# A Lower Bound on Observability for Target Tracking with Range Sensors and its Application to Sensor Assignment

## TL;DR

This paper develops approximation algorithms for sensor assignment problems in multi-target tracking, providing bounds on observability and proposing measures that improve tracking performance.

## Contribution

It introduces a 1/3-approximation algorithm for assigning sensor pairs and a 1/2-approximation for team formation, along with new measures for observability in range-only sensing.

## Key findings

- The inverse condition number is not monotone or submodular.
- New monotone, submodular measures for observability are proposed.
- Algorithms achieve near-optimal sensor assignment with proven approximation bounds.

## Abstract

We study two sensor assignment problems for multi-target tracking with the goal of improving the observability of the underlying estimator. In the restricted version of the problem, we focus on assigning unique pairs of sensors to each target. We present a 1/3-approximation algorithm for this problem. We use the inverse of the condition number as the value function. If the target's motion model is not known, the inverse cannot be computed exactly. Instead, we present a lower bound for range-only sensing. In the general version, the sensors must form teams to track individual targets. We do not force any specific constraints on the size of each team, instead assume that the value function is monotonically increasing and is submodular. A greedy algorithm that yields a 1/2-approximation. However, we show that the inverse of the condition number is neither monotone nor submodular. Instead, we present other measures that are monotone and submodular. In addition to theoretical results, we evaluate our results empirically through simulations.