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Computational Complexity of the Recoverable Robust Shortest Path Problem with Discrete Recourse

Marcel Jackiewicz, Adam Kasperski, Paweł Zieliński

TL;DR

The paper investigates the computational complexity of the recoverable robust shortest path problem under discrete budgeted uncertainty, focusing on three recourse neighborhoods. It employs reductions from quantified-Boolean problems to prove higher-level hardness results: $\Sigma^p_3$-hardness for arc exclusion and arc symmetric-difference neighborhoods, and $\Pi^p_2$-hardness for the inner adversarial problem. By constructing decision-variants and leveraging standard reductions (including $2$-vertex-disjoint paths), it shows that both the adversarial and recoverable robust formulations are intractable at the higher levels of the polynomial hierarchy. The findings delineate the precise complexity, including hardness to approximate in some cases, thereby clarifying the computational limits of robust routing with discrete recourse.

Abstract

In this paper the recoverable robust shortest path problem is investigated. Discrete budgeted interval uncertainty representation is used to model uncertain second-stage arc costs. The known complexity results for this problem are strengthened. It is shown that it is Sigma_3^p-hard for the arc exclusion and the arc symmetric difference neighborhoods. Furthermore, it is also proven that the inner adversarial problem for these neighborhoods is Pi_2^p-hard.

Computational Complexity of the Recoverable Robust Shortest Path Problem with Discrete Recourse

TL;DR

The paper investigates the computational complexity of the recoverable robust shortest path problem under discrete budgeted uncertainty, focusing on three recourse neighborhoods. It employs reductions from quantified-Boolean problems to prove higher-level hardness results: -hardness for arc exclusion and arc symmetric-difference neighborhoods, and -hardness for the inner adversarial problem. By constructing decision-variants and leveraging standard reductions (including -vertex-disjoint paths), it shows that both the adversarial and recoverable robust formulations are intractable at the higher levels of the polynomial hierarchy. The findings delineate the precise complexity, including hardness to approximate in some cases, thereby clarifying the computational limits of robust routing with discrete recourse.

Abstract

In this paper the recoverable robust shortest path problem is investigated. Discrete budgeted interval uncertainty representation is used to model uncertain second-stage arc costs. The known complexity results for this problem are strengthened. It is shown that it is Sigma_3^p-hard for the arc exclusion and the arc symmetric difference neighborhoods. Furthermore, it is also proven that the inner adversarial problem for these neighborhoods is Pi_2^p-hard.
Paper Structure (4 sections, 11 theorems, 9 equations, 2 figures)

This paper contains 4 sections, 11 theorems, 9 equations, 2 figures.

Table of Contents

  1. Preliminaries
  2. New results
  3. Hardness of the adversarial problem
  4. Hardness of the recoverable robust problem

Key Result

Theorem 1

$\forall (\Gamma) \exists$CNF-SAT is $\Pi^p_2$-complete, even for $|\mathbb{Y}|=|\mathbb{Z}|$.

Figures (2)

  • Figure 1: A graph $G$ in an instance of Decision-Adv SP$(\Gamma^d)$ (the solid and dashed arcs) and an embedded instance of 2-Vertex-Disjoint Paths (the solid arcs) corresponding to the formula $\digamma=(\overline{y}_1\vee y_2 \vee \overline{z}_1) \wedge (\overline{y}_n\vee \overline{z}_1 \vee z_n) \wedge \cdots \wedge \mathcal{C}_m$.
  • Figure 2: The $n$$\pmb{x}$-variable subgraphs.

Theorems & Definitions (18)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • Theorem 3
  • proof
  • Corollary 2
  • Lemma 2
  • ...and 8 more