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Title: Online Disjoint Spanning Trees and Polymatroid Bases
Finding the maximum number of disjoint spanning trees in a given graph is a well-studied problem with several applications and connections. The Tutte-Nash-Williams theorem provides a min-max relation for this problem which also extends to disjoint bases in a matroid and leads to efficient algorithms [Schrijver, 2003]. Several other packing problems such as element disjoint Steiner trees, disjoint set covers, and disjoint dominating sets are NP-Hard but admit an O(log n)-approximation [Feige et al., 2002; Cheriyan and Salavatipour, 2007]. Călinescu, Chekuri, and Vondrák [G. Călinescu et al., 2009] viewed all these packing problems as packing bases of a polymatroid and provided a unified perspective. Motivated by applications in wireless networks, recent works have studied the problem of packing set covers in the online model [Pananjady et al., 2015; Emek et al., 2019; Bienkowski et al., 2025]. The online model poses new challenges for packing problems. In particular, it is not clear how to pack a maximum number of disjoint spanning trees in a graph when edges arrive online. Motivated by these applications and theoretical considerations, we formulate an online model for packing bases of a polymatroid, and describe a randomized algorithm with a polylogarithmic competitive ratio. Our algorithm is based on interesting connections to the notion of quotients of a polymatroid that has recently seen applications in polymatroid sparsification [Quanrud, 2024]. We generalize the previously known result for the online disjoint set cover problem [Emek et al., 2019] and also address several other packing problems in a unified fashion. For the special case of packing disjoint spanning trees in a graph (or a hypergraph) whose edges arrive online, we provide an alternative to our general algorithm that is simpler and faster while achieving the same poly-logarithmic competitive ratio.  more » « less
Award ID(s):
2402667
PAR ID:
10682520
Author(s) / Creator(s):
; ;
Editor(s):
Censor-Hillel, Keren; Grandoni, Fabrizio; Ouaknine, Joel; Puppis, Gabriele
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
334
ISSN:
1868-8969
ISBN:
978-3-95977-372-0
Page Range / eLocation ID:
44:1-44:20
Subject(s) / Keyword(s):
Disjoint Spanning Trees Base Packing Polymatroids Online Algorithms Theory of computation → Approximation algorithms analysis Theory of computation → Online algorithms
Format(s):
Medium: X Size: 20 pages; 999344 bytes Other: application/pdf
Size(s):
20 pages 999344 bytes
Right(s):
Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
Sponsoring Org:
National Science Foundation
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