Scalable Parameterised Algorithms for two Steiner Problems

In the Steiner Problem, we are given as input (i) a connected graph with nonnegative integer weights associated with the edges; and (ii) a subset of vertices called terminals. The task is to find a minimum-weight subgraph connecting all the terminals. In the Group Steiner Problem, we are given as input (i) a connected graph with nonnegative integer weights associated with the edges; and (ii) a collection of subsets of vertices called groups. The task is to find a minimum-weight subgraph that contains at least one vertex from each group. Even though the Steiner Problem and the Group Steiner Problem are NP-complete, they are known to admit parameterised algorithms that run in linear time in the size of the input graph and the exponential part can be restricted to the number of terminals and the number of groups, respectively. In this thesis, we discuss two parameterised algorithms for solving the Steiner Problem, and by reduction, the Group Steiner Problem: (a) a dynamic programming algorithm presented by Dreyfus and Wagner in 1971; and (b) an improvement of the Dreyfus-Wagner algorithm presented by Erickson, Monma and Veinott in 1987 that runs in linear time in the size of the input graph. We develop a parallel implementation of the Erickson-Monma-Veinott algorithm, and carry out extensive experiments to study the scalability of our implementation with respect to its runtime, memory bandwidth, and memory usage. Our experimental results demonstrate that the implementation can scale up to a billion edges on a single modern compute node provided that the number of terminals is small. For example, using our parallel implementation a Steiner tree for a graph with hundred million edges and ten terminals can be found in approximately twenty minutes. For an input graph with one hundred million edges and ten terminals, our parallel implementation is at least fifteen times faster than its serial counterpart on a Haswell compute node with two processors and twelve cores in each processor. Our implementation of the Erickson-Monma-Veinott algorithm is available as open source

Engineering Motif Search for Large Motifs

Given a vertex-colored graph H and a multiset M of colors as input, the graph motif problem asks us to decide whether H has a connected induced subgraph whose multiset of colors agrees with M. The graph motif problem is NP-complete but known to admit randomized algorithms based on constrained multilinear sieving over GF(2^b) that run in time O(2^kk^2m {M({2^b})}) and with a false-negative probability of at most k/2^{b-1} for a connected m-edge input and a motif of size k. On modern CPU microarchitectures such algorithms have practical edge-linear scalability to inputs with billions of edges for small motif sizes, as demonstrated by Björklund, Kaski, Kowalik, and Lauri [ALENEX'15]. This scalability to large graphs prompts the dual question whether it is possible to scale to large motif sizes. We present a vertex-localized variant of the constrained multilinear sieve that enables us to obtain, in time O(2^kk^2m{M({2^b})}) and for every vertex simultaneously, whether the vertex participates in at least one matchwith the motif, with a per-vertex probability of at most k/2^{b-1} for a false negative. Furthermore, the algorithm is easily vector-parallelizable for up to 2^k threads, and parallelizable for up to 2^kn threads, where n is the number of vertices in H. Here {M({2^b})} is the time complexity to multiply in GF(2^b). We demonstrate with an open-source implementation that our variant of constrained multilinear sieving can be engineered for vector-parallel microarchitectures to yield hardware utilization that is bound by the available memory bandwidth. Our main engineering contributions are (a) a version of the recurrence for tightly labeled arborescences that can be executed as a sequence of memory-and-arithmetic coalescent parallel workloads on multiple GPUs, and (b) a bit-sliced low-level implementation for arithmetic in characteristic 2 to support (a).Peer reviewe