Skip to main content

Stochastic Makespan Minimization in Structured Set Systems (Extended Abstract)

  • Conference paper
  • First Online:
Integer Programming and Combinatorial Optimization (IPCO 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12125))

Abstract

We study stochastic combinatorial optimization problems where the objective is to minimize the expected maximum load (a.k.a. the makespan). In this framework, we have a set of n tasks and m resources, where each task j uses some subset of the resources. Tasks have random sizes \(X_j\), and our goal is to non-adaptively select t tasks to minimize the expected maximum load over all resources, where the load on any resource i is the total size of all selected tasks that use i. For example, given a set of intervals in time, with each interval j having random load \(X_j\), how do we choose t intervals to minimize the expected maximum load at any time? Our technique is also applicable to other problems with some geometric structure in the relation between tasks and resources; e.g., packing paths, rectangles, and “fat” objects. Specifically, we give an \(O(\log \log m)\)-approximation algorithm for all these problems.

Our approach uses a strong LP relaxation using the cumulant generating functions of the random variables. We also show an LP integrality gap of \(\varOmega (\log ^* m)\) even for the problem of selecting intervals on a line.

All missing proofs and details can be found in the full version [11].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    The support of vector \(z\in \mathbb {R}^n_+\) is \(\{j\in [n] : z_j>0\}\) which corresponds to its positive entries.

References

  1. Adamaszek, A., Chalermsook, P., Wiese, A.: How to tame rectangles: solving independent set and coloring of rectangles via shrinking. In: APPROX/RANDOM, pp. 43–60 (2015)

    Google Scholar 

  2. Agarwal, P.K., Mustafa, N.H.: Independent set of intersection graphs of convex objects in 2D. Comput. Geom. 34(2), 83–95 (2006)

    Article  MathSciNet  Google Scholar 

  3. Chakrabarti, A., Chekuri, C., Gupta, A., Kumar, A.: Approximation algorithms for the unsplittable flow problem. Algorithmica 47(1), 53–78 (2007)

    Article  MathSciNet  Google Scholar 

  4. Chalermsook, P.: Coloring and maximum independent set of rectangles. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) APPROX/RANDOM 2011. LNCS, vol. 6845, pp. 123–134. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22935-0_11

    Chapter  MATH  Google Scholar 

  5. Chalermsook, P., Chuzhoy, J.: Maximum independent set of rectangles. In: SODA, pp. 892–901 (2009)

    Google Scholar 

  6. Chan, T.M.: A note on maximum independent sets in rectangle intersection graphs. Inf. Process. Lett. 89(1), 19–23 (2004)

    Article  MathSciNet  Google Scholar 

  7. Chan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. Discret. Comput. Geom. 48(2), 373–392 (2012)

    Article  MathSciNet  Google Scholar 

  8. Chekuri, C., Mydlarz, M., Shepherd, F.B.: Multicommodity demand flow in a tree and packing integer programs. ACM Trans. Algorithms 3(3), 27 (2007)

    Article  MathSciNet  Google Scholar 

  9. Elwalid, A.I., Mitra, D.: Effective bandwidth of general markovian traffic sources and admission control of high speed networks. IEEE/ACM Trans. Netw. 1(3), 329–343 (1993)

    Article  Google Scholar 

  10. Gupta, A., Kumar, A., Nagarajan, V., Shen, X.: Stochastic load balancing on unrelated machines. In: SODA, pp. 1274–1285. Society for Industrial and Applied Mathematics (2018)

    Google Scholar 

  11. Gupta, A., Kumar, A., Nagarajan, V., Shen, X.: Stochastic makespan minimization in structured set systems. arXiv (2020). https://arxiv.org/abs/2002.11153

  12. Hui, J.Y.: Resource allocation for broadband networks. IEEE J. Sel. Areas Commun. 6(3), 1598–1608 (1988)

    Article  Google Scholar 

  13. Kelly, F.P.: Notes on effective bandwidths. In: Stochastic Networks: Theory and Applications, pp. 141–168. Oxford University Press (1996)

    Google Scholar 

  14. Kleinberg, J., Rabani, Y., Tardos, E.: Allocating bandwidth for bursty connections. SIAM J. Comput. 30(1), 191–217 (2000)

    Article  MathSciNet  Google Scholar 

  15. Molinaro, M.: Stochastic \(\ell _p\) load balancing and moment problems via the l-function method. In: SODA, pp. 343–354 (2019)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Viswanath Nagarajan .

