A Distributed Mincut/Maxflow Algorithm Combining Path Augmentation and Push-Relabel

We develop a novel distributed algorithm for the minimum cut problem. We primarily aim at solving large sparse problems. Assuming vertices of the graph are partitioned into several regions, the algorithm performs path augmentations inside the regions and updates of the push-relabel style between the regions. The interaction between regions is considered expensive (regions are loaded into the memory one-by-one or located on separate machines in a network). The algorithm works in sweeps - passes over all regions. Let $B$ be the set of vertices incident to inter-region edges of the graph. We present a sequential and parallel versions of the algorithm which terminate in at most $2|B|^2+1$ sweeps. The competing algorithm by Delong and Boykov uses push-relabel updates inside regions. In the case of a fixed partition we prove that this algorithm has a tight $O(n^2)$ bound on the number of sweeps, where $n$ is the number of vertices. We tested sequential versions of the algorithms on instances of maxflow problems in computer vision. Experimentally, the number of sweeps required by the new algorithm is much lower than for the Delong and Boykov's variant. Large problems (up to $10^8$ vertices and $6\cdot 10^8$ edges) are solved using under 1GB of memory in about 10 sweeps.Comment: 40 pages, 15 figure

Analyzing Data-Centric Properties for Graph Contrastive Learning

Recent analyses of self-supervised learning (SSL) find the following data-centric properties to be critical for learning good representations: invariance to task-irrelevant semantics, separability of classes in some latent space, and recoverability of labels from augmented samples. However, given their discrete, non-Euclidean nature, graph datasets and graph SSL methods are unlikely to satisfy these properties. This raises the question: how do graph SSL methods, such as contrastive learning (CL), work well? To systematically probe this question, we perform a generalization analysis for CL when using generic graph augmentations (GGAs), with a focus on data-centric properties. Our analysis yields formal insights into the limitations of GGAs and the necessity of task-relevant augmentations. As we empirically show, GGAs do not induce task-relevant invariances on common benchmark datasets, leading to only marginal gains over naive, untrained baselines. Our theory motivates a synthetic data generation process that enables control over task-relevant information and boasts pre-defined optimal augmentations. This flexible benchmark helps us identify yet unrecognized limitations in advanced augmentation techniques (e.g., automated methods). Overall, our work rigorously contextualizes, both empirically and theoretically, the effects of data-centric properties on augmentation strategies and learning paradigms for graph SSL.Comment: Accepted to NeurIPS 202

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions

Approximating Minimum Cost Connectivity Orientation and Augmentation

We investigate problems addressing combined connectivity augmentation and orientations settings. We give a polynomial-time 6-approximation algorithm for finding a minimum cost subgraph of an undirected graph $G$ that admits an orientation covering a nonnegative crossing $G$-supermodular demand function, as defined by Frank. An important example is $(k,\ell)$-edge-connectivity, a common generalization of global and rooted edge-connectivity. Our algorithm is based on a non-standard application of the iterative rounding method. We observe that the standard linear program with cut constraints is not amenable and use an alternative linear program with partition and co-partition constraints instead. The proof requires a new type of uncrossing technique on partitions and co-partitions. We also consider the problem setting when the cost of an edge can be different for the two possible orientations. The problem becomes substantially more difficult already for the simpler requirement of $k$-edge-connectivity. Khanna, Naor, and Shepherd showed that the integrality gap of the natural linear program is at most $4$ when $k=1$ and conjectured that it is constant for all fixed $k$. We disprove this conjecture by showing an $\Omega(|V|)$ integrality gap even when $k=2$

Fully Dynamic Matching in Bipartite Graphs

Maximum cardinality matching in bipartite graphs is an important and well-studied problem. The fully dynamic version, in which edges are inserted and deleted over time has also been the subject of much attention. Existing algorithms for dynamic matching (in general graphs) seem to fall into two groups: there are fast (mostly randomized) algorithms that do not achieve a better than 2-approximation, and there slow algorithms with \O(\sqrt{m}) update time that achieve a better-than-2 approximation. Thus the obvious question is whether we can design an algorithm -- deterministic or randomized -- that achieves a tradeoff between these two: a $o(\sqrt{m})$ approximation and a better-than-2 approximation simultaneously. We answer this question in the affirmative for bipartite graphs. Our main result is a fully dynamic algorithm that maintains a 3/2 + \eps approximation in worst-case update time O(m^{1/4}\eps^{/2.5}). We also give stronger results for graphs whose arboricity is at most \al, achieving a (1+ \eps) approximation in worst-case time O(\al (\al + \log n)) for constant \eps. When the arboricity is constant, this bound is $O(\log n)$ and when the arboricity is polylogarithmic the update time is also polylogarithmic. The most important technical developement is the use of an intermediate graph we call an edge degree constrained subgraph (EDCS). This graph places constraints on the sum of the degrees of the endpoints of each edge: upper bounds for matched edges and lower bounds for unmatched edges. The main technical content of our paper involves showing both how to maintain an EDCS dynamically and that and EDCS always contains a sufficiently large matching. We also make use of graph orientations to help bound the amount of work done during each update.Comment: Longer version of paper that appears in ICALP 201

Dynamic Maxflow via Dynamic Interior Point Methods

In this paper we provide an algorithm for maintaining a $(1-\epsilon)$-approximate maximum flow in a dynamic, capacitated graph undergoing edge additions. Over a sequence of $m$-additions to an $n$-node graph where every edge has capacity $O(\mathrm{poly}(m))$ our algorithm runs in time $\widehat{O}(m \sqrt{n} \cdot \epsilon^{-1})$. To obtain this result we design dynamic data structures for the more general problem of detecting when the value of the minimum cost circulation in a dynamic graph undergoing edge additions obtains value at most $F$ (exactly) for a given threshold $F$. Over a sequence $m$-additions to an $n$-node graph where every edge has capacity $O(\mathrm{poly}(m))$ and cost $O(\mathrm{poly}(m))$ we solve this thresholded minimum cost flow problem in $\widehat{O}(m \sqrt{n})$. Both of our algorithms succeed with high probability against an adaptive adversary. We obtain these results by dynamizing the recent interior point method used to obtain an almost linear time algorithm for minimum cost flow (Chen, Kyng, Liu, Peng, Probst Gutenberg, Sachdeva 2022), and introducing a new dynamic data structure for maintaining minimum ratio cycles in an undirected graph that succeeds with high probability against adaptive adversaries.Comment: 30 page