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Approximation Algorithms for Hop Constrained and Buy-at-Bulk Network Design via Hop Constrained Oblivious Routing

Chandra Chekuri, Rhea Jain

TL;DR

The paper advances two-cost network design by delivering polylogarithmic approximations for MC-BaB in the nonuniform setting and for hop-constrained network design problems, all relative to LP relaxations. It introduces LP-based reductions that link Buy-at-Bulk to Hop-Constrained Network Design and extends to Set Connectivity, leveraging hop-constrained congestion-based tree embeddings and oblivious routing. A novel fault-tolerant treatment yields polylog bicriteria approximations for single-source HCND with any fixed number of failures and, for the multicommodity case at k=2, a junction-structure approach achieving competitive randomized guarantees. The work unifies LP relaxations, sophisticated tree-embedding techniques, and augmentation/junction strategies to tackle previously intractable fault-tolerant and nonuniform variants, with implications for practical network design under uncertainty and failures.

Abstract

We consider two-cost network design models in which edges of the input graph have an associated cost and length. We build upon recent advances in hop-constrained oblivious routing to obtain two sets of results. We address multicommodity buy-at-bulk network design in the nonuniform setting. Existing poly-logarithmic approximations are based on the junction tree approach [CHKS09,KN11]. We obtain a new polylogarithmic approximation via a natural LP relaxation. This establishes an upper bound on its integrality gap and affirmatively answers an open question raised in [CHKS09]. The rounding is based on recent results in hop-constrained oblivious routing [GHZ21], and this technique yields a polylogarithmic approximation in more general settings such as set connectivity. Our algorithm for buy-at-bulk network design is based on an LP-based reduction to hop constrained network design for which we obtain LP-based bicriteria approximation algorithms. We also consider a fault-tolerant version of hop constrained network design where one wants to design a low-cost network to guarantee short paths between a given set of source-sink pairs even when k-1 edges can fail. This model has been considered in network design [GL17,GML18,AJL20] but no approximation algorithms were known. We obtain polylogarithmic bicriteria approximation algorithms for the single-source setting for any fixed k. We build upon the single-source algorithm and the junction-tree approach to obtain an approximation algorithm for the multicommodity setting when at most one edge can fail.

Approximation Algorithms for Hop Constrained and Buy-at-Bulk Network Design via Hop Constrained Oblivious Routing

TL;DR

The paper advances two-cost network design by delivering polylogarithmic approximations for MC-BaB in the nonuniform setting and for hop-constrained network design problems, all relative to LP relaxations. It introduces LP-based reductions that link Buy-at-Bulk to Hop-Constrained Network Design and extends to Set Connectivity, leveraging hop-constrained congestion-based tree embeddings and oblivious routing. A novel fault-tolerant treatment yields polylog bicriteria approximations for single-source HCND with any fixed number of failures and, for the multicommodity case at k=2, a junction-structure approach achieving competitive randomized guarantees. The work unifies LP relaxations, sophisticated tree-embedding techniques, and augmentation/junction strategies to tackle previously intractable fault-tolerant and nonuniform variants, with implications for practical network design under uncertainty and failures.

Abstract

We consider two-cost network design models in which edges of the input graph have an associated cost and length. We build upon recent advances in hop-constrained oblivious routing to obtain two sets of results. We address multicommodity buy-at-bulk network design in the nonuniform setting. Existing poly-logarithmic approximations are based on the junction tree approach [CHKS09,KN11]. We obtain a new polylogarithmic approximation via a natural LP relaxation. This establishes an upper bound on its integrality gap and affirmatively answers an open question raised in [CHKS09]. The rounding is based on recent results in hop-constrained oblivious routing [GHZ21], and this technique yields a polylogarithmic approximation in more general settings such as set connectivity. Our algorithm for buy-at-bulk network design is based on an LP-based reduction to hop constrained network design for which we obtain LP-based bicriteria approximation algorithms. We also consider a fault-tolerant version of hop constrained network design where one wants to design a low-cost network to guarantee short paths between a given set of source-sink pairs even when k-1 edges can fail. This model has been considered in network design [GL17,GML18,AJL20] but no approximation algorithms were known. We obtain polylogarithmic bicriteria approximation algorithms for the single-source setting for any fixed k. We build upon the single-source algorithm and the junction-tree approach to obtain an approximation algorithm for the multicommodity setting when at most one edge can fail.
Paper Structure (27 sections, 27 theorems, 7 equations, 1 figure, 3 algorithms)

This paper contains 27 sections, 27 theorems, 7 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Overview of Results
  3. Overview of Techniques
  4. Related Work
  5. Group Steiner Tree and Set Connectivity
  6. Buy-at-Bulk Network Design
  7. Hop-Constrained Network Design
  8. Preliminaries
  9. Linear Programming Relaxations
  10. Hop-Constrained Congestion-Based Embeddings
  11. Group Connectivity and Oblivious Rounding
  12. LP-Based Reduction from Buy-at-Bulk to Hop-Constraints
  13. Length and Demand Assumptions
  14. Approximation Algorithms for Hop-Constrained Network Design Problems via Oblivious Routing
  15. Algorithm Overview
  16. ...and 12 more sections

Key Result

Theorem 1.1

There is a randomized $O(\log D\log^3 n \log r)$-approximation for multicommodity Buy-at-Bulk with respect to $\textnormal{OPT}_{LP}$, where $D = \max_{i \in [r]} \delta(i)$ is the maximum demand. This extends to Buy-at-Bulk Set Connectivity and the approximation ratio is $O(\log D\log^7(nr))$.

Figures (1)

  • Figure 1: Each $p_i$ has $\textnormal{hop}(p_i) = h/2$. $s$ and $t$ are connected by a path of hop-length $h$ despite the failure of any one edge. However, one needs hop-length $\Omega(h^2)$ to obtain two edge disjoint $s$-$t$ paths.

Theorems & Definitions (85)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: hop_congestion21
  • Lemma 2.4
  • proof
  • ...and 75 more