Degree-constrained Subgraph Reconfiguration is in P

The degree-constrained subgraph problem asks for a subgraph of a given graph such that the degree of each vertex is within some specified bounds. We study the following reconfiguration variant of this problem: Given two solutions to a degree-constrained subgraph instance, can we transform one solution into the other by adding and removing individual edges, such that each intermediate subgraph satisfies the degree constraints and contains at least a certain minimum number of edges? This problem is a generalization of the matching reconfiguration problem, which is known to be in P. We show that even in the more general setting the reconfiguration problem is in P.Comment: Full version of the paper published at Mathematical Foundations of Computer Science (MFCS) 201

Perfectly Secure Communication, based on Graph-Topological Addressing in Unique-Neighborhood Networks

We consider network graphs $G=(V,E)$ in which adjacent nodes share common secrets. In this setting, certain techniques for perfect end-to-end security (in the sense of confidentiality, authenticity (implying integrity) and availability, i.e., CIA+) can be made applicable without end-to-end shared secrets and without computational intractability assumptions. To this end, we introduce and study the concept of a unique-neighborhood network, in which nodes are uniquely identifiable upon their graph-topological neighborhood. While the concept is motivated by authentication, it may enjoy wider applicability as being a technology-agnostic (yet topology aware) form of addressing nodes in a network

Fully Dynamic Matching in Bipartite Graphs

Maximum cardinality matching in bipartite graphs is an important and well-studied problem. The fully dynamic version, in which edges are inserted and deleted over time has also been the subject of much attention. Existing algorithms for dynamic matching (in general graphs) seem to fall into two groups: there are fast (mostly randomized) algorithms that do not achieve a better than 2-approximation, and there slow algorithms with \O(\sqrt{m}) update time that achieve a better-than-2 approximation. Thus the obvious question is whether we can design an algorithm -- deterministic or randomized -- that achieves a tradeoff between these two: a $o(\sqrt{m})$ approximation and a better-than-2 approximation simultaneously. We answer this question in the affirmative for bipartite graphs. Our main result is a fully dynamic algorithm that maintains a 3/2 + \eps approximation in worst-case update time O(m^{1/4}\eps^{/2.5}). We also give stronger results for graphs whose arboricity is at most \al, achieving a (1+ \eps) approximation in worst-case time O(\al (\al + \log n)) for constant \eps. When the arboricity is constant, this bound is $O(\log n)$ and when the arboricity is polylogarithmic the update time is also polylogarithmic. The most important technical developement is the use of an intermediate graph we call an edge degree constrained subgraph (EDCS). This graph places constraints on the sum of the degrees of the endpoints of each edge: upper bounds for matched edges and lower bounds for unmatched edges. The main technical content of our paper involves showing both how to maintain an EDCS dynamically and that and EDCS always contains a sufficiently large matching. We also make use of graph orientations to help bound the amount of work done during each update.Comment: Longer version of paper that appears in ICALP 201

Routing on the Visibility Graph

We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let $P$ be a set of $n$ points in the plane and let $S$ be a set of non-crossing line segments whose endpoints are in $P$. We present two deterministic 1-local $O(1)$-memory routing algorithms that are guaranteed to find a path of at most linear size between any pair of vertices of the \emph{visibility graph} of $P$ with respect to a set of constraints $S$ (i.e., the algorithms never look beyond the direct neighbours of the current location and store only a constant amount of additional information). Contrary to {\em all} existing deterministic local routing algorithms, our routing algorithms do not route on a plane subgraph of the visibility graph. Additionally, we provide lower bounds on the routing ratio of any deterministic local routing algorithm on the visibility graph.Comment: An extended abstract of this paper appeared in the proceedings of the 28th International Symposium on Algorithms and Computation (ISAAC 2017). Final version appeared in the Journal of Computational Geometr

Finding long cycles in graphs

We analyze the problem of discovering long cycles inside a graph. We propose and test two algorithms for this task. The first one is based on recent advances in statistical mechanics and relies on a message passing procedure. The second follows a more standard Monte Carlo Markov Chain strategy. Special attention is devoted to Hamiltonian cycles of (non-regular) random graphs of minimal connectivity equal to three