# Polynomial-time Capacity Calculation and Scheduling for Half-Duplex 1-2-1 Networks

## TL;DR

This paper introduces polynomial-time algorithms to approximate the capacity and determine optimal scheduling in 1-2-1 half-duplex networks, addressing the complexity of capacity computation with innovative algorithmic solutions.

## Contribution

It presents the first polynomial-time algorithms for approximate capacity calculation and scheduling in 1-2-1 half-duplex networks, using advanced combinatorial optimization techniques.

## Key findings

- Algorithms efficiently approximate network capacity.
- Optimal schedules are computed in polynomial time.
- The approach leverages perfect matching polytopes and Gomory-Hu trees.

## Abstract

This paper studies the 1-2-1 half-duplex network model, where two half-duplex nodes can communicate only if they point `beams' at each other; otherwise, no signal can be exchanged or interference can be generated. The main result of this paper is the design of two polynomial-time algorithms that: (i) compute the approximate capacity of the 1-2-1 half-duplex network and, (ii) find the network schedule optimal for the approximate capacity. The paper starts by expressing the approximate capacity as a linear program with an exponential number of constraints. A core technical component consists of building a polynomial-time separation oracle for this linear program, by using algorithmic tools such as perfect matching polytopes and Gomory-Hu trees.