# Tight Approximation Algorithms for Bichromatic Graph Diameter and Related Problems

## TL;DR

This paper develops tight approximation algorithms for bichromatic graph diameter and related problems, providing new bounds and algorithms that are nearly optimal under the Strong Exponential Time Hypothesis.

## Contribution

It introduces the first nontrivial approximation algorithms for various bichromatic graph parameters and establishes tight bounds under SETH.

## Key findings

- Presented a 5/3-approximation algorithm for Bichromatic Diameter in weighted graphs.
- Established tight bounds under SETH for approximation factors and running times.
- Provided comprehensive analysis of approximability for multiple related graph parameters.

## Abstract

Some of the most fundamental and well-studied graph parameters are the Diameter (the largest shortest paths distance) and Radius (the smallest distance for which a "center" node can reach all other nodes). The natural and important $ST$-variant considers two subsets $S$ and $T$ of the vertex set and lets the $ST$-diameter be the maximum distance between a node in $S$ and a node in $T$, and the $ST$-radius be the minimum distance for a node of $S$ to reach all nodes of $T$. The bichromatic variant is the special case in which $S$ and $T$ partition the vertex set. In this paper we present a comprehensive study of the approximability of $ST$ and Bichromatic Diameter, Radius, and Eccentricities, and variants, in graphs with and without directions and weights. We give the first nontrivial approximation algorithms for most of these problems, including time/accuracy trade-off upper and lower bounds. We show that nearly all of our obtained bounds are tight under the Strong Exponential Time Hypothesis (SETH), or the related Hitting Set Hypothesis. For instance, for Bichromatic Diameter in undirected weighted graphs with $m$ edges, we present an $\tilde{O}(m^{3/2})$ time $5/3$-approximation algorithm, and show that under SETH, neither the running time, nor the approximation factor can be significantly improved while keeping the other unchanged.