Editor information

Editors and Affiliations

A Analysis Outline

A Analysis Outline

We now show that the expected makespan for the solution produced by the rounding algorithm above is \(O(\alpha \lambda \rho )\), where \(\rho =\log \log m\) is the number of classes. In particular, we show that the expected makespan (taken over all resources) due to tasks of each class \(\ell \) is \(O(\alpha \lambda )\).

Using the terminating condition in line 5, we can show:

Lemma 6

For any class \(\ell \), \(0\le \ell \le \rho \), and resource \(i\in [m]\),

$$\sum _{j \in J_\ell \cap L_i}\beta _{r}(X_j')\cdot y_j\le 2b,\quad \text{ where } r=2^{2^\ell }. $$

Next, the sets \(D_\ell \) cannot become too large (as a function of \(\ell \)).

Lemma 7

For any \(\ell \), \(0 \le \ell \le \rho ,\) \(|D_\ell |\le k^2\), where \(k=2^{2^{\ell }}\). So \(|M_\ell |\le k^p\) for some constant p.

Proof

The claim is trivial for the last class \(\ell =\rho \) as \(k\ge m\) in this case. Now consider any class \(\ell <\rho \). For each \(i \in D_\ell \), we know \(\sum _{j \in \widetilde{L_i}}\beta _{k^2}(X_j')\cdot y_j> 2b\), where \(\widetilde{L_i}\) is as defined in line 7. Moreover, the subsets \(\{\widetilde{L_i} : i\in D_\ell \}\) are disjoint as the set J gets updated (in line 7) after adding each \(i\in D_\ell \). Suppose, for the sake of contradiction, that \(|D_\ell |>k^2\). Then let \(K\subseteq D_\ell \) be any set of size \(k^2\). By the LP constraint (8) on this subset K,

$$2b \cdot k^2 < \sum _{i\in K}\sum _{j \in \widetilde{L_i}}\beta _{k^2}(X_j')\cdot y_j \le \sum _{j\in L(K)} \beta _{k^2}(X_j')\cdot y_j \le b|K| = b\cdot k^2,$$

which is a contradiction. This proves the first part of the claim. Finally, the \(\lambda \)-safe property implies that \(|M_\ell |\) is polynomially bounded by \(|D_\ell |\).    \(\blacksquare \)

Using the definition of the DetCost instance and Lemma 6, we can show:

Lemma 8

The fractional solution \(\bar{y}\) is feasible for the LP relaxation (10) corresponding to the DetCost instance \({\mathcal {C}}\). Moreover, \(\theta \ge \max _j s_j\) and \(\psi \ge \max _j c_j\).

The above lemmas show that the algorithm is well-defined, so we can indeed use Theorem 4 to round \(\bar{y}\) into an integer solution. Recall that our final solution is \(N = N_H\cup N_L\). The next two lemmas follow from this rounding step.

Lemma 9

\(|N_H|+|N_L|\ge t\).

Lemma 10

For any class \(\ell \le \rho \) and resource \(i\in M_\ell \),

$$\begin{aligned} \sum _{j\in N_\ell \cap L_i} \beta _k(X'_j) \le 4\bar{\alpha } b, \quad \text{ where } k=2^{2^\ell }. \end{aligned}$$

We now focus on a particular class \(\ell \le \rho \) and show that the expected makespan due to tasks in \(N\cap J_\ell \) is small. Recall that \(k=2^{2^\ell }\). Let \(N_\ell := N\cap J_\ell \) and let \(\mathsf {Load}^{(\ell )}_i := \sum _{j\in N_\ell \cap L_i} X'_{j}\) be the load on any resource \(i\in [m]\) due to class-\(\ell \) tasks. We can now bound the makespan due to the truncated random variables.

