Abstract
In the well-studied Unsplittable Flow on a Path problem (UFP), we are given a path graph with edge capacities. Furthermore, we are given a collection of n tasks, each one characterized by a subpath, a weight, and a demand. Our goal is to select a maximum weight subset of tasks so that the total demand of selected tasks using each edge is upper bounded by the corresponding capacity. Chakaravarthy et al. [ESA’14] studied a generalization of UFP, bagUFP, where tasks are partitioned into bags, and we can select at most one task per bag. Intuitively, bags model jobs that can be executed at different times (with different duration, weight, and demand). They gave a \(O(\log n)\) approximation for bagUFP. This is also the best known ratio in the case of uniform weights. In this paper we achieve the following main results:
\(\bullet \) We present an LP-based \(O(\log n/\log \log n)\) approximation for bagUFP. We remark that, prior to our work, the best known integrality gap (for a non-extended formulation) was \(O(\log n)\) even in the special case of UFP [Chekuri et al., APPROX’09].
\(\bullet \) We present an LP-based O(1) approximation for uniform-weight bagUFP. This also generalizes the integrality gap bound for uniform-weight UFP by Anagnostopoulos et al. [IPCO’13].
\(\bullet \) We consider a relevant special case of bagUFP, twUFP, where tasks in a bag model the possible ways in which we can schedule a job with a given processing time within a given time window. We present a QPTAS for twUFP with quasi-polynomial demands and under the Bounded Time-Window Assumption, i.e. assuming that the time window size of each job is within a constant factor from its processing time. This generalizes the QPTAS for UFP by Bansal et al. [STOC’06].
This work is partially supported by the ERC StG project NEWNET no. 279352.
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Notes
- 1.
Unless differently stated, \(\varepsilon \) denotes an arbitrarily small positive constant parameter. Where needed, we also assume that \(1/\varepsilon \) is integral and sufficiently large.
- 2.
By non-extended we mean that it contains only decision variables for tasks. In the same paper the authors present an extended formulation with O(1) integrality gap.
- 3.
The same gap is proved by Chekuri et al. [12]. The authors claim a \(O(\log ^2 n)\) gap, and then refine it to \(O(\log n)\) in an unpublished manuscript.
- 4.
We recall that a Quasi-Polynomial-Time Approximation Scheme (QPTAS) is an algorithm that, given a constant parameter \(\varepsilon >0\), computes a \(1+\varepsilon \) approximation in Quasi-Polynomial Time (QPT), i.e. in time \(2^{poly\log (s)}\) where s is the input size.
- 5.
We remark that Batra et al. [7] recently managed to remove this assumption on the demands for UFP. Their approach does not seem to be compatible with our randomized dissection technique (at least not trivially).
- 6.
Throughout this paper, by guessing we mean trying all the possibilities.
- 7.
In the guessing we of course guarantee that \(r^*=\sum _{f,a,b}r^{f,a,b}\).
- 8.
Here we exploit a property of twUFP not satisfied by bagUFP.
- 9.
Intuitively, the first term in the outer max corresponds to the case that the best solution does not use job k, and the second term to the weight obtained by including some task \(i\in \mathcal{B}_k\) in the solution.
- 10.
We call good the jobs of level \(\ell \) by definition.
- 11.
For our goals, it is convenient to consider two rectangles as overlapping iff they overlap on a positive value area. In particular, overlapping on rectangle boundaries is allowed.
- 12.
Observe that, by construction, \(APX_{max}\) is a maximal independent set w.r.t \(\mathcal{R}'\). This might not be the case w.r.t. \(\mathcal{R}\) since bag constraints might prevent some non-overlapping rectangle to be included in the maximal solution.
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Acknowledgements
The authors wish to thank Andreas Wiese for very helpful discussions about UFP and related problems.
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Grandoni, F., Ingala, S., Uniyal, S. (2015). Improved Approximation Algorithms for Unsplittable Flow on a Path with Time Windows. In: Sanità, L., Skutella, M. (eds) Approximation and Online Algorithms. WAOA 2015. Lecture Notes in Computer Science(), vol 9499. Springer, Cham. https://doi.org/10.1007/978-3-319-28684-6_2
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