Approximation Algorithms for Discounted Graph Search with Norm Objectives

2026-07-13

AI Summary

This study addresses the graph traversal problem with a discount factor α and a p-norm objective, aiming to efficiently find a path that visits all vertices while minimizing the α-discounted latency p-norm. The authors propose the first unified framework that integrates discounted traversal costs with p-norm objectives, subsuming classical models such as path search, orienteering, minimum spanning trees, and the traveling salesman problem. They develop combinatorial optimization and approximation algorithms: for p = 1, they present a polynomial-time constant-factor approximation algorithm; for arbitrary p ≥ 1, they design a randomized constant-factor approximation algorithm and derandomize it to obtain a deterministic pseudo-polynomial-time algorithm, achieving theoretical guarantees across the full parameter range.

Abstract

We introduce a unified framework for classical search and routing problems, including pathwise search, expanding search, the minimum spanning tree problem, and the traveling salesperson problem. The framework is based on two parameters. The first is a discount factor $α\in [0,1]$: the first traversal of an edge incurs its full cost, whereas each subsequent traversal incurs only an $α$-fraction of this cost. For a path starting at a designated root vertex, the $α$-latency of a vertex is the discounted cost accumulated until the vertex is first visited. The second parameter is a norm parameter $p\geq 1$. The objective is to find a root-starting path that visits all vertices and minimizes the $p$-norm of the resulting vector of $α$-latencies. The model interpolates between several well-studied objectives. For $p=1$ and $α=1$, it recovers pathwise search; for $p=1$ and $α=0$, it recovers expanding search. As $p$ tends to infinity, the objective converges to a makespan-type criterion. At the endpoints $α=1$ and $α=0$, this limiting objective corresponds to TSP-type and MST-type behavior, respectively. For $p=1$, we give polynomial-time constant-factor approximation algorithms for all $α\in[0,1]$, matching the best known guarantees for expanding search at $α=0$ and pathwise search at $α=1$. For general $p\geq 1$, we obtain a randomized constant-factor approximation algorithm and a derandomized pseudo-polynomial-time algorithm with the same guarantee.

Problem

• discounted graph search
• approximation algorithms
• latency
• norm objectives
• combinatorial optimization

Innovation

• discounted graph search
• approximation algorithms
• norm objectives
• latency minimization
• unified framework