We present a general framework for constructing cut sparsifiers in undirected graphs---weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative factor of $(1 \pm \epsilon)$. Using this framework, we simplify, unify, and improve upon previous sparsification results. As simple instantiations of this framework, we show that sparsifiers can be constructed by sampling edges according to their strength (a result of Benczúr and Karger [ Approximating s-t minimum cuts in o͂(n$^2$) time, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, ACM, New York, 1996, pp. 47--55], [ SIAM J. Comput., 44 (2015), pp. 290--319]), effective resistance (a result of Spielman and Srivastava [ SIAM J. Comput., 40 (2011), pp. 1913--1926]), or edge connectivity. Sampling according to edge connectivity is the most aggressive method, and the most challenging to analyze. Our proof that this method produces sparsifiers resolves an open question of Benczúr and Karger. While the above results are interesting from a combinatorial standpoint, we also prove new algorithmic results. In particular, we give the first (optimal) $O(m)$-time sparsification algorithm for unweighted graphs. Our algorithm has a running time of $O(m) + \tilde{O}(n/\epsilon^2)$ for weighted graphs, which is also linear unless the input graph is very sparse itself. In both cases, this improves upon the previous best running times (due to Benczúr and Karger [ Approximating s-t minimum cuts in o͂(n$^2$) time, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, ACM, New York, 1996, pp. 47--55], [ SIAM J. Comput., 44 (2015), pp. 290--319]) of $O(m\log^2 n)$ (for the unweighted case) and $O(m\log^3 n)$ (for the weighted case), respectively. Our algorithm constructs sparsifiers that contain $O(n\log n/\epsilon^2)$ edges in expectation. A key ingredient of our proofs is a natural generalization of Karger's bound on the number of small cuts in an undirected graph. Given the numerous applications of Karger's bound, we suspect that our generalization will also be of independent interest.

Graphs with bounded highway dimension were introduced by Abraham et al. [ Proceedings of SODA 2010, pp. 782--793] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph $G=(V,E)$ of constant highway dimension, we show how to randomly compute a weighted graph $H=(V,E')$ that distorts shortest path distances of $G$ by at most a $1+\varepsilon$ factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of $G$. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location. To construct our embedding for low highway dimension graphs we extend Talwar's [ Proceedings of STOC 2004, pp. 281--290] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several nontrivial ingredients to Talwar's techniques, and in particular thoroughly analyze the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics.

We present a general framework for constructing cut sparsifiers in undirected graphs --- weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative factor of (1 ε). Using this framework, we simplify, unify and improve upon previous sparsification results. As simple instantiations of this framework, we show that sparsifiers can be constructed by sampling edges according to their strength (a result of Benczur and Karger), effective resistance (a result of Spielman and Srivastava), edge connectivity, or by sampling random spanning trees. Sampling according to edge connectivity is the most aggressive method, and the most challenging to analyze. Our proof that this method produces sparsifiers resolves an open question of Benczur and Karger.

Given a complete undirected graph, a cost function on edges, and a degree bound B, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree B satisfying given connectivity requirements. Even for a simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NP-hard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies the triangle inequality, there are constant factor approximation algorithms for various degree bounded network design problems. In global edge-connectivity, there is a $(2+\frac{1}{k})$-approximation algorithm for the minimum bounded degree k-edge-connected subgraph problem. In local edge-connectivity, there is a 4-approximation algorithm for the minimum bounded degree Steiner network problem when $r_{\max}$ is even, and a 5.5-approximation algorithm when $r_{\max}$ is odd, where $r_{\max}$ is the maximum connectivity requirement. In global vertex-connectivity, there is a $(2+\frac{k-1}{n}+\frac{1}{k})$-approximation algorithm for the minimum bounded degree k-vertex-connected subgraph problem when $n\geq2k$, where n is the number of vertices. For spanning tree, there is a $(1+\frac{1}{B-1})$-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with the smallest possible maximum degree, and in most cases the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs. Our algorithms can be seen as a generalization of Christofides' algorithm for the metric traveling salesman problem. The main technical tool is a simplicity-preserving edge splitting-off operation, which is used to “short-cut” vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions.

Given a complete undirected graph, a cost function on edges and a degree bound B, the degree bounded network design problem is to find a minimum cost simple subgraph with maximum degree B satisfying given connectivity requirements. Even for simple connectivity requirement such as finding a spanning tree, computing a feasible solution for the degree bounded network design problem is already NP-hard, and thus there is no polynomial factor approximation algorithm for this problem. In this paper, we show that when the cost function satisfies triangle inequalities, there are constant factor approximation algorithms for various degree bounded network design problems.Global edge-connectivity: There is a (2+1/k)-approximation algorithm for the minimum bounded degree k-edge-connected subgraph problem. Local edge-connectivity: There is a 6-approximation algorithm for the minimum bounded degree Steiner network problem. Global vertex-connectivity: there is a (2+(k-1)/n+1/k)-approximation algorithm for the minimum bounded degree k-vertex-connected subgraph problem. Spanning tree: there is an (1+1/(d-1))-approximation algorithm for the minimum bounded degree spanning tree problem. These approximation algorithms return solutions with smallest possible maximum degree, and the cost guarantee is obtained by comparing to the optimal cost when there are no degree constraints. This demonstrates that degree constraints can be incorporated into network design problems with metric costs.Our algorithms can be seen as a generalization of Christofides' algorithm for metric TSP. The main technical tool is a simplicity-preserving edge splitting-off operation, which is used to "short-cut" vertices with high degree while maintaining connectivity requirements and preserving simplicity of the solutions.