We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

In this chapter, you will see how the simplex method simplifies when it is applied to a class of optimization problems that are known as “network flow models.” You will also see that if a network flow model has “integer-valued data,” the simplex method finds an optimal solution that is integer-valued.

Flows in Networks

We give a deterministic algorithm for triangulating a simple polygon in linear time. The basic strategy is to build a coarse approximation of a triangulation in a bottom-up phase and then use the information computed along the way to refine the triangulation in a top-down phase. The main tools used are the polygon-cutting theorem, which provides us with a balancing scheme, and the planar separator theorem, whose role is essential in the discovery of new diagonals. Only elementary data structures are required by the algorithm. In particular, no dynamic search trees, of our algorithm.

Triangulating a simple polygon in linear time

We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.

/pdf/route-planning-in-transportation-networks-k0typz8i4k.pdf

Route Planning in Transportation Networks

A set of vertices $S$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. This paper studies the metric dimension of cartesian products $G\,\square\,H$. We prove that the metric dimension of $G\,\square\,G$ is tied in a strong sense to the minimum order of a so-called doubly resolving set in $G$. Using bounds on the order of doubly resolving sets, we establish bounds on $G\,\square\,H$ for many examples of $G$ and $H$. One of our main results is a family of graphs $G$ with bounded metric dimension for which the metric dimension of $G\,\square\,G$ is unbounded.

https://users.monash.edu.au/~davidwo/papers/MetricDimension-arXiv.pdf

On the Metric Dimension of Cartesian Products of Graphs

Due to its fast, dynamic, and distributed growth process, it is hard to obtain an accurate map of the Internet. In many cases, such a map-representing the structure of the Internet as a graph with nodes and links-is a prerequisite when investigating properties of the Internet. A common way to obtain such maps is to make certain local measurements at a small subset of the nodes, and then to combine these in order to "discover" (an approximation of) the actual graph. Each of these measurements is potentially quite costly. It is thus a natural objective to minimize the number of measurements which still discover the whole graph. We formalize this problem as a combinatorial optimization problem and consider it for two different models characterized by different types of measurements. We give several upper and lower bounds on the competitive ratio (for the online network discovery problem) and the approximation ratio (for the offline network verification problem) in both models. Furthermore, for one of the two models, we compare four simple greedy strategies in an experimental analysis

/pdf/network-discovery-and-verification-4ftpo68xso.pdf

Network Discovery and Verification

For a given graph with weighted vertices, the goal of the minimum-weight dominating set problem is to compute a vertex subset of smallest weight such that each vertex of the graph is contained in the subset or has a neighbor in the subset. A unit disk graph is a graph in which each vertex corresponds to a unit disk in the plane and two vertices are adjacent if and only if their disks have a non-empty intersection. We present the first constant-factor approximation algorithm for the minimum-weight dominating set problem in unit disk graphs, a problem motivated by applications in wireless ad-hoc networks. The algorithm is obtained in two steps: First, the problem is reduced to the problem of covering a set of points located in a small square using a minimum-weight set of unit disks. Then, a constant-factor approximation algorithm for the latter problem is obtained using enumeration and dynamic programming techniques exploiting the geometry of unit disks. Furthermore, we also show how to obtain a constant-factor approximation algorithm for the minimum-weight connected dominating set problem in unit disk graphs.

/pdf/constant-factor-approximation-for-minimum-weight-connected-35t45a5ze3.pdf

Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs

Consider the problem of discovering (or verifying) the edges and non-edges of a network, modelled as a connected undirected graph, using a minimum number of queries. A query at a vertex v discovers (or verifies) all edges and non-edges whose endpoints have different distance from v. In the network discovery problem, the edges and non-edges are initially unknown, and the algorithm must select the next query based only on the results of previous queries. We study the problem using competitive analysis and give a randomized on-line algorithm with competitive ratio O(sqrt(n*log n)) for graphs with n vertices. We also show that no deterministic algorithm can have competitive ratio better than 3. In the network verification problem, the graph is known in advance and the goal is to compute a minimum number of queries that verify all edges and non-edges. This problem has previously been studied as the problem of placing landmarks in graphs or determining the metric dimension of a graph. We show that there is no approximation algorithm for this problem with ratio o(log n) unless P=NP. Furthermore, we prove that the optimal number of queries for d-dimensional hypercubes is Theta(d/log d).

In this survey we present a selection of commonly used and new train classification methods from an algorithmic perspective.

Shunting for Dummies: An Introductory Algorithmic Survey

We present a (4+e)-approximation algorithm for the problem of computing a minimum-weight dominating set in unit disk graphs, where e is an arbitrarily small constant. The previous best known approximation ratio was 5+e. The main result of this paper is a 4-approximation algorithm for the problem restricted to constant-size areas. To obtain the (4+e)-approximation algorithm for the unrestricted problem, we then follow the general framework from previous constant-factor approximations for the problem: We consider the problem in constant-size areas, and combine the solutions obtained by our 4-approximation algorithm for the restricted case to get a feasible solution for the whole problem. Using the shifting technique (selecting a best solution from several considered partitionings of the problem into constant-size areas) we obtain the claimed (4+e)-approximation algorithm. By combining our algorithm with a known algorithm for node-weighted Steiner trees, we obtain a 7.875-approximation for the minimum-weight connected dominating set problem in unit disk graphs.

Matúš Mihalák

Papers

Network Discovery and Verification

Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs

Network Discovery and Verification

Shunting for Dummies: An Introductory Algorithmic Survey

A (4+ε)-approximation for the minimum-weight dominating set problem in unit disk graphs