A Matter of Degree: Improved Approximation Algorithms for Degree-Bounded Minimum Spanning Trees
SIAM Journal on Computing
Minimum Bounded Degree Spanning Trees
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
On the L∞-norm of extreme points for crossing supermodular directed network LPs
Mathematical Programming: Series A and B
Survivable network design with degree or order constraints
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Approximating minimum bounded degree spanning trees to within one of optimal
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Journal of Graph Theory
Degree bounded matroids and submodular flows
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
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We present polynomial-time approximation algorithms for some degree-bounded directed network design problems. Our main result is for intersecting supermodular connectivity requirements with degree bounds: given a directed graph $G=(V,E)$ with nonnegative edge-costs, a connectivity requirement specified by an intersecting supermodular function $f$, and upper bounds $\{a_v,b_v\}_{v\in V}$ on in-degrees and out-degrees of vertices, find a minimum-cost $f$-connected subgraph of $G$ that satisfies the degree bounds. We give a bicriteria approximation algorithm for this problem using the natural LP relaxation and show that our guarantee is the best possible relative to this LP relaxation. We also obtain similar results for the (more general) class of crossing supermodular requirements. In the absence of edge-costs, our result gives the first additive $O(1)$-approximation guarantee for degree-bounded intersecting/crossing supermodular connectivity problems. We also consider the minimum crossing spanning tree problem: Given an undirected edge-weighted graph $G$, edge-subsets $\{E_i\}_{i=1}^k$, and nonnegative integers $\{b_i\}_{i=1}^k$, find a minimum-cost spanning tree (if it exists) in $G$ that contains at most $b_i$ edges from each set $E_i$. We obtain a $+(r-1)$ additive approximation for this problem, when each edge lies in at most $r$ sets. A special case of this problem is the degree-bounded minimum spanning tree, and our techniques give a substantially shorter proof of the recent $+1$ approximation of Singh and Lau [in Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2007, pp. 661-670].