Approximating the shortest path problem with scenarios

TL;DR

An $\widetilde{O}(\sqrt{n})$ flow LP-based approximation algorithm for min-max shortest path in general graphs is constructed and it is shown that the approximation ratio obtained is close to an integrality gap of the corresponding flow LP relaxation.

Abstract

This paper discusses the shortest path problem in a general directed graph with $n$ nodes and $K$ cost scenarios (objectives). In order to choose a solution, the min-max criterion is applied. The min-max version of the problem is hard to approximate within $Ω(\log^{1-ε} K)$ for any $ε>0$ unless NP$\subseteq \text{DTIME}(n^{\text{polylog} \,n})$ even for arc series-parallel graphs and within $Ω(\log n/\log\log n)$ unless NP$\subseteq \text{ZPTIME}(n^{\log\log n})$ for acyclic graphs. The best approximation algorithm for the min-max shortest path problem in general graphs, known to date, has an approximation ratio of~$K$. In this paper, an $\widetilde{O}(\sqrt{n})$ flow LP-based approximation algorithm for min-max shortest path in general graphs is constructed. It is also shown that the approximation ratio obtained is close to an integrality gap of the corresponding flow LP relaxation.

Figures (3)

• Figure 1: (a) The cut-sets determined in the first round: $(S_1,\overline{S}_1)$, $(S_2,\overline{S}_2)$, $(S_3,\overline{S}_3)$ and $L_P=3$, and (b) the cut-sets determined in the second round: $(S_1,\overline{S}_1)$, $(S_2,\overline{S}_2)$ and $L_P=2$. The dashed arcs are selected and have $l_e=0$, and the solid arcs are not selected and have $l_e=1$. The numbers assigned to the nodes are the shortest path distances $d(v)$, $v\in V$.
• Figure 2: An instance of Min-Max SP with the integrality gap of at least 2.
• Figure 3: An instance of Min-Max SP with the integrality gap of at least 4

Theorems & Definitions (14)

• Theorem 1
• Proposition 1
• proof
• Theorem 2: Flow decomposition
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof