Coordinator: Naveen Garg
The Algorithms Group at IIT Delhi, India was designated a Partner-Group of Max Planck Institute for Informatics for research in the area of ”Approximation Algorithms”. The group started its operations in July 2005.
2.11.1 Scheduling to Minimize Flow Time
Investigators: Naveen Garg, Amit Kumar, V. N. Muralidhara, Jivitej S. Chadha, and Vinayaka Pandit
We consider the problem of scheduling jobs on multiple machines so as to minimize the total flow time. The flow time of a job is the total time it spends in the system and equals the difference between its completion time and release time.
Unrelated Machines
A job j has a processing time pij on machine i; this is the most general model of processing times and is known as the unrelated machines model.
In [3] we consider a special case of scheduling on unrelated machines. The machines are identical except that each job can be assigned only to a specified subset of the machines.
We show that no online algorithm can have a bounded competitive ratio. We provide an
O(log P )-approximation algorithm by modifying the single-source unsplittable flow algorithm of Dinitz et.al.[2]. Here P is the ratio of the maximum to the minimum processing times.
We also establish an Ω(log P )-integrality gap for our LP-relaxation and use this to show an Ω(log P/ log log P ) lower bound on the approximability of the problem.
For the unrelated machines setting, we introduce a notion of (α, β) variability to capture settings where processing times of jobs on machines are not completely arbitrary. We say that processing times have an (α, β)-variability if the processing time of job j on machine i can be expressed as pij = aj· bij· si where bij ∈ B, |B| = β and 1 ≤ aj ≤ α. Our main result in [4] is a simple O(β log α) approximation for this setting. As special cases, we get
(a) an O(k) approximation when there are only k different processing times.
(b) an O(log P )-approximation if each job can only go on a specified subset of machines, but has the same processing requirement on each such machine. Further, the machines can have different speeds. Here P is the ratio of the largest to the smallest processing requirement.
(c) an O(−1log −1)- approximation algorithm for unrelated machines if we assume that our algorithm has machines which are an (1 + )-factor faster than the optimum algorithm’s machines.
We also extend the hardness results to the problem of minimizing flow time on parallel machines. We show that the problem cannot be approximated to within Ω(log1−P ) for any
> 0.
Unrelated Machines with Speed Augmentation
We consider the online problem of scheduling jobs on unrelated machines so as to minimize the total weighted flow time. This problem has an unbounded competitive ratio even for very restricted settings. In [1] we show that if we allow the machines of the online algorithm to have more speed than those of the offline algorithm then we can get an O((1 + −1)2 )-competitive algorithm.
Our algorithm schedules jobs preemptively but without migration. However, we compare our solution to an offline algorithm which allows migration. Our analysis uses a potential function argument which can also be extended to give a simpler and better proof of the randomized immediate dispatch algorithm of Chekuri-Goel-Khanna-Kumar for minimizing average flow time on parallel machines.
Order Scheduling
In [5] we consider scheduling problems in which a job consists of components of different types to be processed on m machines. Each machine is capable of processing components of a single type. Different components of a job are independent and can be processed in parallel on different machines. A job is considered as completed only when all its components have been completed. We study both completion and flow time aspects of such problems.
We show both lower bounds and upper bounds for the completion time problem. We first show that even the unweighted completion time with single release date is MAX-SNP hard.
We give an approximation algorithm based on linear programming which has an approxi-mation ratio of 3 for weighted completion time with multiple release dates. We give online algorithms for the weighted completion time which are constant factor competitive.
For the flow time, we give only lower bounds in both the offline and online settings. We show that it is NP-hard to approximate flow time within Ω(log m) in the offline setting. We show that no online algorithm for the flow time can have a competitive ratio better than Ω(√
m).
References
[1] J. S. Chadha, N. Garg, A. Kumar, and V. N. Muralidhara. A competitive algorithm for minimizing weighted flow time on unrelated machines with speed augmentation. In STOC, 2009.
[2] Y. Dinitz, N. Garg, and M. X. Goemans. On the single-source unsplittable flow problem. Com-binatorica, 19(1):17–41, 1999.
