New Structures and Algorithms for Length-Constrained Expander Decompositions

Expander decompositions form the basis of one of the most flexible paradigms for close-to-linear-time graph algorithms. Length-constrained expander de-compositions generalize this paradigm to better work for problems with lengths, distances and costs. Roughly, an (h,s) -length \phi -expander decompo...

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Veröffentlicht in:Proceedings / annual Symposium on Foundations of Computer Science S. 1634 - 1645
Hauptverfasser: Haeupler, Bernhard, Hershkowitz, D Ellis, Tan, Zihan
Format: Tagungsbericht
Sprache:Englisch
Veröffentlicht: IEEE 27.10.2024
Schlagworte:
Computer science
Costs
expander decompositions
graph algorithms
Graph theory
length-constrained expander decompositions
linear time algorithms
Routing
ISSN:2575-8454
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Zusammenfassung:Expander decompositions form the basis of one of the most flexible paradigms for close-to-linear-time graph algorithms. Length-constrained expander de-compositions generalize this paradigm to better work for problems with lengths, distances and costs. Roughly, an (h,s) -length \phi -expander decomposition is a small collection of length increases to a graph so that nodes within distance h can route flow over paths of length hs with congestion at most 1/\phi . In this work, we give a close-to-linear time algorithm for computing length-constrained expander decompositions in graphs with general lengths and capacities. Notably, and unlike previous works, our algorithm allows for one to trade off off between the size of the decomposition and the length of routing paths: for any \epsilon > 0 not too small, our algorithm computes in close-to-linear time an (h, s) -length \phi -expander decomposition of size m\cdot\phi\cdot n^{\epsilon} where s = exp(poly (1/\epsilon) ). The key foundations of our algorithm are: (1) a simple yet powerful structural theorem which states that the union of a sequence of sparse length-constrained cuts is itself sparse and (2) new algorithms for efficiently computing sparse length-constrained flows.
ISSN:2575-8454
DOI:10.1109/FOCS61266.2024.00102