5 research outputs found
Balancing problems in acyclic networks
A directed acyclic network with nonnegative integer arc lengths is called balanced if any two paths with common endpoints have equal lengths. In the buffer assignment problem such a network is given, and the goal is to balance it by increasing arc lengths by integer amounts (called buffers), so that the sum of the amounts added is minimal. This problem arises in VLSI design, and was recently shown to be polynomial for rooted networks. Here we give simple procedures which solve several generalizations of this problem in strongly polynomial time, using ideas from network flow theory. In particular, we solve a weighted version of the problem, extend the results to nonrooted networks, and allow upper bounds on buffers. We also give a strongly polynomial algorithm for solving the min-max buffer assignment problem, based on a strong proximity result between fractional and integer balanced solutions. Finally, we show that the problem of balancing a network while minimizing the number of arcs with positive buffers is NP-hard
Topological Additive Numbering of Directed Acyclic Graphs
We propose to study a problem that arises naturally from both Topological
Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as
Lucky Labeling). Let be a digraph and a labeling of its vertices with
positive integers; denote by the sum of labels over all neighbors of
each vertex . The labeling is called \emph{topological additive
numbering} if for each arc of the digraph. The problem
asks to find the minimum number for which has a topological additive
numbering with labels belonging to , denoted by
.
We characterize when a digraph has topological additive numberings, give a
lower bound for , and provide an integer programming formulation for
our problem, characterizing when its coefficient matrix is totally unimodular.
We also present some families for which can be computed in
polynomial time. Finally, we prove that this problem is \np-Hard even when its
input is restricted to planar bipartite digraphs
On cardinality constrained cycle and path polytopes
Given a directed graph D = (N, A) and a sequence of positive integers 1 <=
c_1 < c_2 < ... < c_m <= |N|, we consider those path and cycle polytopes that
are defined as the convex hulls of simple paths and cycles of D of cardinality
c_p for some p, respectively. We present integer characterizations of these
polytopes by facet defining linear inequalities for which the separation
problem can be solved in polynomial time. These inequalities can simply be
transformed into inequalities that characterize the integer points of the
undirected counterparts of cardinality constrained path and cycle polytopes.
Beyond we investigate some further inequalities, in particular inequalities
that are specific to odd/even paths and cycles.Comment: 24 page
Facets of the p-cycle polytope
The purpose of this study is to provide a polyhedral analysis of the p-cycle polytope, which is the convex hull of the incidence vectors of all the p-cycles (simple directed cycles consisting of p arcs) of the complete directed graph Kn. We first determine the dimension of the p-cycle, polytope, characterize the bases of its equality set, and prove two lifting results. We then describe several classes of valid inequalities for the case 2<p<n, together with necessary and sufficient conditions for these inequalities to induce facets of the p-cycle polytope. We also briefly discuss the complexity of the associated separation problems. Finally, we investigate the relationship between the p-cycle polytope and related polytopes, including the p-circuit polytope. Since the undirected versions of symmetric inequalities which induce facets of the p-cycle polytope are facet-inducing for the p-circuit polytope, we obtain new classes of facet-inducing inequalities for the p-circuit polytope