## 1 Introduction  
A bibliographic study concerning the problems considered in this study showed that no ant colony algorithms have been published yet to address the problem of determining the maximum number of cliques with the maximum sum of their weights in d-partite incomplete weighted graphs; however, in this paper, such an ant algorithm is presented. To address the problem of determining all maximum cliques with the minimum sum of their weights in complete weighted graphs, solutions for real-world applications were presented in [1, 2, 3, 4]. Studies by the same author [1, 2, 3] investigated problems of small sizes, up to seven vertices. Meanwhile, the authors of [4, 5] demonstrated the weakness of the algorithms presented in [1, 2, 3] against problems of large sizes while presenting other formulations of the objective function. The problem of determining the maximum number of maximum cliques in unweighted incomplete graphs was addressed in [5]. In [4, 5], the effects of different forms of desire functions on the obtained results considering the look-ahead procedure was not studied. New ant algorithms for the above two problems are presented in this paper that allow us to obtain a smaller sum of weights of the maximum cliques and a greater number of maximum cliques. The problem of determining all maximum cliquesin complete and incomplete graphs can be presented as a multidimensional assignment problem [1; 2]. This problem is NP-hard [6]. The multidimensional assignment problem can be encountered in many industrial applications, for example, in vision systems [7; 8; 9; 10] or in control problems that occur in textile production systems [11; 12]. To solve the problem of searching for the maximum number of cliques with the minimum sum of their weights in d-partite graphs the General Maximum-Minimal Clique Problem (GMMCP) trace tool was recently proposed [7; 13]. The problem of assigning measurements to objects can be transformed to a 0-1 optimization problem, where the total distance/benefit of assigning targets to measurements is minimized/maximized [14; 15]. The multidimensional assignment problem is NP-hard [16; 17; 18]. Some heuristic algorithms have been developed to solve these problems [19; 20; 21]. In [19], data association is used, which incorporates both motion and appearance and solves the data association problem for one object at a time. Tracking multiple targets is achieved through the minimization of a continuous energy function. The global energy function depends on all vertices in all parts of the graph. Object identification is carried out based on a single frame by associating data with the corresponding object [20; 21]. In [15], recursive algorithms using the Hungarian method were studied. In [22], the problem was constructed as a network flow graph with cost edges. All edges are assigned weights with different costs depending on the degree of connection between the vertices, so the minimum-cost flow solution involves globally finding optimal paths for tracking multiple objects. The multi-target tracking problem in frame-by-frame detections was reduced to standard linear programming, and the problem was solved by the k-shortest path algorithm [23]. The data association problem is mapped to a cost flow network, and the optimal object detection is found by a min-cost flow algorithm in the network [24]. The authors of [25] presented the multi-target tracking problem as a flow network, and a solution was found using the min-cost flow algorithm. In [26], a dynamic version of the successive shortest path algorithm was introduced, which solves the multi-target tracking problem formulated as a min-cost flow problem. In [27], multi-object tracking was formulated as a min-cost flow optimization problem and solved by an algorithm that incorporates quadratic pairwise costs into the min-cost flow network. The problem can thus be expressed as an integer quadratic program. In [28], the multi-target tracking problem was formulated as an evidence graph.