The problem of finding a minimum dominating set (MinDS) in a graph is a well known combinatorial optimization problem with wide applications. Due to its NP-hardness, for large and hard instances one must resort to heuristic approaches to obtain good solutions within reasonable time. This paper develops an efficient local search algorithm for MinDS, which has two main ideas. The first one is a novel local search framework, while the second is a construction procedure with inference rules. Our algorithm named FastDS is evaluated on 4 standard benchmarks and 3 massive graphs benchmarks. FastDS obtains the best performance for almost all benchmarks. Particularly, FastDS obtains significantly better solutions than state-of-the-art algorithms on massive graphs.

/pdf/two-goal-local-search-and-inference-rules-for-minimum-f403sxfv8c.pdf

Two-goal Local Search and Inference Rules for Minimum Dominating Set.

We study the parameterized complexity of various graph theoretic problems in the dynamic framework where the input graph is being updated by a sequence of edge additions and deletions. Vertex subset problems on graphs typically deal with finding a subset of vertices having certain properties. In real world applications, the graph under consideration often changes over time and due to this dynamics, the solution at hand might lose the desired properties. The goal in the area of dynamic graph algorithms is to efficiently maintain a solution under these changes. Recomputing a new solution on the new graph is an expensive task especially when the number of modifications made to the graph is significantly smaller than the size of the graph. In the context of parameterized algorithms, two natural parameters are the size k of the symmetric difference of the edge sets of the two graphs (on n vertices) and the size r of the symmetric difference of the two solutions. We study the Dynamic $$\Pi $$
 -Deletion problem which is the dynamic variant of the classical $$\Pi $$
 -Deletion problem and show NP-hardness, fixed-parameter tractability and kernelization results. For specific cases of Dynamic $$\Pi $$
 -Deletion such as Dynamic Vertex Cover and Dynamic Feedback Vertex Set, we describe improved algorithms and linear kernels. Specifically, we show that Dynamic Vertex Cover has a deterministic algorithm with $$1.0822^k n^{\mathcal {O}(1)}$$
 running time and Dynamic Feedback Vertex Set has a randomized algorithm with $$1.6667^k n^{\mathcal {O}(1)}$$
 running time. We also show that Dynamic Connected Feedback Vertex Set can be solved in $$2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}$$
 time. For each of Dynamic Connected Vertex Cover, Dynamic Dominating Set and Dynamic Connected Dominating Set, we describe an algorithm with $$2^k n^{\mathcal {O}(1)}$$
 running time and show that this is the optimal running time (up to polynomial factors) assuming the Set Cover Conjecture.

/pdf/dynamic-parameterized-problems-5er77ehx5j.pdf

Dynamic Parameterized Problems

The paper demonstrates the potential of chaining two distinct methodologies in service of topic modelling. The first, as of recent years, is more-or-less standard natural language processing (NLP) with word2vec; the second is graph-theoretical or combinatorial algorithm. Together, we show how they may be used to help classify documents into distinct, but perhaps not disjointed, classes. The procedure is demonstrated on a collection of Twitter feeds, or tweets. Heuristics is the basis for this procedure; it is not presumed to perfectly work in every situation, or for every input, and, in fact, the authors believe that the procedure will yield better results in a more homogeneous corpora written in some standardized fashion, as written in, e.g., legal or medical documents.

Combinatorial Text Classification: the Effect of Multi-Parameterized Correlation Clustering

We introduce a dynamic version of the NP-hard Cluster Editing problem. The essential point here is to take into account dynamically evolving input graphs: Having a cluster graph (that is, a disjoint union of cliques) that represents a solution for a first input graph, can we cost-efficiently transform it into a "similar" cluster graph that is a solution for a second ("subsequent") input graph? This model is motivated by several application scenarios, including incremental clustering, the search for compromise clusterings, or also local search in graph-based data clustering. We thoroughly study six problem variants (edge editing, edge deletion, edge insertion; each combined with two distance measures between cluster graphs). We obtain both fixed-parameter tractability as well as parameterized hardness results, thus (except for two open questions) providing a fairly complete picture of the parameterized computational complexity landscape under the perhaps two most natural parameterizations: the distance of the new "similar" cluster graph to (i) the second input graph and to (ii) the input cluster graph.

