Graph Minors and Parameterized Algorithm Design

Abstract. The Graph Minors Theory, developed by Robertson and Sey-mour, has been one of the most influential mathematical theories in pa-rameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct meta-algorithmic consequences, we present the algorithmic appli-cations of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex technique

Orienting Fully Dynamic Graphs with Worst-Case Time Bounds ⋆

Abstract. In edge orientations, the goal is usually to orient (direct) the edges of an undirected network (modeled by a graph) such that all outdegrees are bounded. When the network is fully dynamic, i.e., admits edge insertions and deletions, we wish to maintain such an orientation while keeping a tab on the update time. Low out-degree orientations turned out to be a surprisingly useful tool for managing networks. Brodal and Fagerberg (1999) initiated the study of the edge orientation problem in terms of the graph’s arboricity, which is very natural in this context. Their solution achieves a constant out-degree and a logarithmic amortized update time for all graphs with constant arboricity, which include all planar and excluded-minor graphs. It remained an open question – first proposed by Brodal and Fagerberg, later by Erickson and others – to obtain similar bounds with worst-case update time. We address this 15 year old question by providing a simple algorithm with worst-case bounds that nearly match the previous amortized bounds. Our algorithm is based on a new approach of maintaining a combinatorial invariant, and achieves a logarithmic out-degree with logarithmic worst-case update times. This result has applications to various dynamic network problems such as maintaining a maximal matching, where we obtain logarithmic worst-case update time compared to a similar amortized update time of Neiman and Solomon (2013).

Strong orientations of planar graphs with bounded stretch factor

We study the problem of orienting some edges of given planar graph such that the resulting subdigraph is strongly connected and spans all vertices of the graph. We are interested in orientations with minimum number of arcs and such that they produce a digraph with bounded stretch factor. Such orientations have applications into the problem of establishing strongly connected sensor network when sensors are equipped with directional antennae. We present three constructions for such orientations. Let G = (V, E) be a connected planar graph without cut edges and let Φ(G) be the degree of largest face in G. Our constructions are based on a face coloring, say with λ colors. First construction gives a strong orientation with at most (2 - 4λ-6/ λ(λ-1) |E| arcs and stretch factor at most Φ(G) - 1. The second construction gives a strong orientation with at most |E| arcs and stretch factor at most (Φ(G) - 1)⌈λ+1/2⌈. The third construction can be applied to planar graphs which are 3-edge connected. It uses a particular 6-face coloring and for any integer k ≤ 1 produces a strong orientation with at most (1 - k/10(k+1)) |E| arcs and stretch factor at most Φ2(G)(Φ(G) - 1)2k+4. These are worst-case upper bounds. In fact the stretch factors depend on the faces being traversed by a path

Bounded Arboricity to Determine the Local Structure of Sparse Graphs

The original publication is available at www.springerlink.comInternational audienceA known approach of detecting dense subgraphs communities in large sparse graphs involves first computing the probability~vectors for short random~walks on the graph, and then using these probability vectors to detect the communities. In this paper we focus on the first part of such an approach i.e. the computation of the probability vectors for the random walks, and propose a more efficient algorithm for computing these vectors in time complexity that is linear in the size of the output, in case the input graphs are restricted to a family of graphs of bounded arboricity. Such classes of graphs cover a large number of cases of interest, e.g all minor closed graph classes (planar graphs, graphs of bounded treewidth etc) and random graphs within the preferential attachment model. Our approach is extensible to other models of computation (PRAM, BSP or out-of-core computation) and also w.h.p. stays within the same complexity bounds for Erdös~Renyi graphs

Efficient Edge Splitting-Off Algorithms Maintaining All-Pairs Edge-Connectivities

In this paper we present new edge splitting-off results maintaining all-pairs edge-connectivities of a graph. We first give an alternate proof of Mader’s theorem, and use it to obtain a deterministic Õ(rmax 2 · n 2)-time complete edge splitting-off algorithm for unweighted graphs, where rmax denotes the maximum edge-connectivity requirement. This improves upon the best known algorithm by Gabow by a factor of ˜ Ω(n). We then prove a new structural property, and use it to further speedup the algorithm to obtain a randomized Õ(m + rmax 3 · n)-time algorithm. These edge splitting-off algorithms can be used directly to speedup various graph algorithms

Forward Analysis of Depth-Bounded Processes

Depth-bounded processes form the most expressive known fragment of the π-calculus for which interesting verification problems are still decidable. In this paper we develop an adequate domain of limits for the well-structured transition systems that are induced by depth-bounded processes. An immediate consequence of our result is that there exists a forward algorithm that decides the covering problem for this class. Unlike backward algorithms, the forward algorithm terminates even if the depth of the process is not known a priori. More importantly, our result suggests a whole spectrum of forward algorithms that enable the effective verification of a large class of mobile systems