An FPTAS for Shortest-Longest Path Problem

TL;DR

The paper defines the Shortest-Longest Path (SLP) problem in multicriteria routing for multi-domain networks, where a path must satisfy $w_S(p)\le W_S$ and $w_L(p)\ge W_L$. It introduces an FPTAS based on dynamic programming and scaling-and-rounding, using an auxiliary graph and edge repetition detection to enforce resource constraints. It proves a $(1+\epsilon,1-\epsilon)$-approximation with time $O(\tau n \log(\tau n) + \tau m^2)$, where $\tau = m/\epsilon$, under feasibility assumptions. The work advances multicriteria optimization for QoS routing and multi-domain NFV/SFC resource allocation, while acknowledging limitations of approximate feasibility and potential solution-space reduction in the auxiliary construction.

Abstract

Motivated by multi-domain service function chain (SFC) orchestration, we define the shortest-longest path (SLP) problem, prove its hardness, and design an efficient fully polynomial time approximation scheme (FPTAS) using the dynamic programming (DP) and scaling and rounding (SR) techniques to compute an approximation solution with provable performance guarantee. The SLP problem and its solution algorithm have theoretical significance in multicriteria optimization and also have application potential in QoS routing and multi-domain network resource allocation scenarios.

Figures (4)

• Figure 1: Feasible region and approximation region.
• Figure 2: Motivation example. Only TA for domain B is shown.
• Figure 3: Illustrative example. $r=(s, t, 6, 6)$. $\epsilon=1$.
• Figure 4: The necessity of ERD. $r=(s, t, 3, 3)$, $\epsilon=2/3$.