Computing maximum weight independent sets in graphs is an important NP-hard optimization problem. The problem is particularly difficult to solve in large graphs for which data reduction techniques do not work well. To be more precise, state-of-the-art branch-and-reduce algorithms can solve many large-scale graphs if reductions are applicable. Otherwise, their performance quickly degrades due to branching requiring exponential time. In this paper, we develop an advanced memetic algorithm to tackle the problem, which incorporates recent data reduction techniques to compute near-optimal weighted independent sets in huge sparse networks. More precisely, we use a memetic approach to recursively choose vertices that are likely to be in a large-weight independent set. We include these vertices into the solution, and further reduce the graph. We show that identifying and removing vertices likely to be in large-weight independent sets opens up the reduction space and speeds up the computation of large-weight independent sets remarkably. Our experimental evaluation indicates that we are able to outperform state-of-the-art algorithms. For example, our two algorithm configurations compute the best results among all competing algorithms for 205 out of 207 instances. Thus can be seen as a useful tool when large-weight independent sets need to be computed in practice.

Finding the connected components of an undirected graph is one of the most fundamental graph problems. Connected components are used in a wide spectrum of applications including VLSI design, machine learning and image analysis. Sequentially, one can easily find all connected components in linear time using breadth-first traversal. However, in a massively distributed setting, finding connected components in a scalable way becomes much harder due to data irregularities and the overhead associated with the increased need for communication. In this work, we present a communication-efficient distributed graph algorithm for finding connected components that scales to massively parallel machines. Our algorithm is based on a recent linear-work shared-memory parallel algorithm by Blelloch et al. [1] and refines it for a distributed memory setting. This includes a communication-efficient graph contraction procedure, as well as a distributed variant of the low diameter decomposition by Miller et al. [2]. We tackle the data irregularities introduced by high degree vertices by using an efficient procedure for distributing their incident edges. Our experimental evaluation on up to 16384 cores indicates a good weak scaling behavior that outperforms current state-of-the-art algorithms.

Given a vertex-weighted graph, the maximum weight independent set problem asks for a pair-wise non-adjacent set of vertices such that the sum of their weights is maximum. The branch-and-reduce paradigm is the de facto standard approach to solve the problem to optimality in practice. In this paradigm, data reduction rules are applied to decrease the problem size. These data reduction rules ensure that given an optimum solution on the new (smaller) input, one can quickly construct an optimum solution on the original input. We introduce new generalized data reduction and transformation rules for the problem. A key feature of our work is that some transformation rules can increase the size of the input. Surprisingly, these so-called increasing transformations can simplify the problem and also open up the reduction space to yield even smaller irreducible graphs later throughout the algorithm. In experiments, our algorithm computes significantly smaller irreducible graphs on all except one instance, solves more instances to optimality than previously possible, is up to two orders of magnitude faster than the best state-of-the-art solver, and finds higher-quality solutions than heuristic solvers DynWVC and HILS on many instances. While the increasing transformations are only efficient enough for preprocessing at this time, we see this as a critical initial step towards a new branch-and-transform paradigm.

Finding a maximum independent set is a fundamental NP-hard problem that is used in many real-world applications. Given an unweighted graph, this problem asks for a maximum cardinality set of pairwise non-adjacent vertices. Some of the most successful algorithms for this problem are based on the branch-and-bound or branch-and-reduce paradigms. In particular, branch-and-reduce algorithms, which combine branch-and-bound with reduction rules, achieved substantial results, solving many previously infeasible instances. These results were to a large part achieved by developing new, more practical reduction rules. However, other components that have been shown to have an impact on the performance of these algorithms have not received as much attention. One of these is the branching strategy, which determines what vertex is included or excluded in a potential solution. The most commonly used strategy selects vertices based on their degree and does not take into account other factors that contribute to the performance. In this work, we develop and evaluate several novel branching strategies for both branch-and-bound and branch-and-reduce algorithms. Our strategies are based on one of two approaches. They either (1) aim to decompose the graph into two or more connected components which can then be solved independently, or (2) try to remove vertices that hinder the application of a reduction rule. Our experiments on a large set of real-world instances indicate that our strategies are able to improve the performance of the state-of-the-art branch-and-reduce algorithms. To be more specific, our reduction-based packing branching rule is able to outperform the default branching strategy of selecting a vertex of highest degree on 65% of all instances tested. Furthermore, our decomposition-based strategy based on edge cuts is able to achieve a speedup of 2.29 on sparse networks (1.22 on all instances).

Kernelization is a general theoretical framework for preprocessing instances of NP-hard problems into (generally smaller) instances with bounded size, via the repeated application of data reduction rules. For the fundamental Max Cut problem, kernelization algorithms are theoretically highly efficient for various parameterizations. However, the efficacy of these reduction rules in practice—to aid solving highly challenging benchmark instances to optimality—remains entirely unexplored. We engineer a new suite of efficient data reduction rules that subsume most of the previously published rules, and demonstrate their significant impact on benchmark data sets, including synthetic instances, and data sets from the VLSI and image segmentation application domains. Our experiments reveal that current state-of-the-art solvers can be sped up by up to multiple orders of magnitude when combined with our data reduction rules. On social and biological networks in particular, kernelization enables us to solve four instances that were previously unsolved in a ten-hour time limit with state-of-the-art solvers; three of these instances are now solved in less than two seconds.

One powerful technique to solve NP-hard optimization problems in practice is branch-and-reduce search—which is branch-and-bound that intermixes branching with reductions to decrease the input size. While this technique is known to be very effective in practice for unweighted problems, very little is known for weighted problems, in part due to a lack of known effective reductions. In this work, we develop a full suite of new reductions for the maximum weight independent set problem and provide extensive experiments to show their effectiveness in practice on real-world graphs of up to millions of vertices and edges. Our experiments indicate that our approach is able to outperform existing state-of-the-art algorithms, solving many instances that were previously infeasible. In particular, we show that branch-and-reduce is able to solve a large number of instances up to two orders of magnitude faster than existing (inexact) local search algorithms—and is able to solve the majority of instances within 15 minutes. For those instances remaining infeasible, we show that combining kernelization with local search produces higher-quality solutions than local search alone.

We consider the problem of sampling n numbers from the range { 1,… , N } without replacement on modern architectures. The main result is a simple divide-and-conquer scheme that makes sequential algorithms more cache efficient and leads to a parallel algorithm running in expected time O ( n / p +log p ) on p processors, i.e., scales to massively parallel machines even for moderate values of n . The amount of communication between the processors is very small (at most O (log p )) and independent of the sample size. We also discuss modifications needed for load balancing, online sampling, sampling with replacement, Bernoulli sampling, and vectorization on SIMD units or GPUs.

We present the design and a first performance evaluation of Thrill - a prototype of a general purpose big data processing framework with a convenient data-flow style programming interface. Thrill is somewhat similar to Apache Spark and Apache Flink with at least two main differences. First, Thrill is based on C++ which enables performance advantages due to direct native code compilation, a more cache-friendly memory layout, and explicit memory management. In particular, Thrill uses template meta-programming to compile chains of subsequent local operations into a single binary routine without intermediate buffering and with minimal indirections. Second, Thrill uses arrays rather than multisets as its primary data structure which enables additional operations like sorting, prefix sums, window scans, or combining corresponding fields of several arrays (zipping). We compare Thrill with Apache Spark and Apache Flink using five kernels from the HiBench suite. Thrill is consistently faster and often several times faster than the other frameworks. At the same time, the source codes have a similar level of simplicity and abstraction.