Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Faster Approximation Algorithms for Restricted Shortest Paths in Directed Graphs

Vikrant Ashvinkumar, Aaron Bernstein, Adam Karczmarz

TL;DR

Two randomized bicriteria $(1+\varepsilon, 1+\varepsilon)$-approximation algorithms are shown that give an affirmative answer to the restricted shortest paths problem: one suited to dense graphs, and the other that works better for sparse graphs.

Abstract

In the restricted shortest paths problem, we are given a graph $G$ whose edges are assigned two non-negative weights: lengths and delays, a source $s$, and a delay threshold $D$. The goal is to find, for each target $t$, the length of the shortest $(s,t)$-path whose total delay is at most $D$. While this problem is known to be NP-hard [Garey and Johnson, 1979] $(1+\varepsilon)$-approximate algorithms running in $\tilde{O}(mn)$ time [Goel et al., INFOCOM'01; Lorenz and Raz, Oper. Res. Lett.'01] given more than twenty years ago have remained the state-of-the-art for directed graphs. An open problem posed by [Bernstein, SODA'12] -- who gave a randomized $m\cdot n^{o(1)}$ time bicriteria $(1+\varepsilon, 1+\varepsilon)$-approximation algorithm for undirected graphs -- asks if there is similarly an $o(mn)$ time approximation scheme for directed graphs. We show two randomized bicriteria $(1+\varepsilon, 1+\varepsilon)$-approximation algorithms that give an affirmative answer to the problem: one suited to dense graphs, and the other that works better for sparse graphs. On directed graphs with a quasi-polynomial weights aspect ratio, our algorithms run in time $\tilde{O}(n^2)$ and $\tilde{O}(mn^{3/5})$ or better, respectively. More specifically, the algorithm for sparse digraphs runs in time $\tilde{O}(mn^{(3 - α)/5})$ for graphs with $n^{1 + α}$ edges for any real $α\in [0,1/2]$.

Faster Approximation Algorithms for Restricted Shortest Paths in Directed Graphs

TL;DR

Two randomized bicriteria -approximation algorithms are shown that give an affirmative answer to the restricted shortest paths problem: one suited to dense graphs, and the other that works better for sparse graphs.

Abstract

In the restricted shortest paths problem, we are given a graph whose edges are assigned two non-negative weights: lengths and delays, a source , and a delay threshold . The goal is to find, for each target , the length of the shortest -path whose total delay is at most . While this problem is known to be NP-hard [Garey and Johnson, 1979] -approximate algorithms running in time [Goel et al., INFOCOM'01; Lorenz and Raz, Oper. Res. Lett.'01] given more than twenty years ago have remained the state-of-the-art for directed graphs. An open problem posed by [Bernstein, SODA'12] -- who gave a randomized time bicriteria -approximation algorithm for undirected graphs -- asks if there is similarly an time approximation scheme for directed graphs. We show two randomized bicriteria -approximation algorithms that give an affirmative answer to the problem: one suited to dense graphs, and the other that works better for sparse graphs. On directed graphs with a quasi-polynomial weights aspect ratio, our algorithms run in time and or better, respectively. More specifically, the algorithm for sparse digraphs runs in time for graphs with edges for any real .

Paper Structure

This paper contains 75 sections, 22 theorems, 69 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Complexity of exact $\mathsf{RSP}$.
  3. Approximate $\mathsf{RSP}$.
  4. Our Results
  5. Techniques and Organization
  6. Techniques.
  7. Related Work.
  8. Organization.
  9. Preliminaries
  10. Numbers and Sets.
  11. Graphs.
  12. Pareto Frontier.
  13. Directed Low-Diameter Decomposition.
  14. A Frequency-Driven Dynamic Program
  15. Examples of Useful Frequencies
  16. ...and 60 more sections

Key Result

Theorem 1

There is a Monte Carlo randomized $(1 + \varepsilon, 1 + \varepsilon)$-approximate algorithm for $\mathsf{Directed}\textrm{-}\mathsf{RSP}$ that runs in $\widetilde{O}(n^2\log(W)/\varepsilon^3)$ time, where $W$ is the aspect ratioFormally, the aspect ratio $W$ is the quantity $(\max_{e\in E_{\ell+}}\

Figures (5)

  • Figure 1: Length-delay plots of $(s,t)$-paths for a fixed $s$ and $t$. Green line indicates the Pareto Frontier. Left: $p_1$ Pareto dominates $p_2$ since $p_2$ lies to the northeast of $p_1$, and (weakly) Pareto dominates $p_3$ since $p_3$ lies to the east of $p_1$. Neither $p_2$ nor $p_3$ Pareto dominate each other as neither are in the northeast region of the other. Center: The gold region and blue points indicate valid answers a $(1, 1+\varepsilon)$-approximation can give. Right: The gold region and blue points indicate valid answers a $(1+\varepsilon, 1+\varepsilon)$-approximation can give.
  • Figure 2: Diagram of a $2$ level LDD Hierarchy, with the root in level $0$ (for convenience), and leaves in level $2$. Each node in levels $i > 0$ are SCCs contained in their parent. On the right, the same LDD Hierarchy, flattened. Star-edges have not been depicted.
  • Figure 3: Blocks respecting the structure of the LDD Hierarchy. Unfilled circles represent non-small SCCs and gray blobs represent small SCCs. Rectangular boxes represent blocks. We can then talk about the ancestors of blocks or, for a fixed level, the topological ordering of blocks and non-small SCCs.
  • Figure 4: A (not drawn-to-scale) depiction of finely chopped level $i$ blocks. Green blobs represent level $i - 1$ SCCs, orange blobs represent level $i$ SCCs that are not small, and black dots represent small level $i$ SCCs. Blocks drawn with thick rectangular boxes.
  • Figure 5: A depiction of the level $i$ block construction process. Green blobs represent level $i - 1$ SCCs, orange blobs represent level $i$ SCCs that are not small, and black dots represent small level $i$ SCCs. Two initial blocks from step (2) are drawn with thick rectangular boxes. Step (3) splits the initial blocks along large dashed lines. Step (4) splits the blocks from step (3) along smaller dotted lines. The resulting blocks at the end of step (4) are the same as in \ref{['fig:chopped']}.

Theorems & Definitions (68)

  • Definition 1.1
  • Theorem 1: $\mathsf{Directed}\textrm{-}\mathsf{RSP}$ on Dense Graphs; simplified version of \ref{['thm:dense']} in \ref{['sec:dense']}
  • Theorem 2: $\mathsf{Directed}\textrm{-}\mathsf{RSP}$ on Sparse Graphs; simplified version of \ref{['thm:sparse']} in \ref{['sec:sparse']}
  • Theorem 3: All-Pairs $\mathsf{Directed}\textrm{-}\mathsf{RSP}$; \ref{['thm:all-pairs']} in \ref{['sec:all-pairs']} simplified
  • Proposition 2.1: Theorems 13, 14 in tarjan1972depth and tarjan1976edge
  • Definition 2.2: (Strong) Directed Low-Diameter Decomposition
  • Proposition 2.3: Theorem 5 in bringmann2023negative
  • Theorem 4
  • proof : Proof of Recurrence
  • proof
  • ...and 58 more