# Discrete Convex Functions on Graphs and Their Algorithmic Applications

## TL;DR

This paper explores a theory of discrete convex functions on graphs, highlighting its algorithmic applications in combinatorial optimization, and extending discrete convex analysis to new graph structures.

## Contribution

It introduces a novel framework of discrete convex functions on graphs, expanding the scope of discrete convex analysis and its applications in optimization problems.

## Key findings

- Developed a theory of discrete convex functions on graphs
- Applied the theory to combinatorial optimization problems
- Provided algorithmic solutions for facility location and multiflow problems

## Abstract

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems.