Tight Lower Bounds for Directed Cut Sparsification and Distributed Min-Cut

TL;DR

This work establishes tight ε-dependent lower bounds for two core cut-approximation problems on directed and local-graph models. Using communication-complexity reductions (notably Index and Gap-Hamming) and carefully designed β-balanced graph constructions, the authors obtain a $\tilde{\Omega}(n \sqrt{\beta}/\varepsilon)$ lower bound for for-each cut sketches and a $\Omega(n \beta/\varepsilon^2)$ lower bound for for-all cut sketches, resolving open questions in the directed setting. They also derive a tight lower bound for the local-query min-cut problem, proving $\Omega(\min\{m, m/(\varepsilon^2 k)\})$ queries are necessary, and complement this with an almost matching upper bound that achieves $\tilde{O}(m/(\varepsilon^2 k))$ queries. Collectively, these results close the parameter regimes for ε-dependence and demonstrate near-optimal trade-offs between sketch size, accuracy, and computational access models, with implications for distributed minimum-cut estimation and graph sketching in large-scale systems.

Abstract

In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is to approximate cuts in balanced directed graphs. In this problem, the goal is to build a data structure that $(1 \pm ε)$-approximates cut values in graphs with $n$ vertices. For arbitrary directed graphs, such a data structure requires $Ω(n^2)$ bits even for constant $ε$. To circumvent this, recent works study $β$-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most $β$ times that in the other direction. We consider two models: the {\em for-each} model, where the goal is to approximate each cut with constant probability, and the {\em for-all} model, where all cuts must be preserved simultaneously. We improve the previous $Ω(n \sqrt{β/ε})$ lower bound to $\tildeΩ(n \sqrtβ/ε)$ in the for-each model, and we improve the previous $Ω(n β/ε)$ lower bound to $Ω(n β/ε^2)$ in the for-all model. This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is to approximate the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We improve the previous $Ω\bigl(\frac{m}{k}\bigr)$ query complexity lower bound to $Ω\bigl(\min\{m, \frac{m}{ε^2 k}\}\bigr)$ for this problem, where $m$ is the number of edges, $k$ is the size of the minimum cut, and we seek a $(1+ε)$-approximation. In addition, we show that existing upper bounds with slight modifications match our lower bound up to logarithmic factors.

Figures (6)

• Figure 1: For $S = A \cup(R \setminus B)$, the (directed) edges from $S$ to $(V \setminus S)$ consist of the following: the forward edges from $A$ to $B$, each with weight $\Theta(\log(1/\varepsilon))$, and the backward edges from $(R \setminus B)$ to $(L \setminus A)$, each with weight $1/\beta$.
• Figure 2: Example of $G_{x,y}(V,E)$ where $x = 000000100$ and $y = 100010100$. The red edges represent the intersection at $x_{31} = y_{31} = 1$. The green edges represent all the non-intersections in $x$ and $y$.
• Figure 3: $u, v \in A$. We omit all the $(a_i, b_j')$, $(b_i, b_j')$, and $(b_i, a_j')$ edges.
• Figure 4: $u \in A, v \in A'$. We omit all the $(a_i, b_j')$, $(b_i, b_j')$, and $(b_i, a_j')$ edges. The green edges exist since $v$ has at least $2\gamma$ neighbors in $A$. The orange edges exist since $u^{A}_i$ and $u$ have at least $2\gamma$ common neighbors in $A'$.
• Figure 5: $u \in A, v \in B'$. The first set of paths $S_1$ goes from $u \rightarrow u_i \rightarrow u'_i \rightarrow w_i \rightarrow x_i$. We omit the paths from $x_i$ to $v$, as they are symmetric to the paths from $w_i$ to $u$. Once we extend the paths from $x_i$ to $v$, we have $\gamma$ edge-disjoint paths from $u$ to $v$. Note that the $w_i$ and $x_i$ may not be distinct.

Theorems & Definitions (34)

• Theorem 1.1: For-Each Cut Sketch for Balanced Graphs
• Theorem 1.2: For-All Cut Sketch for Balanced Graphs
• Theorem 1.3: Approximating Min-Cut using Local Queries
• Definition 2.1: $\beta$-Balanced Graphs
• Definition 2.2: For-All Cut Sketch
• Definition 2.3: For-Each Cut Sketch
• Theorem 3.1: For-Each Cut Sketch for Balanced Graphs
• Lemma 3.1: KNR01
• Lemma 3.2
• proof