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Recoverable robust shortest path problem under interval budgeted uncertainty representations

Marcel Jackiewicz, Adam Kasperski, Pawel Zielinski

TL;DR

This work studies the recoverable robust shortest path problem under interval and budgeted uncertainty. It establishes polynomial-time solvability for Rec Rob SP on acyclic multidigraphs with traditional interval uncertainty, across several neighborhoods, via reductions to CSP and ASP-based DP techniques, while strengthening hardness results for general digraphs. For budgeted uncertainty, it provides a compact MIP for the continuous budget and analyzes approximability, showing favorable ratios in acyclic graphs under certain degradation assumptions, with strong hardness results in general digraphs. The results advance both exact and approximate methods for robust shortest paths, offering practical tools (MIP formulations, CSP reductions, and ASP algorithms) and outlining key open questions for budgeted uncertainty in acyclic settings. The work has implications for robust routing, logistics, and network design under interval data and budget constraints.

Abstract

In this paper, the recoverable robust shortest path problem under interval uncertainty representations is discussed. This problem is known to be strongly NP-hard and also hard to approximate in general digraphs. In this paper, the class of acyclic digraphs is considered. It is shown that for the traditional interval uncertainty, the problem can be solved in polynomial time for all natural, known from the literature, neighborhoods. Efficient algorithms for various classes of acyclic digraphs are constructed. Some negative results for general digraphs are strengthened. Finally, some exact and approximate methods of solving the problem under budgeted interval uncertainty are proposed.

Recoverable robust shortest path problem under interval budgeted uncertainty representations

TL;DR

This work studies the recoverable robust shortest path problem under interval and budgeted uncertainty. It establishes polynomial-time solvability for Rec Rob SP on acyclic multidigraphs with traditional interval uncertainty, across several neighborhoods, via reductions to CSP and ASP-based DP techniques, while strengthening hardness results for general digraphs. For budgeted uncertainty, it provides a compact MIP for the continuous budget and analyzes approximability, showing favorable ratios in acyclic graphs under certain degradation assumptions, with strong hardness results in general digraphs. The results advance both exact and approximate methods for robust shortest paths, offering practical tools (MIP formulations, CSP reductions, and ASP algorithms) and outlining key open questions for budgeted uncertainty in acyclic settings. The work has implications for robust routing, logistics, and network design under interval data and budget constraints.

Abstract

In this paper, the recoverable robust shortest path problem under interval uncertainty representations is discussed. This problem is known to be strongly NP-hard and also hard to approximate in general digraphs. In this paper, the class of acyclic digraphs is considered. It is shown that for the traditional interval uncertainty, the problem can be solved in polynomial time for all natural, known from the literature, neighborhoods. Efficient algorithms for various classes of acyclic digraphs are constructed. Some negative results for general digraphs are strengthened. Finally, some exact and approximate methods of solving the problem under budgeted interval uncertainty are proposed.
Paper Structure (16 sections, 19 theorems, 28 equations, 4 figures, 2 tables, 5 algorithms)

This paper contains 16 sections, 19 theorems, 28 equations, 4 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Recoverable robust shortest path problem
  3. The Rec Rob SP problem under the interval uncertainty
  4. Recoverable shortest path problem in general multidigraphs
  5. Hardness results
  6. Compact mixed integer programming formulation
  7. Recoverable shortest path problem in acyclic multidigraphs
  8. Layered multidigraphs
  9. General acyclic multidigraphs
  10. Arc series-parallel multidigraph
  11. The Rec Rob SP problem under the budgeted uncertainty
  12. The Rec Rob SP problem under $\mathcal{U}(\Gamma^c)$
  13. The Rec Rob SP problem under $\mathcal{U}(\Gamma^d)$
  14. Approximation algorithms
  15. Conclusion
  16. ...and 1 more sections

Key Result

Lemma 1

There is a polynomial time reduction from $K$-V-DP in a digraph $G$ to Rec SP with $\Phi^{\mathrm{incl}}(X,k)$, $k=K$, in a digraph $G+H$ with costs $(C_e,\overline{c}_e)\in \{(0,0),(1,0),(0,1)\}$ for all arcs $e$ in $G+H$, where digraph $H$ is a simple path containing $2k-1$ arcs, $|A_H|=2k-1$. Mor

Figures (4)

  • Figure 1: A graph $G+H$ used in the reductions in the proofs of Lemmas \ref{['lcrecsp']} and \ref{['lredicsex']}. The bold arcs form the graph $H$.
  • Figure 2: Illustration of the proof of Lemma \ref{['lem1']} for $\Phi^{\mathrm{incl}}(X,k)$ and $\Phi^{\mathrm{excl}}(X,k)$. The dashed circles denote the layers. The pair of paths $X$ and $Y$ in $G$ corresponds to the path $\pi$ in $G'$.
  • Figure 3: Illustration of the proof of Lemma \ref{['lem2']}. The pair of paths $X$ and $Y$ for $\Phi^{\mathrm{incl}}(X,k)$ in $G$ corresponds to the path $\pi$ in $G'$.
  • Figure 4: Two sample graphs $G=(V,A)$ with the first and second-stage arc costs $C_e,\overline{c}_e$, $e\in A$, respectively, where $M$ is a big constant. The paths in bold are the optimal first-stage paths.

Theorems & Definitions (32)

  • Lemma 1
  • proof
  • Theorem 1
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • Theorem 4
  • proof
  • ...and 22 more