40 research outputs found
Perfect Elimination Orderings for Symmetric Matrices
We introduce a new class of structured symmetric matrices by extending the
notion of perfect elimination ordering from graphs to weighted graphs or
matrices. This offers a common framework capturing common vertex elimination
orderings of monotone families of chordal graphs, Robinsonian matrices and
ultrametrics. We give a structural characterization for matrices that admit
perfect elimination orderings in terms of forbidden substructures generalizing
chordless cycles in graphs.Comment: 16 pages, 3 figure
THE MINIMIZATION OF OPEN STACKS PROBLEM
ABSTRACT The Minimization of Open Stacks Problem is a pattern sequencing problem that is based on the premise that the different items obtained from cutting patterns are piled in stacks in the work area until all items of the same size have been cut. Due to space limitations, it is gainful to find a sequence for the patterns that minimizes the number of open stacks. We have developed an integer programming model based on interval graphs that searches for an appropriate edge completion of the given graph of the problem, while defining a suitable coloring of its vertices
Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphs
The class of Worpitzky-compatible subarrangements of a Weyl arrangement
together with an associated Eulerian polynomial was recently introduced by
Ashraf, Yoshinaga and the first author, which brings the characteristic and
Ehrhart quasi-polynomials into one formula. The subarrangements of the braid
arrangement, the Weyl arrangement of type , are known as the graphic
arrangements. We prove that the Worpitzky-compatible graphic arrangements are
characterized by cocomparability graphs. Our main result yields new formulas
for the chromatic and graphic Eulerian polynomials of cocomparability graphs.Comment: 11 pages, comments are welcome
On the heapability of finite partial orders
We investigate the partitioning of partial orders into a minimal number of
heapable subsets. We prove a characterization result reminiscent of the proof
of Dilworth's theorem, which yields as a byproduct a flow-based algorithm for
computing such a minimal decomposition. On the other hand, in the particular
case of sets and sequences of intervals we prove that this minimal
decomposition can be computed by a simple greedy-type algorithm. The paper ends
with a couple of open problems related to the analog of the Ulam-Hammersley
problem for decompositions of sets and sequences of random intervals into
heapable sets