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Width Helps and Hinders Splitting Flows

Authors Manuel Cáceres , Massimo Cairo, Andreas Grigorjew , Shahbaz Khan , Brendan Mumey , Romeo Rizzi, Alexandru I. Tomescu , Lucia Williams

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Author Details

Manuel Cáceres
  • Department of Computer Science, University of Helsinki, Finland
Massimo Cairo
  • Department of Computer Science, University of Helsinki, Finland
Andreas Grigorjew
  • Department of Computer Science, University of Helsinki, Finland
Shahbaz Khan
  • Department of Computer Science and Engineering, Indian Institute of Technology Roorkee, India
Brendan Mumey
  • School of Computing, Montana State University, Bozeman, MT, USA
Romeo Rizzi
  • Department of Computer Science, University of Verona, Italy
Alexandru I. Tomescu
  • Department of Computer Science, University of Helsinki, Finland
Lucia Williams
  • School of Computing, Montana State University, Bozeman, MT, USA

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Manuel Cáceres, Massimo Cairo, Andreas Grigorjew, Shahbaz Khan, Brendan Mumey, Romeo Rizzi, Alexandru I. Tomescu, and Lucia Williams. Width Helps and Hinders Splitting Flows. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 31:1-31:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow X on directed graph G into weighted source-to-sink paths whose superposition equals X. We focus on a common formulation of the problem where the path weights must be non-negative integers and also on a new variant where these weights can be negative. We show that, for acyclic graphs, considering the width of the graph (the minimum number of s-t paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the non-negative version, we show that a popular heuristic is a O(log |X|)-approximation (|X| being the total flow of X) on graphs satisfying two properties related to the width (satisfied by e.g., series-parallel graphs), and strengthen its worst-case approximation ratio from Ω(√m) to Ω(m / log m) for sparse graphs, where m is the number of edges in the graph. For the negative version, we give a (⌈log ║X║⌉+1)-approximation (║X║ being the maximum absolute value of X on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary flows (║X║ ≤ 1) into at most width paths. We also disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [ALENEX 2018], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.

Subject Classification

ACM Subject Classification
  • Theory of computation → Network flows
  • Flow decomposition
  • approximation algorithms
  • graph width


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