down machines to save energy, selectively dropping outliers to improve system performance, handling machines with hard capacity bounds on the number of jobs they can process etc. Hard capacity constraints arise naturally in many other applications and often render a hitherto simple combinatorial optimization problem difficult. In this thesis, we encounter many such instances of hard capacity constraints, namely in budgeted allocation of advertisements for cellular networks, overlay network design, and in classical problems like vertex cover, set cover and k-median
Capacitated Network Design on Outerplanar Graphs
Network design problems model the efficient allocation of resources like routers, optical fibres, roads, canals etc. to effectively construct and operate critical infrastructures. In this thesis, we consider the capacitated network design problem (CapNDP), which finds applications in supply-chain logistics problems and network security. Here, we are given a network and for each edge in the network, several security reinforcement options. In addition, for each pair of nodes in the network, there is a specified level of protection demanded. The objective is to select a minimum-cost set of reinforcements for all the edges so that an adversary with strength less than the protection level of a particular pair of nodes cannot disconnect these nodes. The optimal solution to this problem cannot, in general, be found in reasonable time. One way to tackle such hard problems is to develop approximation algorithms, which are fast algorithms that are guaranteed to find near-optimal solutions; the worst-case ratio between the cost of the solution output by the algorithm and the optimum cost is called the approximation ratio of the algorithm. In this thesis, we investigate CapNDP when the network structure is constrained to belong to a class of graphs called outerplanar graphs. This particular special case was first considered by Carr, Fleischer, Leung and Philips; while they claimed to obtain an approximation ratio arbitrarily close to 1, their algorithm has certain fatal flaws. We build upon some of the ideas they use to approximate CapNDP on general networks to develop a new algorithm for CapNDP on outerplanar graphs. The approximation ratio achieved by our algorithm improves the state-of-the-art by a doubly exponential factor. We also notice that our methods can be applied to a more general class of problems called column-restricted covering integers programs, and be adapted to improve the approximation ratio on more instances of CapNDP if the structure of the network is known. Furthermore, our techniques also yield interesting results for a completely unrelated problem in the area of data structures