Given an undirected graph, a balanced node separator is a set of nodes whose removal splits the graph into connected components of limited size. Balanced node separators are used for graph partitioning, for the constr...
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Given an undirected graph, a balanced node separator is a set of nodes whose removal splits the graph into connected components of limited size. Balanced node separators are used for graph partitioning, for the construction of graph data structures, and for measuring network reliability. It is NP-hard to decide whether a graph has a balanced node separator of size at most k. Therefore, practical algorithms typically try to find small separators in a heuristic fashion. In this paper, we present a branching algorithm that for a given value k either outputs a balanced node separator of size at most k or certifies that. k is a valid lower bound. Using this algorithm iteratively for growing values of k allows us to find a minimum balanced node separator. To make this algorithm scalable to real-world (road) networks of considerable size, we first describe pruning rules to reduce the graph size without affecting the minimum balanced separator size. In addition, we prove several structural properties of minimum balanced node separators which are then used to reduce the branching factor and search depth of our algorithm. We experimentally demonstrate the applicability of our algorithm to graphs with thousands of nodes and edges. We showcase the usefulness of having minimum balanced separators for judging the quality of existing heuristics, for improving preprocessing-based route planning techniques on road networks, and for lower bounding important graph parameters. Finally, we discuss how our ideas and methods can also be leveraged to accelerate the heuristic computation of balanced node separators.
We consider the problem of finding a directed acyclic graph (DAG) that optimizes a decomposable Bayesian network score. While in a favorable case an optimal DAG can be found in polynomial time, in the worst case the f...
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We consider the problem of finding a directed acyclic graph (DAG) that optimizes a decomposable Bayesian network score. While in a favorable case an optimal DAG can be found in polynomial time, in the worst case the fastest known algorithms rely on dynamic programming across the node subsets, taking time and space 2n, to within a factor polynomial in the number of nodes n. In practice, these algorithms are feasible to networks of at most around 30 nodes, mainly due to the large space requirement. Here, we generalize the dynamic programming approach to enhance its feasibility in three dimensions: first, the user may trade space against time; second, the proposed algorithms easily and efficiently parallelize onto thousands of processors; third, the algorithms can exploit any prior knowledge about the precedence relation on the nodes. Underlying all these results is the key observation that, given a partial order P on the nodes, an optimal DAG compatible with P can be found in time and space roughly proportional to the number of ideals of P, which can be significantly less than 2n. Considering sufficiently many carefully chosen partial orders guarantees that a globally optimal DAG will be found. Aside from the generic scheme, we present and analyze concrete tradeoff schemes based on parallel bucket orders.
Given an undirected graph G = (V,E) with edge weights and a positive integer number k, the k-cardinality tree problem consists of finding a subtree T of G with exactly k edges and the minimum possible weight. Many alg...
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Given an undirected graph G = (V,E) with edge weights and a positive integer number k, the k-cardinality tree problem consists of finding a subtree T of G with exactly k edges and the minimum possible weight. Many algorithms have been proposed to solve this NP-hard problem, resulting in mainly heuristic and metaheuristic *** this article, we present an exact ILP-based algorithm using directed cuts. We mathematically compare the strength of our formulation to the previously known ILP formulations of this problem, and show the advantages of our approach. Afterwards, we give an extensive study on the algorithm's practical performance compared to the state-of-the-art *** contrast to the widespread assumption that such a problem cannot be efficiently tackled by exact algorithms for medium and large graphs (between 200 and 5,000 nodes), our results show that our algorithm not only has the advantage of proving the optimality of the computed solution, but also often outperforms the metaheuristic approaches in terms of running time.
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