# Complexity of URLLC Scheduling and Efficient Approximation Schemes

## TL;DR

This paper investigates the complexity of scheduling for URLLC, proving NP-hardness results, and proposes efficient heuristic algorithms for practical resource allocation in low latency, high reliability networks.

## Contribution

It establishes the NP-completeness of URLLC scheduling problems and introduces polynomial-time feasibility checks and heuristics for the continuous SNR model.

## Key findings

- Feasibility checking can be done in polynomial time.
- Scheduling for at least m users is NP-complete.
- Heuristic algorithms perform well in numerical evaluations.

## Abstract

In this paper we address the problem of joint admission control and resource scheduling for \emph{Ultra Reliable Low Latency Communications} (URLLC). We examine two models: (i) the \emph{continuous}, where all allocated resource blocks contribute to the success probability, and (ii) a \emph{binary}, where only resource blocks with strong signal are "active" for each user, and user $k$ needs $d_k$ active resource blocks for a successful URLLC transmission. In situations of congestion, we are interested in finding a subset of users that can be scheduled simultaneously. We show that finding a feasible schedule for at least $m$ URLLC users is NP-complete in the (easier) binary SNR model, hence also in the continuous. Maximizing the reward obtained from a feasible set of URLLC users is NP-hard and inapproximable to within ${(\log_2d)^2}/{d}$ of the optimal, where $d\doteq \max_kd_k$. On the other hand, we prove that checking a candidate set of users for feasibility and finding the corresponding schedule (when feasible) can be done in polynomial time, which we exploit to design an efficient heuristic algorithm for the general continuous SNR model. We complement our theoretical contributions with a numerical evaluation of our proposed schemes.