[3] N. Garg and A. Kumar. Minimizing average flow-time : Upper and lower bounds. In FOCS, 2007, pp. 603–613.
[4] N. Garg, A. Kumar, and V. N. Muralidhara. Minimizing total flow-time: The unrelated case. In ISAAC, 2008, pp. 424–435.
[5] N. Garg, A. Kumar, and V. Pandit. Order scheduling models: Hardness and algorithms. In FSTTCS, 2007, pp. 96–107.
2.11.2 Stochastic Analysis of Online Algorithms
Investigators: Naveen Garg, Anupam Gupta, Stefano Leonardi, and Piotr Sankowski The study of online algorithms has been an extremely popular and successful program in the area of algorithms. This has mainly focused on competitive analysis, where the performance of an online algorithm (that knows nothing about the future) is compared to the optimal so-lution built with hindsight. This model has led to cleanly defined problems, and strong upper and lower bounds on the competitive ratio are known for most problems of interest. There are, however, shortcomings to using competitive analysis: the biggest objection being that the strict definition of competitive ratio does not allow us to make fine-grained distinctions between algorithms.
Over the years, these drawbacks to the competitive analysis framework have caused re-searchers to try and weaken the rigid competitive analysis framework, and return to variants of the fundamental question: Can we do better if we are given access to the input distribu-tion? While the situation is fairly-well understood for classical online problems like paging and k-server, we still know almost nothing for, say, online Steiner tree in this model. In fact, the starting point of our investigation is the following basic question:
Can we beat the Θ(log n) bound known for online Steiner tree if at each time instant, the demand vertex is a uniformly random vertex from the graph?
In [1], we answer this question in the affirmative: we show that if each vertex is an indepen-dent draw from some probability distribution π : V → [0, 1], a slight variant of the natural
greedy algorithm achieves either an O(1), or an O(log log n) performance guarantee, depend-ing on the precise nature of the guarantees desired. Some of these results can be extended to other subadditive problems as well. Furthermore, we show that both assumptions that the input sequence consists of independent draws from π, and that π is known to the algorithm are both essential; we show logarithmic lower bounds if either assumption is violated.
References
[1] N. Garg, A. Gupta, S. Leonardi, and P. Sankowski. Stochastic analyses for online combinatorial optimization problems. In SODA, 2008, pp. 942–951.
2.11.3 Stochastic Network Design
Investigators: Anupam Gupta, Amit Kumar and MohammadTaghi Hajiaghayi
While the Steiner tree problem has been well studied in the model of two stage stochastic optimization with recourse, with several different solutions and extensions to the multistage case, all these algorithms work only in the case when all the edge-costs increase by a uni-form factor. When each edge-cost is allowed to increase by a different factor, nothing was previously known.
In [1] we show that this problem admits a poly-logarithmic approximation guarantee;
moreover, it is as hard as the cost-distance problem, for which we have a Ω(log log n) hard-ness. Also, we show that if the inflation is allowed to vary over scenarios, the problem becomes as hard as Label-cover. Finally, we give a new linear-programming relaxation of the multi-commodity cost-distance problem.
In [2] we consider the stochastic Steiner forest problem: suppose we were given a collection of Steiner forest instances, and were guaranteed that a random one of these instances would appear tomorrow; moreover, the cost of edges tomorrow will be λ times the cost of edges today. Which edges should we buy today so that we can extend it to a solution for the instance arriving tomorrow, to minimize the expected total cost? While very general results have been developed for many problems in stochastic discrete optimization over the past years, the approximation status of the stochastic Steiner Forest problem has remained open, with previous works yielding constant-factor approximations only for special cases. We resolve the status of this problem by giving a constant-factor primal-dual based approximation algorithm.
References
[1] A. Gupta, M. Hajiaghayi, and A. Kumar. Stochastic steiner tree with non-uniform inflation. In APPROX-RANDOM, 2007, pp. 134–148.
[2] A. Gupta and A. Kumar. A constant factor approximation for stochastic steiner forest. In Proceedings on 41st Annual ACM Symposium on Theory of Computing (STOC), 2009.