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Improved Approximation Algorithms for Flexible Graph Connectivity and Capacitated Network Design

Ishan Bansal, Joseph Cheriyan, Sanjeev Khanna, Miles Simmons

TL;DR

The paper advances approximation algorithms for robust connectivity and capacitated design by strengthening LP relaxations with Knapsack-Cover Inequalities and employing Cover-Small-Cuts alongside Jain's iterative rounding. It achieves a $7$-approximation for $(1,q)$-FGC and an $O(\log k)$-approximation for Cap-$k$-ECSS when $k=o(n)$, while establishing $O(1)$-approximate reductions between $(2,q)$-FGC and 2-Cover-Small-Cuts. It also clarifies when $(p,q)$-FGC can be formulated as Cap-$k$-ECSS (iff $p=1$ or $q=1$) and reports a CIP-based $O(p\log n)$-approximation for $(p,q)$-FGC. Collectively, these results unify LP-based rounding, cut-covering techniques, and covering integer programs to yield improved, practically relevant guarantees for robust network design problems.

Abstract

We present improved approximation algorithms for some problems in the related areas of Flexible Graph Connectivity and Capacitated Network Design. In the $(p,q)$-Flexible Graph Connectivity problem, denoted $(p,q)$-FGC, the input is a graph $G(V, E)$ where $E$ is partitioned into safe and unsafe edges, and the goal is to find a minimum cost set of edges $F$ such that the subgraph $G'(V, F)$ remains $p$-edge connected upon removal of any $q$ unsafe edges from $F$. In the related Cap-$k$-ECSS problem, we are given a graph $G(V,E)$ whose edges have arbitrary integer capacities, and the goal is to find a minimum cost subset of edges $F$ such that the graph $G'(V,F)$ is $k$-edge connected. We obtain a $7$-approximation algorithm for the $(1,q)$-FGC problem that improves upon the previous best $(q+1)$-approximation. We also give an $O(\log{k})$-approximation algorithm for the Cap-$k$-ECSS problem, improving upon the previous best $O(\log{n})$-approximation whenever $k = o(n)$. Both these results are obtained by using natural LP relaxations strengthened with the knapsack-cover inequalities, and then during the rounding process utilizing an $O(1)$-approximation algorithm for the problem of covering small cuts. We also show that the the problem of covering small cuts inherently arises in another variant of $(p,q)$-FGC. Specifically, we show $O(1)$-approximate reductions between the $(2,q)$-FGC problem and the 2-Cover$\;$Small$\;$Cuts problem where each small cut needs to be covered twice.

Improved Approximation Algorithms for Flexible Graph Connectivity and Capacitated Network Design

TL;DR

The paper advances approximation algorithms for robust connectivity and capacitated design by strengthening LP relaxations with Knapsack-Cover Inequalities and employing Cover-Small-Cuts alongside Jain's iterative rounding. It achieves a -approximation for -FGC and an -approximation for Cap--ECSS when , while establishing -approximate reductions between -FGC and 2-Cover-Small-Cuts. It also clarifies when -FGC can be formulated as Cap--ECSS (iff or ) and reports a CIP-based -approximation for -FGC. Collectively, these results unify LP-based rounding, cut-covering techniques, and covering integer programs to yield improved, practically relevant guarantees for robust network design problems.

Abstract

We present improved approximation algorithms for some problems in the related areas of Flexible Graph Connectivity and Capacitated Network Design. In the -Flexible Graph Connectivity problem, denoted -FGC, the input is a graph where is partitioned into safe and unsafe edges, and the goal is to find a minimum cost set of edges such that the subgraph remains -edge connected upon removal of any unsafe edges from . In the related Cap--ECSS problem, we are given a graph whose edges have arbitrary integer capacities, and the goal is to find a minimum cost subset of edges such that the graph is -edge connected. We obtain a -approximation algorithm for the -FGC problem that improves upon the previous best -approximation. We also give an -approximation algorithm for the Cap--ECSS problem, improving upon the previous best -approximation whenever . Both these results are obtained by using natural LP relaxations strengthened with the knapsack-cover inequalities, and then during the rounding process utilizing an -approximation algorithm for the problem of covering small cuts. We also show that the the problem of covering small cuts inherently arises in another variant of -FGC. Specifically, we show -approximate reductions between the -FGC problem and the 2-CoverSmallCuts problem where each small cut needs to be covered twice.

Paper Structure

This paper contains 11 sections, 13 theorems, 30 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. A $7$-Approximation Algorithm for $(1,q)\text{-}\mathrm{FGC}$
  3. $O(1)$-Approximate Reductions between $(2,q)\text{-}\mathrm{FGC}$ and $\mathrm{2\text{-}Cover\,Small\,Cuts}$
  4. An $O(\log{k})$-Approximation Algorithm for Cap-$k$-ECSS
  5. Iteration 1
  6. Iteration $i$ Phase 1:
  7. Iteration $i$ Phase 2:
  8. Iteration $T$
  9. Solving the LP Relaxation:
  10. Formulating $(p,q)\text{-}\mathrm{FGC}$ as a Cap-$k$-ECSS Problem
  11. An $O(p\log{n})$-Approximation Algorithm for $(p,q)\text{-}\mathrm{FGC}$ via Covering Integer Programs

Key Result

Proposition 1

Let $G$ be a graph and let $x\in\{0,1\}^{E(G)}$ denote a subgraph of $G$. Let $f: 2^{V(G)} \rightarrow \mathbb Z$ be a weakly $\mathcal{F}$-supermodular function. Then $f(S) - |\delta_x(S)|$ is also a weakly $\mathcal{F}$-supermodular function.

Figures (1)

  • Figure :

Theorems & Definitions (25)

  • Remark
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 15 more