Network flows is an active area of research that has applications in a wide variety of fields. Several network flow problems are reduced to either the maximum flow problem or the minimum cost flow problem. Maximum flow problem involves finding the maximum possible amount of flow between two designated nodes on a network with arcs having flow capacities. Minimum cost flow problem tries to determine a flow with the minimum cost on a network with supply and demand nodes. In this thesis, we propose two parallel algorithms for the maximum flow and the minimum cost flow problems respectively. First, we present a shared memory parallel push-relabel algorithm for the maximum flow problem. Graph coloring is used to avoid collisions between threads for concurrent push and relabel operations. In addition, excess values of target nodes are updated using atomic instructions to prevent race conditions. The experiments show that our algorithm is competitive for wide graphs with low diameters. Second, we contribute a parallel implementation of the network simplex algorithm that is used for the solution of minimum cost flow problem. We propose finding the entering arc in parallel as it often takes the majority of the execution time. Scanning all arcs can take quite some time, so it is common to consider only a fixed number of arcs which is referred as the block search pivoting rule. Arc scans can easily be done in parallel to find the best candidate as the calculations are independent of each other. We used shared memory parallelism using OpenMP along with vectorization using AVX instructions. We also tried adjusting block sizes to increase the parallel portion of the algorithm. Our experiments show speedups up to 4 are possible, though they are typically lower.
