Strong Structural Controllability of Systems on Colored Graphs

This paper deals with structural controllability of leader-follower networks. The system matrix defining the network dynamics is a pattern matrix in which a priori given entries are equal to zero, while the remaining entries take nonzero values. The network is called strongly structurally controllable if for all choices of real values for the nonzero entries in the pattern matrix, the system is controllable in the classical sense. In this paper we introduce a more general notion of strong structural controllability which deals with the situation that given nonzero entries in the system's pattern matrix are constrained to take identical nonzero values. The constraint of identical nonzero entries can be caused by symmetry considerations or physical constraints on the network. The aim of this paper is to establish graph theoretic conditions for this more general property of strong structural controllability.Comment: 13 page

The Edge Group Coloring Problem with Applications to Multicast Switching

This paper introduces a natural generalization of the classical edge coloring problem in graphs that provides a useful abstraction for two well-known problems in multicast switching. We show that the problem is NP-hard and evaluate the performance of several approximation algorithms, both analytically and experimentally. We find that for random $\chi$-colorable graphs, the number of colors used by the best algorithms falls within a small constant factor of $\chi$, where the constant factor is mainly a function of the ratio of the number of outputs to inputs. When this ratio is less than 10, the best algorithms produces solutions that use fewer than $2\chi$ colors. In addition, one of the algorithms studied finds high quality approximate solutions for any graph with high probability, where the probability of a low quality solution is a function only of the random choices made by the algorithm

NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs

In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in $K_{3,3}$-minor-free graphs by building on Kasteleyn's scheme for planar graphs, and stated that this "opens up the possibility of obtaining an NC algorithm for finding a perfect matching in $K_{3,3}$-free graphs." In this paper, we finally settle this 30-year-old open problem. Building on recent NC algorithms for planar and bounded-genus perfect matching by Anari and Vazirani and later by Sankowski, we obtain NC algorithms for perfect matching in any minor-closed graph family that forbids a one-crossing graph. This family includes several well-studied graph families including the $K_{3,3}$-minor-free graphs and $K_5$-minor-free graphs. Graphs in these families not only have unbounded genus, but can have genus as high as $O(n)$. Our method applies as well to several other problems related to perfect matching. In particular, we obtain NC algorithms for the following problems in any family of graphs (or networks) with a one-crossing forbidden minor: $\bullet$ Determining whether a given graph has a perfect matching and if so, finding one. $\bullet$ Finding a minimum weight perfect matching in the graph, assuming that the edge weights are polynomially bounded. $\bullet$ Finding a maximum $st$-flow in the network, with arbitrary capacities. The main new idea enabling our results is the definition and use of matching-mimicking networks, small replacement networks that behave the same, with respect to matching problems involving a fixed set of terminals, as the larger network they replace.Comment: 21 pages, 6 figure