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

J. Könemann, R. Ravi
2005 SIAM journal on computing (Print)  
We present a new bicriteria approximation algorithm for the degree-bounded minimum-cost spanning tree problem: Given an undirected graph with nonnegative edge weights and degree bounds Bv > 1 for all vertices v, find a spanning tree T of minimum total edge-cost such that the maximum degree of each node v in T is at most Bv. Our algorithm finds a tree in which the degree of each node v is O(Bv + log n) and the total edge-cost is at most a constant times the cost of any tree that obeys all degree
more » ... constraints. Our previous algorithm[9] with similar guarantees worked only in the case of uniform degree bounds (i.e. Bv = B for all vertices v). While the new algorithm is based on ideas from Lagrangean relaxation as is our previous work, it does not rely on computing a solution to a linear program. Instead it uses a repeated application of Kruskal's MST algorithm interleaved with a combinatorial update of approximate Lagrangean 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.
doi:10.1137/s0097539702418048 fatcat:dqvqv6sywbczpe2mcmm4nyfgsa