Improved Approximation Algorithms for the Expanding Search Problem

TL;DR

This work studies the expanding search problem on graphs with edge lengths and vertex weights, where a search starts at a root and incrementally clears edges to reach weighted vertices, aiming to minimize the latency-weighted sum of reach times. The authors develop a sequence of improvements: a 2e-approximation for the unweighted case, a general polynomial-time $(2\mathrm{e}+\varepsilon)$-approximation for arbitrary weights (via a binary-weight reduction and a δ-bounded framework), and a PTAS for Euclidean graphs. Central techniques include concatenating $k$-MSTs, constructing an auxiliary path in a directed graph to bound latency, and reducing weighted instances to binary ones, with a careful δ-bounded and κ-segmented segmentation for Euclidean settings aided by portal-based dynamic programming inspired by Arora. They also establish a hardness result showing that no PTAS exists for general graphs unless $\mathsf{P}=\mathsf{NP}$, highlighting a qualitative gap between Euclidean and general instances. Overall, the paper significantly tightens approximation guarantees beyond the prior $8$-approximation and extends tractability to Euclidean graphs, with practical implications for time-sensitive search planning in disaster-relief and related domains.

Abstract

A searcher is tasked with exploring a graph with edge lengths and vertex weights, starting from a designated vertex. Initially, only the starting vertex is considered explored. At each step, the searcher adds an edge to the solution, connecting an unexplored vertex to an explored one. The time required to add an edge equals its length. The objective is to minimize the weighted sum of exploration times for all vertices. We demonstrate that this problem is hard to approximate and present algorithms with improved approximation guarantees. Specifically, we provide a $(2\mathrm{e} + \varepsilon)$-approximation for any $\varepsilon > 0$ for the general case. On instances where the vertex weights are binary, we achieve a $2\mathrm{e}$-approximation. Finally, we develop a polynomial-time approximation scheme (PTAS) for Euclidean graphs. Previously, only an $8$-approximation was known for all these cases.

Figures (1)

• Figure 1: (a) Instance $I$ for ST(1,2). Terminal vertices are shown in red. Blue edges have length $1$ and orange edges have length $2$. (b) Instance $I'$ for the expanding search problem constructed from $I$. Red vertices have weight ${\frac{1}{3}}$ and black vertices have weight $0$. Blue edges have length $1$, orange edges have length $2$, and green edges have length $a=2(|T|-1)$.

Theorems & Definitions (33)

• Theorem 3.1
• Lemma 3.2
• Proof 1
• Lemma 3.3
• Proof 2
• Lemma 3.4
• Proof 3
• Theorem 4.1
• Lemma 4.2
• Proof 4