# On integer network synthesis problem with tree-metric cost

## TL;DR

This paper extends classical network synthesis results to integer capacities, showing that under certain conditions with tree-metric costs, the integer network synthesis problem can be solved efficiently.

## Contribution

It introduces an integer version of the network synthesis problem with tree-metric costs and proves polynomial-time solvability when requirements are at least 2.

## Key findings

- INSP can be solved in polynomial time under specified conditions.
- Tree-metric costs enable efficient solutions for integer network design.
- The result generalizes previous work on half-integral solutions to integer solutions.

## Abstract

Network synthesis problem (NSP) is the problem of designing a minimum-cost network (from the empty network) satisfying a given connectivity requirement. Hau, Hirai, and Tsuchimura showed that if the edge-cost is a tree metric, then a simple greedy-type algorithm solves NSP to obtain a half-integral optimal solution. This is a generalization of the classical result by Gomory and Hu for the uniform edge-cost. In this note, we present an integer version of Hau, Hirai, and Tsuchimura's result for integer network synthesis problem (INSP), where a required network must have an integer capacity. We prove that if each connectivity requirement is at least 2 and the edge-cost is a tree metrc, then INSP can be solved in polynomial time.