On generalizations of network design problems with degree bounds

Abstract

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely laminar crossing spanning tree), and (2) by incorporating 'degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.• Our main result is a (1, b+O(log n))-approximation algorithm for the minimum crossing spanning tree (MCST) problem with laminar degree constraints. The laminar MCST problem is a natural generalization of the well-studied bounded-degree MST, and is a special case of general crossing spanning tree. We also give an additive Ω(log α m) hardness of approximation for general MCST, even in the absence of costs (α > 0 is a fixed constant, and m is the number of degree constraints).• We then consider the crossing contra-polymatroid intersection problem and obtain a (2, 2b + ∆−1)-approximation algorithm, where ∆ is the maximum element frequency. This models for example the degree-bounded spanning-set intersection in two matroids. Finally, we introduce the crossing lattice polyhedra problem, and obtain a (1, b + 2∆ − 1) approximation under certain condition. This result provides a unified framework and common generalization of various problems studied previously, such as degree bounded matroids.

Chain-constrained spanning trees

“…Notice that our hardness result is stronger in terms of the approximation ratio, the underlying constrained spanning tree model, and the complexity assumption. Furthermore, Theorem 2 shows that the additive O(log n)-approximation of Bansal et al [3] for the laminar-constrained spanning tree problem is close to optimal.…”

“…3. Previously, the only hardness result of a similar nature was presented by Bansal et al [3] for a very general constrained spanning tree problem, where constraints |T ∩ E i | ≤ b i ∀i ∈ [k] are given for an arbitrary family of edge sets E 1 , . .…”

Approximation algorithms for network design: A survey

“…. E k are induced by a laminar system of cuts was given in [12]. Checkuri et al [37] recently presented a very elegant randomized technique for rounding a fractional point in a matroid polytope to an integral one.…”

Chain-Constrained Spanning Trees

“…But extending the iterative rounding technique beyond such settings seems to typically be very difficult. Some progress was achieved by Bansal et al [3], who used an iterative approach that iteratively replaces constraints by weaker ones, leading to an additive O(log n)-approximation if the constraints are upper bounds on a laminar family of cuts. They left open whether an additive or multiplicative O(1)-approximation is possible in this setting, even when the cuts form a chain.…”