Abstract

We study the Unsplittable Flow problem (mathsf {UFP}) on trees with a submodular objective function. The input to this problem is a tree with edge capacities and a collection of tasks, each characterized by a source node, a sink node, and a demand. A subset of the tasks is feasible if the tasks can simultaneously send their demands from the source to the sink without violating the edge capacities. The goal is to select a feasible subset of the tasks that maximizes a submodular objective function. Our main result is an O(klog n)-approximation algorithm for Submodular UFP on trees where k denotes the pathwidth of the given tree. Since every tree has pathwidth O(log n), we obtain an O(log ^2 n) approximation for arbitrary trees. This is the first non-trivial approximation guarantee for the problem, matching the best known approximation ratio for UFP on trees with a linear objective function. Our main technical contribution is a new geometric relaxation for UFP on trees that builds on the recent work of Bonsma et al. (2014, FOCS), Anagnostopoulos et al. (Amazing 2+epsilon approximation for unsplittable flow on a path, SIAM, pp 26–41, 2014) for UFP on paths with a linear objective. Our relaxation is very structured, so it can be combined with the contention resolution framework of Chekuri et al. (2009, STOC). Our approach is robust and extends to several related problems, such as UFP with bag constraints and the Storage Allocation Problem. Additionally, we study the special case of UFP on trees with a linear objective and upward instances where, for each task, the source node is a descendant of the sink node. Such instances generalize UFP on paths. We build on the work of Bansal et al. (A quasi-PTAS for unsplittable flow on line graphs, ACM, pp 721–729, 2006) for UFP on paths and obtain a QPTAS for upward instances when the input data is quasi-polynomially bounded. We complement this result by showing that, unlike the path setting, upward instances are mathsf {APX}-hard if the input data is arbitrary.

Highlights

  • Submodular functions are a rich class of functions with many applications both in theory and in practice

  • We study the Unsplittable Flow problem (UFP) on trees with a submodular objective function

  • Our main result is an O(k log n)-approximation algorithm for Submodular UFP on trees where k denotes the pathwidth of the given tree

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Summary

Introduction

Submodular functions are a rich class of functions with many applications both in theory and in practice. There has been a long line of work for UFP on paths with a linear objective that led to a constant factor approximation [5,7]; these approaches combine the standard LP relaxation with dynamic programming techniques. Chakaravarthy et al [10] study the generalization of the UFP with a linear objective, called bagUFP, where tasks are partitioned into bags and a feasible solution is allowed to select at most one task per bag They give O(log n) approximation algorithm for bagUFP on paths based on a primal-dual approach. We obtain an O(log n) approximation for the Submodular Storage Allocation Problem on trees This problem has the same input as UFP, with additional requirements that each selected task gets a private subinterval of width equal to the demand, contained in [0, ue) for each edge e used by the task. The best approximation is a (2 + ε) approximation [2] and a QPTAS [4,6] for UFP on paths, and an O(log n) approximation for UFP on trees [14]

Preliminaries
Geometric relaxation for submodular UFP on trees
A pseudo-polynomial sized relaxation
Partitioning into paths
Geometric viewpoint
LP relaxation
An LP rounding algorithm
A polynomial-sized relaxation
Submodular objective via the CR scheme framework
Applications
An algorithm
Hardness
Proof of Theorem 9
Full Text
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