Additive Guarantees for Degree-Bounded Directed Network Design

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].

[1]  Éva Tardos,et al.  Approximation Algorithms for a Directed Network Design Problem , 1999, IPCO.

[2]  Jørgen Bang-Jensen,et al.  Small Degree Out-branchings , 2001 .

[3]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[4]  Éva Tardos,et al.  Algorithms for a network design problem with crossing supermodular demands , 2004, Networks.

[5]  Eugene L. Lawler,et al.  Matroid intersection algorithms , 1975, Math. Program..

[6]  Harold N. Gabow,et al.  On the L∞-norm of extreme points for crossing supermodular directed network LPs , 2007, Math. Program..

[7]  R. Ravi,et al.  A matter of degree: improved approximation algorithms for degree-bounded minimum spanning trees , 2000, STOC '00.

[8]  Mohit Singh,et al.  Survivable Network Design with Degree or Order Constraints , 2009, SIAM J. Comput..

[9]  Mohit Singh,et al.  Degree bounded matroids and submodular flows , 2008, Comb..

[10]  R. Ravi,et al.  Approximation algorithms for finding low-degree subgraphs , 2004, Networks.

[11]  Gérard Cornuéjols,et al.  The traveling salesman problem on a graph and some related integer polyhedra , 1985, Math. Program..

[12]  Mohit Singh,et al.  Additive approximation for bounded degree survivable network design , 2008, SIAM J. Comput..

[13]  Michel X. Goemans,et al.  Minimum Bounded Degree Spanning Trees , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[14]  András Frank,et al.  Increasing the rooted-connectivity of a digraph by one , 1999, Math. Program..

[15]  R. Ravi,et al.  Many birds with one stone: multi-objective approximation algorithms , 1993, STOC '93.

[16]  Martin Fürer,et al.  Approximating the Minimum-Degree Steiner Tree to within One of Optimal , 1994, J. Algorithms.

[17]  R. Ravi,et al.  A matter of degree: improved approximation algorithms for degree-bounded minimum spanning trees , 2000, STOC '00.

[18]  Mohit Singh,et al.  On the Crossing Spanning Tree Problem , 2004, APPROX-RANDOM.

[19]  Kamal Jain,et al.  A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).