The paper focuses on the minimum flow decomposition (MFD) problem, which is the task of finding a minimum-sized set of weighted paths whose weighted sum equals a given flow on a directed acyclic graph (DAG). MFD is strongly NP-hard, even when the flow values are restricted to {1, 2, 4}.
The authors introduce a new notion called "flow-width" that generalizes the concepts of graph width and parallel-width. They show that flow-width can be used to obtain improved approximation algorithms for the MFD problem on certain classes of graphs.
Specifically, the authors present the following key results:
They show that on width-stable DAGs, the MFD problem can be approximated within a factor of O(log ||f||), where ||f|| is the maximum flow weight. This improves upon the previous O(log Val(f)) approximation factor.
They provide a parameterized approximation algorithm for the MFD problem that uses both the width and parallel-width of the input graph as parameters. This algorithm achieves an approximation ratio of (par-width(G) / width(G)) * (⌊log ||f||⌋ + 1), where par-width(G) is the parallel-width of the graph.
They prove that the MFD problem remains strongly NP-hard even on DAGs of width 3, and NP-hard on DAGs of width 2. This shows that the ratio between parallel-width and width can grow arbitrarily large, and that the width of the graph alone is not a sufficient parameter to make the MFD problem easier.
They show that on constant parallel-width graphs with unary-coded flows, the MFD problem can be solved in quasi-polynomial time.
The paper provides a deeper understanding of the structural properties that affect the complexity and approximability of the MFD problem, which can guide the design of more efficient algorithms for practical applications.
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by Andreas Grig... at arxiv.org 10-01-2024
https://arxiv.org/pdf/2409.20278.pdfDeeper Inquiries