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Approximation Algorithms for Network Design in Non-Uniform Fault Models

Chandra Chekuri, Rhea Jain

TL;DR

Motivated by a conjecture that a constant factor approximation is feasible when the robustness parameters are fixed constants, this work considers two important special cases, namely the single pair case and the global connectivity case and gets constant factor approximations in several parameter ranges of interest.

Abstract

The Survivable Network Design problem (SNDP) is a well-studied problem, motivated by the design of networks that are robust to faults under the assumption that any subset of edges up to a specific number can fail. We consider non-uniform fault models where the subset of edges that fail can be specified in different ways. Our primary interest is in the flexible graph connectivity model, in which the edge set is partitioned into safe and unsafe edges. The goal is to design a network that has desired connectivity properties under the assumption that only unsafe edges up to a specific number can fail. We also discuss the bulk-robust model and the relative survivable network design model. While SNDP admits a 2-approximation, the approximability of problems in these more complex models is much less understood even in special cases. We make two contributions. Our first set of results are in the flexible graph connectivity model. Motivated by a conjecture that a constant factor approximation is feasible when the robustness parameters are fixed constants, we consider two important special cases, namely the single pair case, and the global connectivity case. For both these, we obtain constant factor approximations in several parameter ranges of interest. These are based on an augmentation framework and via decomposing the families of cuts that need to be covered into a small number of uncrossable families. Our second set of results are poly-logarithmic approximations for the bulk-robust model when the "width" of the given instance (the maximum number of edges that can fail in any particular scenario) is fixed. Via this, we derive corresponding approximations for the flexible graph connectivity model and the relative survivable network design model. The results are obtained via two algorithmic approaches and they have different tradeoffs in terms of the approximation ratio and generality.

Approximation Algorithms for Network Design in Non-Uniform Fault Models

TL;DR

Motivated by a conjecture that a constant factor approximation is feasible when the robustness parameters are fixed constants, this work considers two important special cases, namely the single pair case and the global connectivity case and gets constant factor approximations in several parameter ranges of interest.

Abstract

The Survivable Network Design problem (SNDP) is a well-studied problem, motivated by the design of networks that are robust to faults under the assumption that any subset of edges up to a specific number can fail. We consider non-uniform fault models where the subset of edges that fail can be specified in different ways. Our primary interest is in the flexible graph connectivity model, in which the edge set is partitioned into safe and unsafe edges. The goal is to design a network that has desired connectivity properties under the assumption that only unsafe edges up to a specific number can fail. We also discuss the bulk-robust model and the relative survivable network design model. While SNDP admits a 2-approximation, the approximability of problems in these more complex models is much less understood even in special cases. We make two contributions. Our first set of results are in the flexible graph connectivity model. Motivated by a conjecture that a constant factor approximation is feasible when the robustness parameters are fixed constants, we consider two important special cases, namely the single pair case, and the global connectivity case. For both these, we obtain constant factor approximations in several parameter ranges of interest. These are based on an augmentation framework and via decomposing the families of cuts that need to be covered into a small number of uncrossable families. Our second set of results are poly-logarithmic approximations for the bulk-robust model when the "width" of the given instance (the maximum number of edges that can fail in any particular scenario) is fixed. Via this, we derive corresponding approximations for the flexible graph connectivity model and the relative survivable network design model. The results are obtained via two algorithmic approaches and they have different tradeoffs in terms of the approximation ratio and generality.
Paper Structure (26 sections, 41 theorems, 11 equations, 5 figures, 2 algorithms)

This paper contains 26 sections, 41 theorems, 11 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions and Comparison to Existing Work
  3. Related work
  4. Overview of techniques
  5. Preliminaries
  6. LP Relaxations
  7. Augmentation Framework
  8. Uncrossability-Based Approximation Algorithms
  9. An $O(q)$-approximation for $(2, q)$-FGC
  10. Identifying Uncrossable Subfamilies
  11. Approximating $(p, q)$-FGC for $q \leq 4$
  12. An $O(1)$-Approximation for Flex-ST
  13. Approximating Non-Uniform SNDP Problems
  14. Cut-Cover Lemma and Tree Embeddings
  15. Algorithm for Augmentation Problem
  16. ...and 11 more sections

Key Result

Theorem 1.1

For any $q \ge 0$ there is a $(2q+2)$-approximation for $(2,q)$-FGC. For any $p \ge 1$ there is a $(2p+4)$-approximation for $(p,2)$-FGC, and a $(4p+4)$-approximation for $(p,3)$-FGC. Moreover, for all even$p \ge 2$ there is a $(6p+4)$-approximation for $(p,4)$-FGC.

Figures (5)

  • Figure 1: Violated cuts when augmenting from $(3,1)$-FGC to $(3,2)$-FGC are not uncrossable. Red dashed edges are unsafe, green solid edges are safe. Let $A = \{x_1, x_2\}, B = \{x_2, x_3\}$. $A$ and $B$ are violated, since they each are crossed by two safe and two unsafe edges, but $A \cup B = \{x_1, x_2, x_3\}$ and $B - A = \{x_3\}$ are not, since they each are crossed by three safe edges.
  • Figure 2: Violated cuts in augmentation problem from $(p, 3)$-FGC to $(p, 4)$-FGC are not uncrossable when $p$ is odd. The numbers on edges denote the number of parallel edges, red dashed edges are unsafe, green solid edges are safe. Let $A = \{x_1, x_2\}$, $B = \{x_2, x_3\}$; both are crossed by exactly $p-1$ safe edges and $4$ unsafe edges. However, $A \cup B$ and $B - A$ are each crossed by $p$ safe edges, and $A \cap B$ and $A - B$ are each crossed by $p+4$ total edges.
  • Figure 3: Violated cuts in augmentation problem from $(4,4)$-FGC to $(4,5)$-FGC are not uncrossable. The numbers on edges denote the number of parallel edges. Red dashed edges are unsafe, green solid edges are safe. Let $A = \{x_1, x_2\}$, let $B = \{x_2, x_3\}$; both are in $\mathcal{C}_3$. $A \cup B$ and $B - A$ each have at least $4$ safe edges, and $A \cap B$ and $A - B$ each have $9$ total edges, so none are violated.
  • Figure 4: Example where $\mathcal{C}_1$ is not uncrossable when augmenting from $(2,1)$ to $(2,2)$-Flex-ST. Red dashed edges are unsafe, while green solid edges are safe. Let $A = \{s, x_1\}, B = \{s, x_2\}$. $A$ and $B$ are both crossed by exactly one safe and two unsafe edges, but $A \cup B$ is crossed by two safe edges and $A \cap B$ is crossed by four unsafe edges. $A - B$ and $B - A$ are not $s$-$t$ cuts.
  • Figure 5: Integrality gap for $(1,k)$-Flex-ST

Theorems & Definitions (76)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Lemma 2.1
  • ...and 66 more