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Fault-Tolerant Bounded Flow Preservers

Shivam Bansal, Keerti Choudhary, Harkirat Dhanoa, Harsh Wardhan

TL;DR

This work studies single-source fault-tolerant bounded-flow preservers in directed graphs. It delivers a polynomial-time construction of a $(\lambda,k)$-FT-BFP with at most $\lambda 2^k n$ edges and shows a matching lower bound of $\Omega(\min\{2^k\lambda n, n^2\})$ edges, establishing tight size behavior up to polylog factors. The authors prove hardness results: approximating optimal FT-BFP within $O(\log(\lambda^{-1} n))$ is NP-hard for suitable $k$, highlighting limits of algorithmic approaches. They also develop practical applications, including fault-tolerant all-pairs $\lambda$-reachability oracles, and extend the framework to graphs with non-unit capacities via a multigraph reduction. Overall, the paper bridges theory and application by deriving near-optimal sparse preservers and showing their utility for robust flow queries under edge failures.

Abstract

Given a directed graph $G = (V, E)$ with $n$ vertices, $m$ edges and a designated source vertex $s\in V$, we consider the question of finding a sparse subgraph $H$ of $G$ that preserves the flow from $s$ up to a given threshold $λ$ even after failure of $k$ edges. We refer to such subgraphs as $(λ,k)$-fault-tolerant bounded-flow-preserver ($(λ,k)$-FT-BFP). Formally, for any $F \subseteq E$ of at most $k$ edges and any $v\in V$, the $(s, v)$-max-flow in $H \setminus F$ is equal to $(s, v)$-max-flow in $G \setminus F$, if the latter is bounded by $λ$, and at least $λ$ otherwise. Our contributions are summarized as follows: 1. We provide a polynomial time algorithm that given any graph $G$ constructs a $(λ,k)$-FT-BFP of $G$ with at most $λ2^kn$ edges. 2. We also prove a matching lower bound of $Ω(λ2^kn)$ on the size of $(λ,k)$-FT-BFP. In particular, we show that for every $λ,k,n\geq 1$, there exists an $n$-vertex directed graph whose optimal $(λ,k)$-FT-BFP contains $Ω(\min\{2^kλn,n^2\})$ edges. 3. Furthermore, we show that the problem of computing approximate $(λ,k)$-FT-BFP is NP-hard for any approximation ratio that is better than $O(\log(λ^{-1} n))$.

Fault-Tolerant Bounded Flow Preservers

TL;DR

This work studies single-source fault-tolerant bounded-flow preservers in directed graphs. It delivers a polynomial-time construction of a -FT-BFP with at most edges and shows a matching lower bound of edges, establishing tight size behavior up to polylog factors. The authors prove hardness results: approximating optimal FT-BFP within is NP-hard for suitable , highlighting limits of algorithmic approaches. They also develop practical applications, including fault-tolerant all-pairs -reachability oracles, and extend the framework to graphs with non-unit capacities via a multigraph reduction. Overall, the paper bridges theory and application by deriving near-optimal sparse preservers and showing their utility for robust flow queries under edge failures.

Abstract

Given a directed graph with vertices, edges and a designated source vertex , we consider the question of finding a sparse subgraph of that preserves the flow from up to a given threshold even after failure of edges. We refer to such subgraphs as -fault-tolerant bounded-flow-preserver (-FT-BFP). Formally, for any of at most edges and any , the -max-flow in is equal to -max-flow in , if the latter is bounded by , and at least otherwise. Our contributions are summarized as follows: 1. We provide a polynomial time algorithm that given any graph constructs a -FT-BFP of with at most edges. 2. We also prove a matching lower bound of on the size of -FT-BFP. In particular, we show that for every , there exists an -vertex directed graph whose optimal -FT-BFP contains edges. 3. Furthermore, we show that the problem of computing approximate -FT-BFP is NP-hard for any approximation ratio that is better than .
Paper Structure (12 sections, 20 theorems, 3 equations, 2 figures)

This paper contains 12 sections, 20 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Existing Works
  3. Preliminaries
  4. Hardness of logarithmic approximation
  5. Upper bound of $\lambda 2^kn$ Edges
  6. Locality Property for Flow Preservers
  7. Construction of an Improved FTRS
  8. Computing sparse $(\lambda,k)$-FT-BFP
  9. Matching Lower Bound
  10. Applications
  11. Fault-tolerant All-Pairs $\lambda$-reachability oracle
  12. FT-BFP s for graphs with non-unit capacities

Key Result

Theorem 2

There exists an algorithm that for any directed graph ${\cal G}$ on $n$ vertices and $m$ edges, and any integers $\lambda,k\geqslant 1$, computes in $O(\lambda2^k mn)$ time a $(\lambda,k)$-FT-BFP for ${\cal G}$ with at most $\lambda2^k n$ edges.

Figures (2)

  • Figure 1: Depiction of a $(\lambda,k)$-FT-BFP instance obtained from a SET-COVER instance $\langle U,\mathfrak{F} \rangle$.
  • Figure 2: Depiction of lower bound on the size of $(\lambda,k)$-FT-BFP when $k=3$.

Theorems & Definitions (22)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 7
  • Lemma 9: 10.1007/978-3-642-32512-0_24, Theorem 2
  • Lemma 10
  • Lemma 11
  • Lemma 12
  • ...and 12 more