Directed Network Design with Orientation Constraints

We study directed network design problems with orientation constraints. An orientation constraint on a pair of nodes u and v states that a feasible solution may include at most one of the arcs (u,v) and (v,u). Such constraints arise naturally in many network design problems, since link or edge resources such as fibre can be used to support traffic in one of two possible directions only. Our first result is that the directed network design problem with orientation constraints can be solved in polynomial time in the case where the requirement function f is intersecting supermodular. (The case where there no orientation constraints follows from work of Frank [6].) The second main result of the paper is a 4-approximation algorithm for the minimum cost strongly connected subgraph problem with orientation constraints. Our algorithm shows that the problem of enforcing orientation constraints can be reduced to the minimum cost 2-edge connected subgraph problem on undirected graphs. Finally, we study the problem for general crossing supermodular functions and show the following bi-criteria approximation result. Let k denote the maximum requirement of any set under the given requirement function f. We give 2k-approximation algorithm to construct a solution that satisfies a slightly weaker requirement function, namely, f\u27(S) = max{f(S) - 1,0}

The complexity of two graph orientation problems

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 ElsevierWe consider two orientation problems in a graph, namely the minimization of the sum of all the shortest path lengths and the minimization of the diameter. Our main result is that for each positive integer k, there is a linear-time algorithm that decides for a planar graph Gwhether there is an orientation for which the diameter is at most k. We also extend this result from planar graphs to any minor-closed family F not containing all apex graphs. In contrast, it is known to be NP-complete to decide whether a graph has an orientation such that the sum of all the shortest path lengths is at most an integer specified in the input. We give a simpler proof of this result.This work is partially supported by EC Marie Curie programme NET-ACE (MEST-CT-2004-6724), and Heilbronn Institute for Mathematical Research, Bristol

Approximating Minimum Cost Connectivity Orientation and Augmentation

We investigate problems addressing combined connectivity augmentation and orientations settings. We give a polynomial-time 6-approximation algorithm for finding a minimum cost subgraph of an undirected graph $G$ that admits an orientation covering a nonnegative crossing $G$-supermodular demand function, as defined by Frank. An important example is $(k,\ell)$-edge-connectivity, a common generalization of global and rooted edge-connectivity. Our algorithm is based on a non-standard application of the iterative rounding method. We observe that the standard linear program with cut constraints is not amenable and use an alternative linear program with partition and co-partition constraints instead. The proof requires a new type of uncrossing technique on partitions and co-partitions. We also consider the problem setting when the cost of an edge can be different for the two possible orientations. The problem becomes substantially more difficult already for the simpler requirement of $k$-edge-connectivity. Khanna, Naor, and Shepherd showed that the integrality gap of the natural linear program is at most $4$ when $k=1$ and conjectured that it is constant for all fixed $k$. We disprove this conjecture by showing an $\Omega(|V|)$ integrality gap even when $k=2$

Constrained proportional integral control of dynamical distribution networks with state constraints

This paper studies a basic model of a dynamical distribution network, where the network topology is given by a directed graph with storage variables corresponding to the vertices and flow inputs corresponding to the edges. We aim at regulating the system to consensus, while the storage variables remain greater or equal than a given lower bound. The problem is solved by using a distributed PI controller structure with constraints which vary in time. It is shown how the constraints can be obtained by solving an optimization problem.Comment: CDC201

Local Graph Coloring and Index Coding

We present a novel upper bound for the optimal index coding rate. Our bound uses a graph theoretic quantity called the local chromatic number. We show how a good local coloring can be used to create a good index code. The local coloring is used as an alignment guide to assign index coding vectors from a general position MDS code. We further show that a natural LP relaxation yields an even stronger index code. Our bounds provably outperform the state of the art on index coding but at most by a constant factor.Comment: 14 Pages, 3 Figures; A conference version submitted to ISIT 2013; typos correcte