Improved Approximation Algorithms for Balanced Partitioning Problems

We present approximation algorithms for balanced partitioning problems. These problems are notoriously hard and we present new bicriteria approximation algorithms, that approximate the optimal cost and relax the balance constraint. In the first scenario, we consider Min-Max k-Partitioning, the problem of dividing a graph into k equal-sized parts while minimizing the maximum cost of edges cut by a single part. Our approximation algorithm relaxes the size of the parts by (1+epsilon) and approximates the optimal cost by O(log^{1.5}(n) * log(log(n))), for every 0 < epsilon < 1. This is the first nontrivial algorithm for this problem that relaxes the balance constraint by less than 2. In the second scenario, we consider strategies to find a minimum-cost mapping of a graph of processes to a hierarchical network with identical processors at the leaves. This Hierarchical Graph Partitioning problem has been studied recently by Hajiaghayi et al. who presented an (O(log(n)),(1+epsilon)(h+1)) approximation algorithm for constant network heights h. We use spreading metrics to give an improved (O(log(n)),(1+epsilon)h) approximation algorithm that runs in polynomial time for arbitrary network heights

Balanced Crown Decomposition for Connectivity Constraints

We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved approximation algorithms and kernelization for a variety of such problems. In particular, through this structure, we obtain the first constant-factor approximation for the Balanced Connected Partition (BCP) problem, where the task is to partition a vertex-weighted graph into k connected components of approximately equal weight. We derive a 3-approximation for the two most commonly used objectives of maximizing the weight of the lightest component or minimizing the weight of the heaviest component

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions