Approximation Algorithms for Multi-index Transportation Problems with Decomposable Costs

The axial multi-index transportation problem is defined as follows. Given are k sets A,, each set having n, elements, r = 1,. , k. The Cartesian product of the sets A, is denoted by A. To each element a E A a certain cost, c, E R, is associated. Further, a nonnegative demand e,; is associated to each set Ari = {u E A : u(r) = i}. The problem is to find nonnegative real numbers x0 such that each demand is satisfied (that is COCA, x,=eriforr=l ,..., k, i=l,..., n,)and such that total cost (that is xnEA c. . xa) is minimized. In this paper we deal with a special case of this problem where the costs ca are decomposable, that is, given a real-valued function f and a distance d;rxA’ between element i of A, and element j of A,, we assume that c, = f(d$‘,;,$,, . , dtrky ,y!$, ) for all a E A. We present two algorithms for this problem, and we analyze their worst-case behavior without requiring explicit knowledge of the cost-function f. Next, we use these results to derive explicit bounds in the case where .f is the diameter cost-function (that is cn = max,,, d$y,‘&,), and in the case where f is the Hamiltonian path cost-function (that is c, = min{zf,7’ ~J~$$‘$‘~,,, : D is a cyclic permutation of {l,...,k}).