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

Fully dynamic all-pairs shortest paths with worst-case update-time revisited

We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worst-case guarantees on the time needed to process a single update (in contrast to related results, the update time is not amortized over a sequence of updates). Our main result is a simple randomized algorithm that for any parameter $c>1$ has a worst-case update time of $O(cn^{2+2/3} \log^{4/3}{n})$ and answers distance queries correctly with probability $1-1/n^c$, against an adaptive online adversary if the graph contains no negative cycle. The best deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time of $\tilde O(n^{2+3/4})$ and assumes non-negative weights. This is the first improvement for this problem for more than a decade. Conceptually, our algorithm shows that randomization along with a more direct approach can provide better bounds.Comment: To be presented at the Symposium on Discrete Algorithms (SODA) 201

Dynamic Spanning Trees for Connectivity Queries on Fully-dynamic Undirected Graphs (Extended version)

Answering connectivity queries is fundamental to fully dynamic graphs where edges and vertices are inserted and deleted frequently. Existing work proposes data structures and algorithms with worst-case guarantees. We propose a new data structure, the dynamic tree (D-tree), together with algorithms to construct and maintain it. The D-tree is the first data structure that scales to fully dynamic graphs with millions of vertices and edges and, on average, answers connectivity queries much faster than data structures with worst case guarantees

Fast reoptimization for the minimum spanning tree problem

AbstractWe study reoptimization versions of the minimum spanning tree problem. The reoptimization setting can generally be formulated as follows: given an instance of the problem for which we already know some optimal solution, and given some “small” perturbations on this instance, is it possible to compute a new (optimal or at least near-optimal) solution for the modified instance without ex nihilo computation? We focus on two kinds of modifications: node-insertions and node-deletions. When k new nodes are inserted together with their incident edges, we mainly propose a fast strategy with complexity O(kn) which provides a max{2,3−(2/(k−1))}-approximation ratio, in complete metric graphs and another one that is optimal with complexity O(nlogn). On the other hand, when k nodes are deleted, we devise a strategy which in O(n) achieves approximation ratio bounded above by 2⌈|Lmax|/2⌉ in complete metric graphs, where Lmax is the longest deleted path and |Lmax| is the number of its edges. For any of the approximation strategies, we also provide lower bounds on their approximation ratios