Graph Bisection with Pareto Optimization

Abstract

We introduce FlowCutter, a novel algorithm to compute a set of edge cuts or node separators that optimize cut size and balance in the Pareto-sense. Our core algorithm heuristically solves the balanced connected st-edge-cut problem, where two given nodes s and t must be separated by removing edges to obtain two connected parts. Using the core algorithm as subroutine, we build variants that compute node separators which are independent of s and t. From the computed Pareto-set, we can identify cuts with a particularly good tradeoff between cut size and balance that can be used to compute contraction and minimum fill-in orders, which can be used in Customizable Contraction Hierarchies (CCH), a speed-up technique for shortest path computations. Our core algorithm runs in O(c|E|) time where E is the set of edges and c is the size of the largest outputted cut. This makes it wellsuited for separating large graphs with small cuts, such as road graphs, which is the primary application motivating our research. For road graphs, we present an extensive experimental study demonstrating that FlowCutter outperforms the current state-of-the-art both in terms of cut sizes and CCH performance. By evaluating FlowCutter on a standard graph partitioning benchmark, we further show that FlowCutter also finds small, balanced cuts on non-road graphs. Another application is the computation of small tree-decompositions.To evaluate the quality of our algorithm in this context, we entered the PACE 2016 challenge [13] and won the first place in the corresponding sequential competition track. We can therefore conlude that our FlowCutter algorithm finds small, balanced cuts on a wide variety of graphs.

Faster and Better Nested Dissection Orders for Customizable Contraction Hierarchies

“…After introducing preliminaries, we describe the existing biparitioning algorithms FlowCutter and Inertial Flow on a high level, before discussing how to combine them into our new algorithm InertialFlowCutter. We refer the interested reader to [5] for implementation details and a more in-depth discussion of the FlowCutter algorithm. Then we discuss our application Customizable Contraction Hierarchies (CCH), what makes a good CCH order, and how we use recursive bisection to compute them.…”

“…It chooses the node p which minimizes dist(p, t) − dist(s, p), where dist is the hop distance, precomputed via Breadth-First-Search from s and t. Roughly speaking, this attempts to prevent the cut sides from meeting before perfect balance. It also has a geometric interpretation, which is explained in [5].…”

“…Roughly speaking, this stems from performing one graph traversal, e. g. Pseudo-DFS, per unit of flow. The exact details can be found in [5]. Flow-based execution interleaving ensures that no cutter performs more flow augmentations than the other cutters.…”

“…The framework to compute contraction orders is the same as for FlowCutter in [5]. For self-containedness we repeat it here.…”

“…The running time needed for the customization and the shortest path queries depends on the quality of the calculated order. Previously proposed partitioning tools for computing separators in road networks include FlowCutter [5], Inertial Flow [6], KaHiP [7] and Metis [8], PUNCH [9] and Buffoon [10]. KaHiP and Metis are general-purpose graph partitioning tools.…”