Cut Tree Construction from Massive Graphs

The construction of cut trees (also known as Gomory-Hu trees) for a given graph enables the minimum-cut size of the original graph to be obtained for any pair of vertices. Cut trees are a powerful back-end for graph management and mining, as they support various procedures related to the minimum cut, maximum flow, and connectivity. However, the crucial drawback with cut trees is the computational cost of their construction. In theory, a cut tree is built by applying a maximum flow algorithm for $n$ times, where $n$ is the number of vertices. Therefore, naive implementations of this approach result in cubic time complexity, which is obviously too slow for today's large-scale graphs. To address this issue, in the present study, we propose a new cut-tree construction algorithm tailored to real-world networks. Using a series of experiments, we demonstrate that the proposed algorithm is several orders of magnitude faster than previous algorithms and it can construct cut trees for billion-scale graphs.Comment: Short version will appear at ICDM'1

Hierarchies of Predominantly Connected Communities

We consider communities whose vertices are predominantly connected, i.e., the vertices in each community are stronger connected to other community members of the same community than to vertices outside the community. Flake et al. introduced a hierarchical clustering algorithm that finds such predominantly connected communities of different coarseness depending on an input parameter. We present a simple and efficient method for constructing a clustering hierarchy according to Flake et al. that supersedes the necessity of choosing feasible parameter values and guarantees the completeness of the resulting hierarchy, i.e., the hierarchy contains all clusterings that can be constructed by the original algorithm for any parameter value. However, predominantly connected communities are not organized in a single hierarchy. Thus, we develop a framework that, after precomputing at most $2(n-1)$ maximum flows, admits a linear time construction of a clustering \C(S) of predominantly connected communities that contains a given community $S$ and is maximum in the sense that any further clustering of predominantly connected communities that also contains $S$ is hierarchically nested in \C(S). We further generalize this construction yielding a clustering with similar properties for $k$ given communities in $O(kn)$ time. This admits the analysis of a network's structure with respect to various communities in different hierarchies.Comment: to appear (WADS 2013

Minimum Cuts in Geometric Intersection Graphs

Let $\mathcal{D}$ be a set of $n$ disks in the plane. The disk graph $G_\mathcal{D}$ for $\mathcal{D}$ is the undirected graph with vertex set $\mathcal{D}$ in which two disks are joined by an edge if and only if they intersect. The directed transmission graph $G^{\rightarrow}_\mathcal{D}$ for $\mathcal{D}$ is the directed graph with vertex set $\mathcal{D}$ in which there is an edge from a disk $D_1 \in \mathcal{D}$ to a disk $D_2 \in \mathcal{D}$ if and only if $D_1$ contains the center of $D_2$. Given $\mathcal{D}$ and two non-intersecting disks $s, t \in \mathcal{D}$, we show that a minimum $s$-$t$ vertex cut in $G_\mathcal{D}$ or in $G^{\rightarrow}_\mathcal{D}$ can be found in $O(n^{3/2}\text{polylog} n)$ expected time. To obtain our result, we combine an algorithm for the maximum flow problem in general graphs with dynamic geometric data structures to manipulate the disks. As an application, we consider the barrier resilience problem in a rectangular domain. In this problem, we have a vertical strip $S$ bounded by two vertical lines, $L_\ell$ and $L_r$, and a collection $\mathcal{D}$ of disks. Let $a$ be a point in $S$ above all disks of $\mathcal{D}$, and let $b$ a point in $S$ below all disks of $\mathcal{D}$. The task is to find a curve from $a$ to $b$ that lies in $S$ and that intersects as few disks of $\mathcal{D}$ as possible. Using our improved algorithm for minimum cuts in disk graphs, we can solve the barrier resilience problem in $O(n^{3/2}\text{polylog} n)$ expected time.Comment: 11 pages, 4 figure

Polynomial time algorithms for multicast network code construction

The famous max-flow min-cut theorem states that a source node s can send information through a network (V, E) to a sink node t at a rate determined by the min-cut separating s and t. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re-encode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures

A New Push-Relabel Algorithm for Sparse Networks

In this paper, we present a new push-relabel algorithm for the maximum flow problem on flow networks with $n$ vertices and $m$ arcs. Our algorithm computes a maximum flow in $O(mn)$ time on sparse networks where $m = O(n)$. To our knowledge, this is the first $O(mn)$ time push-relabel algorithm for the $m = O(n)$ edge case; previously, it was known that push-relabel implementations could find a max-flow in $O(mn)$ time when $m = \Omega(n^{1+\epsilon})$ (King, et. al., SODA `92). This also matches a recent flow decomposition-based algorithm due to Orlin (STOC `13), which finds a max-flow in $O(mn)$ time on sparse networks. Our main result is improving on the Excess-Scaling algorithm (Ahuja & Orlin, 1989) by reducing the number of nonsaturating pushes to $O(mn)$ across all scaling phases. This is reached by combining Ahuja and Orlin's algorithm with Orlin's compact flow networks. A contribution of this paper is demonstrating that the compact networks technique can be extended to the push-relabel family of algorithms. We also provide evidence that this approach could be a promising avenue towards an $O(mn)$-time algorithm for all edge densities.Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:1309.2525 - This version includes an extension of the result to the O(n) edge cas