Almost-Linear Time Algorithms for Decremental Graphs: Min-Cost Flow and More via Duality

TL;DR

This work addresses the challenge of maintaining minimum-cost flows and related reachability properties in decremental directed graphs with edge deletions, capacity decreases, and cost increases. It introduces an almost-linear time framework built on a dual, cut-centric view, solving a sequence of dynamic min-ratio cut problems via a new fully dynamic tree-cut sparsifier constructed from capacitated expander decompositions, and analyzed through a dual L1 interior-point method. The key contributions include almost-linear total-time algorithms for decremental thresholded min-cost flow, along with deterministic subroutines for decremental SSR/SCC maintenance and decremental s-t distance, and a simplified, cut-geometry-centric toolkit that also yields improved static min-cost flow runtimes. The methods leverage duality to work with cuts rather than distances, enabling robust performance against adaptive adversaries and delivering subpolynomial update-time guarantees for a broad class of flow and cut problems via tree-based reductions and expander hierarchies.

Abstract

We give the first almost-linear total time algorithm for deciding if a flow of cost at most $F$ still exists in a directed graph, with edge costs and capacities, undergoing decremental updates, i.e., edge deletions, capacity decreases, and cost increases. This implies almost-linear time algorithms for approximating the minimum-cost flow value and $s$-$t$ distance on such decremental graphs. Our framework additionally allows us to maintain decremental strongly connected components in almost-linear time deterministically. These algorithms also improve over the current best known runtimes for statically computing minimum-cost flow, in both the randomized and deterministic settings. We obtain our algorithms by taking the dual perspective, which yields cut-based algorithms. More precisely, our algorithm computes the flow via a sequence of $m^{1+o(1)}$ dynamic min-ratio cut problems, the dual analog of the dynamic min-ratio cycle problem that underlies recent fast algorithms for minimum-cost flow. Our main technical contribution is a new data structure that returns an approximately optimal min-ratio cut in amortized $m^{o(1)}$ time by maintaining a tree-cut sparsifier. This is achieved by devising a new algorithm to maintain the dynamic expander hierarchy of [Goranci-Räcke-Saranurak-Tan, SODA 2021] that also works in capacitated graphs. All our algorithms are deterministc, though they can be sped up further using randomized techniques while still working against an adaptive adversary.

Figures (2)

• Figure 1: 1) shows a tree cut sparsifier $T_{i-1}$ (for a graph $G_{i-1}$). Red vertices are the vertices in $B_i$. The grey components show the connected components of $F(T_{i-1}, B_i)$, edges crossing such components are of minimum capacity on a path in $\mathcal{P}_{T_{i-1}, B_i}$. 2) shows the induced core graph $\mathcal{C}(T_{i-1}, T_{i-1}, B_i)$. 3) shows the induced core graph $\widehat{G}_i = \mathcal{C}(T_{i-1} \cup (I_{i-1} \setminus I_i), T_{i-1}, B_i)$, i.e., the previous graph with all edges that are in $G_i$ but not in $G_{i-1}$ (in green). 4) shows a tree cut sparsifier $\widehat{T}_i$ of the graph $\widehat{G}_i$. 5) shows the final tree cut sparsifier $T_i$ of $G_i$ which is formed by the union of $F(T_{i-1}, B_i)$ and the tree cut sparsifier $\widehat{T}_i$ of the induced core graph $\widehat{G}_i$.
• Figure 3: This figure displays a static directed layered tree experiencing two updates $\textsc{AddDelta}(v_6, \delta)$. Edges $(v_i, v_j)$ with $p(v_i, v_j) = 0$ are not labeled. Each update returns one leaf depicted in red, and causes the significance thresholds to be updated. Notice that for leaf vertices $v$ the threshold $t(v)$ is always $s'(v) + r(v) - c(v)$ where $r(v)$ is the value of $c(v)$ when $v$ was last returned. For internal nodes, $t(v) = \min_{(v, u)} t(u) + p(v, u) - c(v)$ at all times. Every internal vertex has a bold outgoing arrow, which points to the child that has the lowest value in its heap and therefore gives rise to its significance threshold.

Theorems & Definitions (119)

• Definition 1.1
• Definition 1.3: $\alpha$-approximate dynamic min-ratio cut
• Definition 1.4: Tree Cut Sparsifier
• Theorem 1.6
• Theorem 1.7
• proof
• Theorem 1.8
• Corollary 1.9
• proof
• Theorem 1.10