ICALP 2015: Automata, Languages, and Programming pp 973-984

# A $$(2+\epsilon )$$-Approximation Algorithm for the Storage Allocation Problem

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)

## Abstract

Packing problems are a fundamental class of problems studied in combinatorial optimization. Three particularly important and well-studied questions in this domain are the Unsplittable Flow on a Path problem (UFP), the Maximum Weight Independent Set of Rectangles problem (MWISR), and the 2-dimensional geometric knapsack problem. In this paper, we study the Storage Allocation Problem (SAP) which is a natural combination of those three questions. Given is a path with edge capacities and a set of tasks that are specified by start and end vertices, demands, and profits. The goal is to select a subset of the tasks that can be drawn as non-overlapping rectangles underneath the capacity profile, the height of a rectangles corresponding to the demand of the respective task. This problem arises naturally in settings where a certain available bandwidth has to be allocated contiguously to selected requests.

While for 2D-knapsack and UFP there are polynomial time $$(2+\epsilon )$$-approximation algorithms known [Jansen and Zhang, SODA 2004] [Anagnostopoulos et al., SODA 2014] the best known approximation factor for SAP is $$9+\epsilon$$ [Bar-Yehuda, SPAA 2013]. In this paper, we level the understanding of SAP and the other two problems above by presenting a polynomial time $$(2+\epsilon )$$-approximation algorithm for SAP. A typically difficult special case of UFP and its variations arises if all input tasks are relatively large compared to the capacity of the smallest edge they are using. For that case, we even obtain a pseudopolynomial time exact algorithm for SAP.

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