Renato F. Werneck

A Hybrid GRASP with Perturbations for the Steiner Problem in Graphs

We propose and describe a hybrid GRASP with weight perturbations and adaptive path-relinking heuristic (HGP + PR) for the Steiner problem in graphs. In this multi-start approach, the greedy randomized construction phase of a GRASP is replaced by the use of several construction heuristics with a weight perturbation strategy that combines intensification and diversification elements, as in a strategic oscillation approach. The improvement phase circularly explores two different local search strategies. The first uses anode-based neighborhood for local search, while the second uses a key-path-based neighborhood. An adaptive path-relinking technique is applied to a set of elite solutions as apost-optimization strategy. Computational results on a broad set of benchmark problems illustrate the effectiveness and the robustness of our heuristic, which is very competitive when compared to other approximate algorithms.

Customizable route planning

We present an algorithm to compute shortest paths on continental road networks with arbitrary metrics (cost functions). The approach supports turn costs, enables real-time queries, and can incorporate a new metric in a few seconds--fast enough to support real-time traffic updates and personalized optimization functions. The amount of metric-specific data is a small fraction of the graph itself, which allows us to maintain several metrics in memory simultaneously.

A hub-based labeling algorithm for shortest paths in road networks

Abraham et al. [SODA 2010] have recently presented a theoretical analysis of several practical point-to-point shortest path algorithms based on modeling road networks as graphs with low highway dimension. They also analyze a labeling algorithm. While no practical implementation of this algorithm existed, it has the best time bounds. This paper describes an implementation of the labeling algorithm that is faster than any existing method on continental road networks.

Hierarchical hub labelings for shortest paths

We study hierarchical hub labelings for computing shortest paths. Our new theoretical insights into the structure of hierarchical labels lead to faster preprocessing algorithms, making the labeling approach practical for a wider class of graphs. We also find smaller labels for road networks, improving the query speed.

Sketch-based Influence Maximization and Computation: Scaling up with Guarantees

Propagation of contagion through networks is a fundamental process. It is used to model the spread of information, influence, or a viral infection. Diffusion patterns can be specified by a probabilistic model, such as Independent Cascade (IC), or captured by a set of representative traces. Basic computational problems in the study of diffusion are influence queries (determining the potency of a specified seed set of nodes) and Influence Maximization (identifying the most influential seed set of a given size). Answering each influence query involves many edge traversals, and does not scale when there are many queries on very large graphs. The gold standard for Influence Maximization is the greedy algorithm, which iteratively adds to the seed set a node maximizing the marginal gain in influence. Greedy has a guaranteed approximation ratio of at least (1-1/e) and actually produces a sequence of nodes, with each prefix having approximation guarantee with respect to the same-size optimum. Since Greedy does not scale well beyond a few million edges, for larger inputs one must currently use either heuristics or alternative algorithms designed for a pre-specified small seed set size. We develop a novel sketch-based design for influence computation. Our greedy Sketch-based Influence Maximization (SKIM) algorithm scales to graphs with billions of edges, with one to two orders of magnitude speedup over the best greedy methods. It still has a guaranteed approximation ratio, and in practice its quality nearly matches that of exact greedy. We also present influence oracles, which use linear-time preprocessing to generate a small sketch for each node, allowing the influence of any seed set to be quickly answered from the sketches of its nodes.

Graph Partitioning with Natural Cuts

We present a novel approach to graph partitioning based on the notion of \emph{natural cuts}. Our algorithm, called PUNCH, has two phases. The first phase performs a series of minimum-cut computations to identify and contract dense regions of the graph. This reduces the graph size, but preserves its general structure. The second phase uses a combination of greedy and local search heuristics to assemble the final partition. The algorithm performs especially well on road networks, which have an abundance of natural cuts (such as bridges, mountain passes, and ferries). In a few minutes, it obtains the best known partitions for continental-sized networks, significantly improving on previous results.

PHAST: Hardware-accelerated shortest path trees☆

Abstract We present a novel algorithm to solve the non-negative single-source shortest path problem on road networks and graphs with low highway dimension. After a quick preprocessing phase, we can compute all distances from a given source in the graph with essentially a linear sweep over all vertices. Because this sweep is independent of the source, we are able to reorder vertices in advance to exploit locality. Moreover, our algorithm takes advantage of features of modern CPU architectures, such as SSE and multiple cores. Compared to Dijkstra’s algorithm, our method needs fewer operations, has better locality, and is better able to exploit parallelism at multi-core and instruction levels. We gain additional speedup when implementing our algorithm on a GPU, where it is up to three orders of magnitude faster than Dijkstra’s algorithm on a high-end CPU. This makes applications based on all-pairs shortest-paths practical for continental-sized road networks. Several algorithms, such as computing the graph diameter, arc flags, or exact reaches, can be greatly accelerated by our method.

Fast local search for the maximum independent set problem

Given a graph G=(V,E), the independent set problem is that of finding a maximum-cardinality subset S of V such that no two vertices in S are adjacent. We introduce two fast local search routines for this problem. The first can determine in linear time whether a maximal solution can be improved by replacing a single vertex with two others. The second routine can determine in O(mΔ) time (where Δ is the highest degree in the graph) whether there are two solution vertices than can be replaced by a set of three. We also present a more elaborate heuristic that successfully applies local search to find near-optimum solutions to a wide variety of instances. We test our algorithms on instances from the literature as well as on new ones proposed in this paper.

Better landmarks within reach

We present significant improvements to a practical algorithm for the point-to-point shortest path problem on road networks that combines A* search, landmark-based lower bounds, and reach-based pruning. Through reach-aware landmarks, better use of cache, and improved algorithms for reach computation, we make preprocessing and queries faster while reducing the overall space requirements. On the road networks of the USA or Europe, the shortest path between two random vertices can be found in about one millisecond after one or two hours of preprocessing. The algorithm is also effective on two-dimensional grids.

VC-dimension and shortest path algorithms

We explore the relationship between VC-dimension and graph algorithm design. In particular, we show that set systems induced by sets of vertices on shortest paths have VC-dimension at most two. This allows us to use a result from learning theory to improve time bounds on query algorithms for the point-to-point shortest path problem in networks of low highway dimension, such as road networks. We also refine the definitions of highway dimension and related concepts, making them more general and potentially more relevant to practice. In particular, we define highway dimension in terms of set systems induced by shortest paths, and give cardinality-based and average case definitions.