Walking Through Waypoints

We initiate the study of a fundamental combinatorial problem: Given a capacitated graph $G=(V,E)$, find a shortest walk ("route") from a source $s\in V$ to a destination $t\in V$ that includes all vertices specified by a set $\mathscr{W}\subseteq V$: the \emph{waypoints}. This waypoint routing problem finds immediate applications in the context of modern networked distributed systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial-time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable

Hamiltonian cycles in faulty random geometric networks

In this paper we analyze the Hamiltonian properties of faulty random networks. This consideration is of interest when considering wireless broadcast networks. A random geometric network is a graph whose vertices correspond to points uniformly and independently distributed in the unit square, and whose edges connect any pair of vertices if their distance is below some specified bound. A faulty random geometric network is a random geometric network whose vertices or edges fail at random. Algorithms to find Hamiltonian cycles in faulty random geometric networks are presented.Postprint (published version

Static Output Feedback: On Essential Feasible Information Patterns

In this paper, for linear time-invariant plants, where a collection of possible inputs and outputs are known a priori, we address the problem of determining the communication between outputs and inputs, i.e., information patterns, such that desired control objectives of the closed-loop system (for instance, stabilizability) through static output feedback may be ensured. We address this problem in the structural system theoretic context. To this end, given a specified structural pattern (locations of zeros/non-zeros) of the plant matrices, we introduce the concept of essential information patterns, i.e., communication patterns between outputs and inputs that satisfy the following conditions: (i) ensure arbitrary spectrum assignment of the closed-loop system, using static output feedback constrained to the information pattern, for almost all possible plant instances with the specified structural pattern; and (ii) any communication failure precludes the resulting information pattern from attaining the pole placement objective in (i). Subsequently, we study the problem of determining essential information patterns. First, we provide several necessary and sufficient conditions to verify whether a specified information pattern is essential or not. Further, we show that such conditions can be verified by resorting to algorithms with polynomial complexity (in the dimensions of the state, input and output). Although such verification can be performed efficiently, it is shown that the problem of determining essential information patterns is in general NP-hard. The main results of the paper are illustrated through examples

Arc-Disjoint Paths and Trees in 2-Regular Digraphs

An out-(in-)branching B_s^+ (B_s^-) rooted at s in a digraph D is a connected spanning subdigraph of D in which every vertex x != s has precisely one arc entering (leaving) it and s has no arcs entering (leaving) it. We settle the complexity of the following two problems: 1) Given a 2-regular digraph $D$, decide if it contains two arc-disjoint branchings B^+_u, B^-_v. 2) Given a 2-regular digraph D, decide if it contains an out-branching B^+_u such that D remains connected after removing the arcs of B^+_u. Both problems are NP-complete for general digraphs. We prove that the first problem remains NP-complete for 2-regular digraphs, whereas the second problem turns out to be polynomial when we do not prescribe the root in advance. We also prove that, for 2-regular digraphs, the latter problem is in fact equivalent to deciding if $D$ contains two arc-disjoint out-branchings. We generalize this result to k-regular digraphs where we want to find a number of pairwise arc-disjoint spanning trees and out-branchings such that there are k in total, again without prescribing any roots.Comment: 9 pages, 7 figure

Finding an induced subdivision of a digraph

We consider the following problem for oriented graphs and digraphs: Given an oriented graph (digraph) $G$, does it contain an induced subdivision of a prescribed digraph $D$? The complexity of this problem depends on $D$ and on whether $G$ must be an oriented graph or is allowed to contain 2-cycles. We give a number of examples of polynomial instances as well as several NP-completeness proofs