/pdf/parameterized-dynamic-cluster-editing-1qay969lxh.pdf

Parameterized Dynamic Cluster Editing

We introduce a dynamic version of the NP-hard graph modification problem Cluster Editing. The essential point here is to take into account dynamically evolving input graphs: having a cluster graph (that is, a disjoint union of cliques) constituting a solution for a first input graph, can we cost-efficiently transform it into a “similar” cluster graph that is a solution for a second (“subsequent”) input graph? This model is motivated by several application scenarios, including incremental clustering, the search for compromise clusterings, or also local search in graph-based data clustering. We thoroughly study six problem variants (three modification scenarios edge editing, edge deletion, edge insertion; each combined with two distance measures between cluster graphs). We obtain both fixed-parameter tractability as well as (parameterized) hardness results, thus (except for three open questions) providing a fairly complete picture of the parameterized computational complexity landscape under the two perhaps most natural parameterizations: the distances of the new “similar” cluster graph to (1) the second input graph and to (2) the input cluster graph.

/pdf/parameterized-dynamic-cluster-editing-1xrypsmtmq.pdf

Turbo-charging is a recent algorithmic technique that is based on the fixed-parameter tractability of the dynamic versions of some problems as a way to improve heuristics. We demonstrate the effectiveness of this technique and develop the turbo-charging idea further. A new hybrid heuristic for the Dominating Set problem that further improves this method is obtained by combining the turbo-charging technique with other standard heuristic tools including Local Search (LS). We implement both the recently proposed “turbo greedy” algorithm of Downey et al. [8] and a new method presented in this paper. The performance of these new heuristics is assessed on three suites of benchmark datasets, namely DIMACS, BHOSLIB and KONECT. Experiments comparing our algorithm to both heuristic and exact algorithms demonstrate its effectiveness. Our algorithm often produced results that were either exact or better than all the other available algorithms.

Turbo-Charging Dominating Set with an FPT Subroutine: Further Improvements and Experimental Analysis

Graphics Processing Units (GPUs) are gaining notable popularity due to their affordable high performance multi-core architecture. They are particularly useful for massive computations that involve large data sets. In this paper, we present a highly scalable approach for the NP-hard Vertex Cover problem. Our method is based on an advanced data structure to reduce memory usage for more parallelism and we propose a load balancing scheme that is effective for multi-GPU architectures. Our parallel algorithm was implemented on multiple AMD GPUs using OpenCL. Experimental results show that our proposed approach can achieve significant speedups on the hard instances of the DIMACS benchmarks as well as the notoriously hard 120-Cell graph and its variants.

Accelerating vertex cover optimization on a GPU architecture

The last decade witnessed an increased interest in exact and parameterized exponential-time algorithms for NP - hard problems. The hardness of polynomial-time approximation of many intractable problems motivated the work on fixed-parameter approximation where polynomial-time is relaxed into FPT -time as long as improved approximation is obtained, most often requiring constant ratio bounds. In this paper we move a step further by investigating the practicality of exponential time approximation (versus FPT-time) as long as obtained solutions are within an additive parameter. The running time of such algorithm would be reduced by some function (factor) of the same parameter. The objective is to obtain a cost-effective trade-off between reduced running time and quality of approximation while providing provably near optimal solutions. We present experimental studies of two problems: Dominating Set and Vertex Cover. Our experiments show that semi-exact algorithms are indeed very promising.

Semi-Exact Exponential-Time Algorithms: an Experimental Study

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Turbo-Charging Dominating Set with an FPT Subroutine: Further Improvements and Experimental Analysis

Accelerating vertex cover optimization on a GPU architecture

Semi-Exact Exponential-Time Algorithms: an Experimental Study