Lemma 11

For any class \(\ell \le \rho \), \(\mathbb {E}\left[ \max _{i\in M_\ell } \mathsf {Load}^{(\ell )}_i\right] \le 4\bar{\alpha } b +O(1)\), and

$$\mathbb {E}\left[ \max _{i=1}^m \mathsf {Load}^{(\ell )}_i\right] \le 4\lambda \bar{\alpha } b +O(\lambda ) = O(\alpha \lambda ).$$

Proof

Consider a resource \(i \in M_\ell \). Lemmas 10 and 3 imply:

$$\Pr \left[ \mathsf {Load}^{(\ell )}_i> 4\bar{\alpha }b +\gamma \right] = \Pr \left[ \sum _{j\in \mathcal{N}_\ell \cap L_i} X'_{j} > 4\bar{\alpha }b +\gamma \right] \le k^{-\gamma },\quad \forall \gamma \ge 0.$$

By a union bound, we get

$$\Pr \left[ \max _{i\in M_\ell } \mathsf {Load}^{(\ell )}_i > 4\bar{\alpha }b +\gamma \right] \le |M_\ell |\cdot k^{-\gamma } \le k^{p-\gamma },\qquad \text{ for } \text{ all } \gamma \ge 0,$$

where p is the constant from Lemma 7. So the expectation

$$\begin{aligned} \mathbb {E}\left[ \max _{i \in M_\ell }\mathsf {Load}^{(\ell )}_i\right]&= \int _{\theta =0}^\infty \Pr \left[ \max _{i \in M_\ell } \mathsf {Load}^{(\ell )}_i > \theta \right] d\theta \\&\le \,\, 4\bar{\alpha } b + p + 2 +\int _{\gamma =p+2}^\infty k^{-\gamma +p}\, d\gamma \,\, \le \,\, 4\bar{\alpha } b + p + 2 + \frac{1}{k(p+1)}, \end{aligned}$$

which completes the proof of the first statement.

We now prove the second statement. Consider any class \(\ell <\rho \): by definition of \(J_\ell \), we know that \(J_\ell \subseteq L(D_\ell )\). So the \(\lambda \)-safe property implies that for every resource i there is a subset \(R_i\subseteq M_\ell \) of size at most \(\lambda \) such that \(L_i\cap J_\ell \subseteq L(R_i)\cap J_\ell \). Because \(N_\ell \subseteq J_\ell \), we also have \(L_i\cap N_\ell \subseteq L(R_i)\cap N_\ell \). Therefore,

$$ \mathsf {Load}^{(\ell )}_i \le \sum _{z \in R_i} \mathsf {Load}^{(\ell )}_z \le \lambda \,\max _{z\in M_\ell } \mathsf {Load}^{(\ell )}_z.$$

Taking expectation on both sides, we obtain the desired result.

Finally, for the last class \(\ell =\rho \), note that any task in \(J_\rho \) loads the resources in \(D_\rho = M_\rho \) only. Therefore, \(\max _{i=1}^m \mathsf {Load}^{(\ell )}_i = \max _{z\in M_\ell } \mathsf {Load}^{(\ell )}_z\). The desired result now follows by taking expectation on both sides.    \(\blacksquare \)

Using Lemma 11 for all \(\rho \) classes, it follows that the expected makespan due to all truncated r.v.s is \(O(\alpha \lambda \rho )\). For the exceptional random variables, we use:

Lemma 12

\(\mathbb {E}\left[ \sum _{j\in N} X''_j \right] = \sum _{j\in N} c_j \le 4 \bar{\alpha }\).

Adding the contributions from truncated and exceptional r.v.s, the overall expected makespan is \(O(\alpha \lambda \rho )\), which completes the proof of Theorem 2.

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gupta, A., Kumar, A., Nagarajan, V., Shen, X. (2020). Stochastic Makespan Minimization in Structured Set Systems (Extended Abstract). In: Bienstock, D., Zambelli, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2020. Lecture Notes in Computer Science(), vol 12125. Springer, Cham. https://doi.org/10.1007/978-3-030-45771-6_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-45771-6_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-45770-9

  • Online ISBN: 978-3-030-45771-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics