Jochen Könemann

Faster and simpler algorithms for multicommodity flow and other fractional packing problems

This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We provide a different approach to these problems which yields faster and much simpler algorithms. Our approach also allows us to substitute shortest path computations for min-cost flow computations in computing maximum concurrent flow and min-cost multicommodity flow; this yields much faster algorithms when the number of commodities is large.

Faster and Simpler Algorithms for Multicommodity Flow and Other Fractional Packing Problems

This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new, faster, and much simpler algorithms for these problems.

A Matter of Degree: Improved Approximation Algorithms for Degree-Bounded Minimum Spanning Trees

In this paper, we present a new bicriteria approximation algorithm for the degree-bounded minimum spanning tree problem. In this problem, we are given an undirected graph, a nonnegative cost function on the edges, and a positive integer B*, and the goal is to find a minimum-cost spanning tree T with maximum degree at most B*. In an n-node graph, our algorithm finds a spanning tree with maximum degree O(B*+logn) and cost O(optB*), where optB* is the minimum cost of any spanning tree whose maximum degree is at most B*. Our algorithm uses ideas from Lagrangean duality. We show how a set of optimum Lagrangean multipliers yields bounds on both the degree and the cost of the computed solution.

Min-max tree covers of graphs

We provide constant factor approximation algorithms for covering the nodes of a graph using trees (rooted or unrooted), under the objective function of minimizing the weight of the maximum weight tree, subject to an upper bound on the number of trees used. These problems are related to location routing and traveling salesperson problems.

Exact geometric computation in LEDA

Almost all geometric algorithms are based on the RealRAM model. Implementors often simply replace the exact real arithmetic of this model by fixed precision arithmetic, thereby making correct algorithms incorrect. Two approaches have been taken to remedy this sit uation. The first approach is redesigning geometric algorithms for fixed precision arithmetic. The redesign is difficult and can inherently not lead to exact results. Moreover it prevents application areas from making use of the rich literature of geometric algorithms developed in computational geometry. The other approach advocat es the use of exact real arithmetic.

Primal-Dual Meets Local Search: Approximating MSTs With Nonuniform Degree Bounds

We present a new bicriteria approximation algorithm for the degree-bounded minimum-cost spanning tree (MST) problem: Given an undirected graph with nonnegative edge weights and a degree bound B, find a spanning tree of maximum node-degree B and minimum total edge-cost. Our algorithm outputs a tree of maximum degree at most a constant times B and total edge-cost at most a constant times that of a minimum-cost degree-B-bounded spanning tree. While our new algorithm is based on ideas from Lagrangian relaxation, as is our previous work [SIAM J. Comput., 31 (2002), pp. 1783--1793], it does not rely on computing a solution to a linear program. Instead, it uses a repeated application of Kruskals MST algorithm interleaved with a combinatorial update of approximate Lagrangian node-multipliers maintained by the algorithm. These updates cause subsequent repetitions of the spanning tree algorithm to run for longer and longer times, leading to overall progress and a proof of the performance guarantee. A second useful feature of our algorithm is that it can handle nonuniform degree bounds on the nodes: Given distinct bounds Bv for every node

Approximating k-hop minimum-spanning trees

v \in V

A matter of degree: improved approximation algorithms for degree-bounded minimum spanning trees

, the output tree has degree at most O(Bv + log|V|) for every

Primal-dual meets local search: approximating MST's with nonuniform degree bounds

v \in V

From primal-dual to cost shares and back: a stronger LP relaxation for the steiner forest problem

. As before, the cost of the tree is at most a constant times that of a minimum-cost tree obeying all degree bounds.