A new class of Heuristic Polynomial Time Algorithms to solve the Multidimensional Assignment Problem F. Perea AWS Department Thales Naval Nederland B.V. The Netherlands federico.perea@nl.thalesgroup.com H. W. de Waard AWS Department Thales Naval Nederland B.V. The Netherlands huub.dewaard@nl.thalesgroup.com Abstract - The multidimensional assignment prob- lem (MAP) is a combinatorial optimization problem arising in many applications, for instance in multi- target multi-sensor tracking problems. It is well- known that the MAP is NP-hard. The objective of a MAP is to match d-tuples of objects in such a way that the solution with the optimum total cost is found. In this paper a new class of approximation algorithms to solve the MAP is presented, named K-SGTS, and its effectiveness in multi-target multi- sensor tracking situations is shown. Its computa- tional complexity is proven to be polynomial. Exper- imental results on the accuracy and speed of K-SGTS are provided in the last section of the paper. Keywords: Tracking, data association, operations re- search. 1 Introduction One of the major difficulties in the application of multi- sensor multi-target tracking involves the problem of associating measurements, also called plots, with the appropriate track hypotheses, especially when there are missing reports, unknown targets and false reports (clutter). In this paper only scanning radars are con- sidered, used to track large numbers of targets in low to moderate clutter. It is assumed that each target in the coverage of the scanning radar can produce a max- imum of one measurement during a radar scan. To determine which measurements are likely candidates to originate from a certain track hypothesis, a correla- tion gate is positioned at the predicted position of the track hypothesis in the measurement space [3]. Define  z k to be a measurement with time stamp t k . Further- more, s(t k ) is the state estimator of a track hypothesis and ¯ P k is the residual error covariance matrix, both predicted to time t k . The measurement is said to cor- relate with the track hypothesis if it falls within the defined correlation gate. In figure 1, the square repre- sents a track t, and the dots p 1 ,p 2 are plots. The circle represents the gate that corresponds to track t. It can be seen that plot p 1 correlates with track t while plot p 2 does not. An example of a correlation gate is the ellipsoidal Figure 1: Correlation gate. gate, defined by [ z k − h( s(t k ))] T B -1 [ z k − h( s(t k ))] ≤ r 2 (1) where the difference  z k − h( s(t k )) is the residual or innovation vector, h(·) is the transformation function from Cartesian to polar coordinates, n is the dimension of the measurement vector and B = H k ¯ P k (H k ) T + R k is the residual covariance matrix. R k is the measure- ment error covariance matrix corresponding with the sensor which produced the measurement and H k is the Jacobian of h(·) taken at the position of the predicted state estimator in state space. Normally, it is assumed that the random variable  z k is Gaussian distributed, which means that r 2 is χ 2 -distributed, see [9]. Using the dimension of the measurement vector and accept- ing a certain risk to make an erroneous decision, the factor r 2 can be read from the χ 2 -distribution table. The data association problem for a number of data sets ≥ 3, coming from single or multiple sensors, is mathematically termed NP-hard ([5], [19]). Here a data set or frame from a sensor corresponds with the data collected during a full scan. More precisely, the computational cost to determine an optimal solution, which assigns measurements to track hypotheses, can grow at a rate much faster than polynomial as the num- ber of observations contained in the data sets increases. Since the pioneering work of Sittler [18], who coined the term data association, a number of algorithms has been developed over the past 35 years to solve the data association problem. An overview of different algorith- mic approaches is given by Pattipati [11]. An elegant and efficient approach to solve the data association problem is based on the application of as-