The minimum cut and minimum length linear arrangement problems usually occur in solving wiring problems and have a lot in common with job sequencing questions. Both problems are NP-complete for general graphs and in P for trees. We present here two parallel algorithms for the CREW PRAM. The rst solves the minimum length linear arrangement problem for trees and the second solves the minimum cut ...

This thesis explores two specific topics of discrete geometry, the multitriangulations and the polytopal realizations of products, whose connection is the problem of finding polytopal realizations of a given combinatorial structure. A k-triangulation is a maximal set of chords of the convex n-gon such that no k+1 of them mutually cross. We propose a combinatorial and geometric study of multitri...

A linear arrangement is a mapping π from the n vertices of graph G to distinct consecutive integers. Linear arrangements can be represented by drawing along horizontal line and edges as semicircles above said line. In this setting, length an edge defined absolute value difference between positions its two in arrangement, cost sum all lengths. Here we study variants Maximum Arrangement problem (...

The bipartite crossing number problem is studied, and a connection between this problem and the linear arrangement problem is established. It is shown that when the arboricity is close to the minimum degree and the graph is not too sparse, then the optimal number of crossings has the same order of magnitude as the optimal arrangement value times the arboricity. The application of the results to...

The minimum linear arrangement problem is widely used and studied in many practical and theoretical applications. In this paper we present a linear-time algorithm for the problem inspired by the algebraic multigrid approach which is based on weighted edge contraction rather than simple contraction. Our results turned out to be better than every known result in almost all cases, while the short ...

A linear arrangement (LA) is an assignment of distinct integers to the vertices of a graph. The cost of an LA is the sum of lengths of the edges of the graph, where the length of an edge is defined as the absolute value of the difference of the integers assigned to its ends. For many application one hopes to find an LA with small cost. However, it is a classical NP-complete problem to decide wh...

A new algorithm for the linear arrangement problem is described. The goal is to produce linear arrangements of software model diagrams such that the total length of all connections is reduced as much as possible. The algorithm uses the same general numbering strategy as existing algorithms for the highly related problems of bandwidth and profile reduction but is based on a new heuristic that ad...