We consider the following general network design problem on directed graphs. The input is an asymmetric metric (V, c), root r in V, monotone submodular function f and budget B. The goal is to find an rrooted arborescence T of cost at most B that maximizes f(T). Our main result is a very simple quasipolynomial time approximation algorithm for this problem, where k ≤ V is the number of vertices in an optimal solution. To the best of our knowledge, this is the first nontrivial approximation ratio for this problem. As a consequence we obtain an O(log^2 k / loglog k) approximation algorithm for directed (polymatroid) Steiner tree in quasipolynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved approximation algorithms for the singlesource buyatbulk and priority Steiner tree problems. For the usual directed Steiner tree problem, our result matches the best previous approximation ratio, but improves significantly on the running time. For polymatroid Steiner tree and singlesource buyatbulk, our result improves prior approximation ratios by a logarithmic factor. For directed priority Steiner tree, our result seems to be the first nontrivial approximation ratio. Under certain complexity assumptions, our approximation ratios are best possible (up to constant factors).
more »
« less
Lasserre Integrality Gaps for Graph Spanners and Related Problems
There has been significant recent progress on algorithms for approximating graph spanners, i.e., algorithms which approximate the best spanner for a given input graph. Essentially all of these algorithms use the same basic LP relaxation, so a variety of papers have studied the limitations of this approach and proved integrality gaps for this LP. We extend these results by showing that even the strongest liftandproject methods cannot help significantly, by proving polynomial integrality gaps even for n^{\Omega(\epsilon)} levels of the Lasserre hierarchy, for both the directed and undirected spanner problems. We also extend these integrality gaps to related problems, notably Directed Steiner Network and ShallowLight Steiner Network.
more »
« less
 Award ID(s):
 1909111
 NSFPAR ID:
 10295388
 Date Published:
 Journal Name:
 International Workshop on Approximation and Online Algorithms (WAOA)
 Page Range / eLocation ID:
 97112
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this


We consider the following general network design problem. The input is an asymmetric metric (V, c), root [Formula: see text], monotone submodular function [Formula: see text], and budget B. The goal is to find an rrooted arborescence T of cost at most B that maximizes f(T). Our main result is a simple quasipolynomial time [Formula: see text]approximation algorithm for this problem, in which [Formula: see text] is the number of vertices in an optimal solution. As a consequence, we obtain an [Formula: see text]approximation algorithm for directed (polymatroid) Steiner tree in quasipolynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved [Formula: see text]approximation algorithms for the singlesource buyatbulk and priority Steiner tree problems. For the usual directed Steiner tree problem, our result matches the best previous approximation ratio but improves significantly on the running time. For polymatroid Steiner tree and singlesource buyatbulk, our result improves prior approximation ratios by a logarithmic factor. For directed priority Steiner tree, our result seems to be the first nontrivial approximation ratio. Under certain complexity assumptions, our approximation ratios are the best possible (up to constant factors).more » « less

Columnsparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (nonuniform) attenuation and multiplechance algorithms, to obtain improved approximation algorithms for some wellknown families of such problems. As three main examples, we attain the integrality gap, up to lowerorder terms, for known LP relaxations for kcolumn sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic kset packing (Bansal et al., Algorithmica, 2012), and go “half the remaining distance” to optimal for a major integralitygap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).more » « less

Given a weighted graph G(V, E) and t ≥ 1, a subgraph H is a t–spanner of G if the lengths of shortest paths in G are preserved in H up to a multiplicative factor of t. The subsetwise spanner problem aims to preserve distances in G for only a subset of the vertices. We generalize the minimumcost subsetwise spanner problem to one where vertices appear on multiple levels, which we call the multilevel graph spanner (MLGS) problem, and describe two simple heuristics. Applications of this problem include road/network building and multilevel graph visualization, especially where vertices may require different grades of service. We formulate a 0–1 integer linear program (ILP) of size O(EV 2) for the more general minimum pairwise spanner problem, which resolves an open question by Sigurd and Zachariasen on whether this problem admits a useful polynomialsize ILP. We extend this ILP formulation to the MLGS problem, and evaluate the heuristic and ILP performance on random graphs of up to 100 vertices and 500 edges.more » « less

Given a graph G = (V, E) and a subset T ⊆ V of terminals, a Steiner tree of G is a tree that spans T. In the vertexweighted Steiner tree (VST) problem, each vertex is assigned a nonnegative weight, and the goal is to compute a minimum weight Steiner tree of G. Vertexweighted problems have applications in network design and routing, where there are different costs for installing or maintaining facilities at different vertices. We study a natural generalization of the VST problem motivated by multilevel graph construction, the vertexweighted gradeofservice Steiner tree problem (VGSST), which can be stated as follows: given a graph G and terminals T, where each terminal v ∈ T requires a facility of a minimum grade of service R(v) ∈ {1, 2, . . . `}, compute a Steiner tree G0 by installing facilities on a subset of vertices, such that any two vertices requiring a certain grade of service are connected by a path in G 0 with the minimum grade of service or better. Facilities of higher grade are more costly than facilities of lower grade. Multilevel variants such as this one can be useful in network design problems where vertices may require facilities of varying priority. While similar problems have been studied in the edgeweighted case, they have not been studied as well in the more general vertexweighted case. We first describe a simple heuristic for the VGSST problem whose approximation ratio depends on `, the number of grades of service. We then generalize the greedy algorithm of [Klein & Ravi, 1995] to show that the VGSST problem admits a (2 ln T)approximation, where T is the set of terminals requiring some facility. This result is surprising, as it shows that the (seemingly harder) multigrade problem can be approximated as well as the VST problem, and that the approximation ratio does not depend on the number of grades of service. Finally, we show that this problem is a special case of the directed Steiner tree problem and provide an integer linear programming (ILP) formulation for the VGSST problem.more » « less