Planar 3-dimensional assignment problems with Monge-like cost arrays

Given an $n\times n\times p$ cost array $C$ we consider the problem $p$-P3AP which consists in finding $p$ pairwise disjoint permutations $\varphi_1,\varphi_2,\ldots,\varphi_p$ of $\{1,\ldots,n\}$ such that $\sum_{k=1}^{p}\sum_{i=1}^nc_{i\varphi_k(i)k}$ is minimized. For the case $p=n$ the planar 3-dimensional assignment problem P3AP results. Our main result concerns the $p$-P3AP on cost arrays $C$ that are layered Monge arrays. In a layered Monge array all $n\times n$ matrices that result from fixing the third index $k$ are Monge matrices. We prove that the $p$-P3AP and the P3AP remain NP-hard for layered Monge arrays. Furthermore, we show that in the layered Monge case there always exists an optimal solution of the $p$-3PAP which can be represented as matrix with bandwidth $\le 4p-3$. This structural result allows us to provide a dynamic programming algorithm that solves the $p$-P3AP in polynomial time on layered Monge arrays when $p$ is fixed.Comment: 16 pages, appendix will follow in v

Geometric versions of the 3-dimensional assignment problem under general norms

We discuss the computational complexity of special cases of the 3-dimensional (axial) assignment problem where the elements are points in a Cartesian space and where the cost coefficients are the perimeters of the corresponding triangles measured according to a certain norm. (All our results also carry over to the corresponding special cases of the 3-dimensional matching problem.) The minimization version is NP-hard for every norm, even if the underlying Cartesian space is 2-dimensional. The maximization version is polynomially solvable, if the dimension of the Cartesian space is fixed and if the considered norm has a polyhedral unit ball. If the dimension of the Cartesian space is part of the input, the maximization version is NP-hard for every $L_p$ norm; in particular the problem is NP-hard for the Manhattan norm $L_1$ and the Maximum norm $L_{\infty}$ which both have polyhedral unit balls.Comment: 21 pages, 9 figure

Polygons with inscribed circles and prescribed side lengths

AbstractWe prove NP-completeness of the following problem: For n given input numbers, decide whether there exists an n-sided, plane, convex polygon that has an inscribed circle and that has the input numbers as side lengths

The Northwest corner rule revisited

Under which conditions can one permute the rows and columns in an instance of the transportation problem, such that the Northwest corner rule solves the resulting permuted instance to optimality? And under which conditions will the Northwest corner rule find the optimal solution for every possible permutation of the instance? We show that the first question touches the area of NP-completeness, and we answer the second question by a simple characterization of such instances

Minimum-cost dynamic flows: The series-parallel case

A dynamic network consists of a directed graph with capacities, costs, and integral transit times on the arcs. In the minimum-cost dynamic flow problem (MCDFP), the goal is to compute, for a given dynamic network with source s, sink t, and two integers v and T, a feasible dynamic flow from s to t of value v, obeying the time bound T, and having minimum total cost. MCDFP contains as subproblems the minimum-cost maximum dynamic flow problem, where v is fixed to the maximum amount of flow that can be sent from s to t within time T and the minimum-cost quickest flow problem, where is T is fixed to the minimum time needed for sending v units of flow from s to t. We first prove that both subproblems are NP-hard even on two-terminal series-parallel graphs with unit capacities. As main result, we formulate a greedy algorithm for MCDFP and provide a full characterization via forbidden subgraphs of the class of graphs, for which this greedy algorithm always yields an optimum solution (for arbitrary choices of problem parameters). G is a subclass of the class of two-terminal series-parallel graphs. We show that the greedy algorithm solves MCDFP restricted to graphs in G in polynomial time