Approximation algorithms for multi-dimensional assignment problems with decomposable costs

The k-dimensional assignment problem with decomposable costs is formulated as follows. Given is a complete k-partite graph G = (X₀ ∪ ⋯ ∪ X_{k - 1}, E), with |X_i| = p for each i, and a nonnegative length function defined on the edges of G. A clique of G is a subset of vertices meeting each X_i in exactly one vertex. The cost of a clique is a function of the lengths of the edges induced by the clique. Four specific cost functions are considered in this paper; namely, the cost of a clique is either the sum of the lengths of the edges induced by the clique (sum costs), or the minimum length of a spanning star (star costs) or of a traveling salesman tour (tour costs) or of a spanning tree (tree costs) of the induced subgraph. The problem is to find a minimum-cost partition of the vertex set of G into cliques. We propose several simple heuristics for this problem, and we derive worst-case bounds on the ratio between the cost of the solutions produced by these heuristics and the cost of an optimal solution. The worst-case bounds are stated in terms of two parameters, viz. k and τ, where the parameter τ indicates how close the edge length function comes to satisfying the triangle inequality.