Graph Sparsification by Edge-Connectivity and Random Spanning Trees

We present new approaches to constructing graph sparsifiers --- weighted subgraphs for which every cut has the same value as the original graph, up to a factor of $(1 \pm \epsilon)$. Our first approach independently samples each edge $uv$ with probability inversely proportional to the edge-connectivity between $u$ and $v$. The fact that this approach produces a sparsifier resolves a question posed by Bencz\'ur and Karger (2002). Concurrent work of Hariharan and Panigrahi also resolves this question. Our second approach constructs a sparsifier by forming the union of several uniformly random spanning trees. Both of our approaches produce sparsifiers with $O(n \log^2(n)/\epsilon^2)$ edges. Our proofs are based on extensions of Karger's contraction algorithm, which may be of independent interest

Vertex Sparsifiers for Hyperedge Connectivity

Recently, Chalermsook et al. [SODA'21(arXiv:2007.07862)] introduces a notion of vertex sparsifiers for $c$-edge connectivity, which has found applications in parameterized algorithms for network design and also led to exciting dynamic algorithms for $c$-edge st-connectivity [Jin and Sun FOCS'21(arXiv:2004.07650)]. We study a natural extension called vertex sparsifiers for $c$-hyperedge connectivity and construct a sparsifier whose size matches the state-of-the-art for normal graphs. More specifically, we show that, given a hypergraph $G=(V,E)$ with $n$ vertices and $m$ hyperedges with $k$ terminal vertices and a parameter $c$, there exists a hypergraph $H$ containing only $O(kc^{3})$ hyperedges that preserves all minimum cuts (up to value $c$) between all subset of terminals. This matches the best bound of $O(kc^{3})$ edges for normal graphs by [Liu'20(arXiv:2011.15101)]. Moreover, $H$ can be constructed in almost-linear $O(p^{1+o(1)} + n(rc\log n)^{O(rc)}\log m)$ time where $r=\max_{e\in E}|e|$ is the rank of $G$ and $p=\sum_{e\in E}|e|$ is the total size of $G$, or in $\text{poly}(m, n)$ time if we slightly relax the size to $O(kc^{3}\log^{1.5}(kc))$ hyperedges.Comment: submitted to ESA 202

Degree-3 Treewidth Sparsifiers

We study treewidth sparsifiers. Informally, given a graph $G$ of treewidth $k$, a treewidth sparsifier $H$ is a minor of $G$, whose treewidth is close to $k$, $|V(H)|$ is small, and the maximum vertex degree in $H$ is bounded. Treewidth sparsifiers of degree $3$ are of particular interest, as routing on node-disjoint paths, and computing minors seems easier in sub-cubic graphs than in general graphs. In this paper we describe an algorithm that, given a graph $G$ of treewidth $k$, computes a topological minor $H$ of $G$ such that (i) the treewidth of $H$ is $\Omega(k/\text{polylog}(k))$; (ii) $|V(H)| = O(k^4)$; and (iii) the maximum vertex degree in $H$ is $3$. The running time of the algorithm is polynomial in $|V(G)|$ and $k$. Our result is in contrast to the known fact that unless $NP \subseteq coNP/{\sf poly}$, treewidth does not admit polynomial-size kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves node-disjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small good-quality vertex-cut sparsifiers that are also minors of the original graph.Comment: Extended abstract to appear in Proceedings of ACM-SIAM SODA 201

Vertex Sparsifiers: New Results from Old Techniques

Given a capacitated graph $G = (V,E)$ and a set of terminals $K \subseteq V$, how should we produce a graph $H$ only on the terminals $K$ so that every (multicommodity) flow between the terminals in $G$ could be supported in $H$ with low congestion, and vice versa? (Such a graph $H$ is called a flow-sparsifier for $G$.) What if we want $H$ to be a "simple" graph? What if we allow $H$ to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier $H$ that maintains congestion up to a factor of $O(\log k/\log \log k)$, where $k = |K|$, (b) a convex combination of trees over the terminals $K$ that maintains congestion up to a factor of $O(\log k)$, and (c) for a planar graph $G$, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in $G$. Moreover, this result extends to minor-closed families of graphs. Our improved bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010. Final version to appear in SIAM J. Computin

Sketching Cuts in Graphs and Hypergraphs

Sketching and streaming algorithms are in the forefront of current research directions for cut problems in graphs. In the streaming model, we show that $(1-\epsilon)$-approximation for Max-Cut must use $n^{1-O(\epsilon)}$ space; moreover, beating $4/5$-approximation requires polynomial space. For the sketching model, we show that $r$-uniform hypergraphs admit a $(1+\epsilon)$-cut-sparsifier (i.e., a weighted subhypergraph that approximately preserves all the cuts) with $O(\epsilon^{-2} n (r+\log n))$ edges. We also make first steps towards sketching general CSPs (Constraint Satisfaction Problems)