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Online Disjoint Spanning Trees and Polymatroid Bases

Karthekeyan Chandrasekaran, Chandra Chekuri, Weihao Zhu

TL;DR

This work introduces an online model for packing disjoint bases of a polymatroid and achieves a polylogarithmic competitive ratio by leveraging quotients and strength decompositions. The main result is a randomized algorithm with $O(\\log^2 f(\\mathcal{N}))$ competitive ratio without prior knowledge of $k^*(f)$, bridging online set packing problems such as disjoint set covers, spanning trees, and connected subhypergraphs within a unified framework. A polynomial-time variant uses strength decomposition to approximate quotients, preserving the same performance, while a simpler online scheme for Disj-Conn-Spanning-Subhypergraphs (and Disj-Spanning-Trees) reaches the same $O(\\log^2 n)$ bound with faster, practical implementation. The findings offer a theoretically grounded, broadly applicable approach to online base packing and motivate future work on tighter bounds, constant-factor online algorithms for specific cases, and relaxing the $f(\\mathcal{N})$ knowledge assumption.

Abstract

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. 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. Călinescu, Chekuri, and Vondrák 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. 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. We generalize the previously known result for the online disjoint set cover problem 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.

Online Disjoint Spanning Trees and Polymatroid Bases

TL;DR

This work introduces an online model for packing disjoint bases of a polymatroid and achieves a polylogarithmic competitive ratio by leveraging quotients and strength decompositions. The main result is a randomized algorithm with competitive ratio without prior knowledge of , bridging online set packing problems such as disjoint set covers, spanning trees, and connected subhypergraphs within a unified framework. A polynomial-time variant uses strength decomposition to approximate quotients, preserving the same performance, while a simpler online scheme for Disj-Conn-Spanning-Subhypergraphs (and Disj-Spanning-Trees) reaches the same bound with faster, practical implementation. The findings offer a theoretically grounded, broadly applicable approach to online base packing and motivate future work on tighter bounds, constant-factor online algorithms for specific cases, and relaxing the knowledge assumption.

Abstract

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. Several other packing problems such as element disjoint Steiner trees, disjoint set covers, and disjoint dominating sets are NP-Hard but admit an -approximation. Călinescu, Chekuri, and Vondrák 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. 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. We generalize the previously known result for the online disjoint set cover problem 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.

Paper Structure

This paper contains 29 sections, 28 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Online Disj-Polymatroid-Bases Model.
  3. Applications.
  4. Prior work in the offline setting.
  5. Prior work in the online setting.
  6. Our Results
  7. Technical Background and Overview of Algorithms
  8. Preliminaries
  9. An $O(\log^2 r)$-Competitive Algorithm
  10. Algorithm Description
  11. Ordered Min-sized Quotients Property
  12. Structural Property
  13. Competitive Ratio Analysis
  14. Combined Algorithm
  15. Approximation of $q_t$ in Polynomial Time
  16. ...and 14 more sections

Key Result

Theorem 1

For Disj-Polymatroid-Bases, there exists a randomized online algorithm with expected competitive ratio $O(\log^2 {f({\mathcal{N}})})$. The runtime of the algorithm at each timestep $t\in [n]$ is $poly(t, \log{f({\mathcal{N}})})$.

Theorems & Definitions (52)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Lemma 1
  • Theorem 3
  • Lemma 2
  • Lemma 3
  • proof
  • Remark 1
  • Lemma 4
  • ...and 42 more