Deterministic Local Minimum Cut Algorithm

• The topic 'Deterministic Local Minimum Cut Algorithm' is a framework that deterministically enumerates or approximates minimum cuts near a given vertex in undirected graphs with rigorous runtime guarantees.
• It employs methods like deterministic forest packing and explicit color coding to isolate cut sets, replacing randomness with exhaustive yet efficient enumeration.
• By eliminating randomization, the approach offers predictable, verifiable performance and integrates with fast dynamic min-cut and global cut algorithms in graph theory.

A deterministic local minimum cut algorithm is an algorithmic framework for enumerating or approximating all minimum cuts near a given vertex in an undirected graph, with rigorous runtime and correctness guarantees, where all sources of randomness are replaced by explicit deterministic constructions. This approach lies at the core of recent advances in fast fully-dynamic min-cut algorithms, overcoming the reliance on randomization that historically dominated both classical and local min-cut techniques (El-Hayek et al., 15 Dec 2025, Henzinger et al., 2017, Kawarabayashi et al., 2014, Gawrychowski et al., 2020).

1. Formal Problem Statement and Preliminaries

Given an undirected, unweighted graph $G=(V,E)$, a vertex $v\in V$, a volume bound $\nu\geq1$, a cut-size bound $s\geq1$, and approximation parameter $\beta\geq1$, the local minimum cut problem seeks all connected vertex-sets $S\subseteq V$ with $v\in S$, $G[S]$ connected, $$\operatorname{vol}_G(S):=\sum_{u\in S}\deg_G(u)\leq\nu$$, $|E(S,V\setminus S)|\leq s$, and the cut is within a $\beta$-approximation of the minimum cut separating $S$ from $V\setminus S$. The task is to efficiently enumerate all such sets $S$ (El-Hayek et al., 15 Dec 2025).

Auxiliary definitions include:

• $S$ $k$-respects a spanning tree $T$ if $|E(S,V\setminus S)\cap E(T)|\leq k$.
• $(1-\delta)$-boundary-sparse $S$ in cluster $C$: $$|E(S,C\setminus S)| < (1-\delta)\cdot \min\{|E(S,V\setminus C)|, |E(C\setminus S,V\setminus C)|\}$$ (El-Hayek et al., 15 Dec 2025).

2. Algorithmic Structure: Deterministic LocalKCut

The current deterministic state-of-the-art local minimum cut procedure, as introduced by El-Hayek, Henzinger, and Li, proceeds via the following elements (El-Hayek et al., 15 Dec 2025):

• Forest Packing: Maintain a dynamic set $\mathcal{F}$ of $k=O(s\log m)$ minimum spanning forests computed greedily (per Karger–Thorup). This ensures that all $\beta$-approximate minimum cuts are $2\beta$-respected by some $T\in\mathcal{F}$.
• Color Coding via Explicit Set Families: Construct two 2-coloring set families $F_1$ (red/blue, size $2^{O(\beta\log\nu)}\log m$) and $F_2$ (green/yellow, size $2^{O(\beta\log s)}\log m$), following Chitnis et al., ensuring for all relevant small subsets a coloring isolates the cut/interior edges.
• Traversal: For each $T\in\mathcal{F}$, and each $(\chi_1,\chi_2)\in F_1\times F_2$, build subgraph $H$ of $G'$ (the relevant dynamic subgraph) containing blue edges of $T$ under $\chi_1$ and green non-tree edges under $\chi_2$. Perform truncated DFS/BFS from $v$ up to volume $\nu$. All connected $S$ found with $|E_{G'}(S,V\setminus S)|\leq s$ and $\operatorname{vol}_{G'}(S)\leq\nu$ are output.

The key determinism comes from exhaustive, but quasi-polynomially bounded, enumeration of all (forest, coloring pair) combinations, entirely removing randomness. For every $\beta$-approximate minimum cut near $v$, some configuration deterministically isolates it via this process (El-Hayek et al., 15 Dec 2025).

Simplified Pseudocode

def LocalKCutQuery(v): S = set() for T in F: for χ1 in F1: for χ2 in F2: H = blue_edges(T, χ1) ∪ green_edges(G' \ T, χ2) S_candidate = DFS(H, v, volume_limit=ν) if is_connected(G'[S_candidate]) and vol_G'(S_candidate) <= ν and cut_size(S_candidate) <= s: S.add(S_candidate) return S

3. Key Theoretical Foundations and Guarantees

The deterministic local minimum cut algorithm builds on several combinatorial and data-structural principles:

Tree-respecting cuts: For any $\beta$-approximate minimum cut in $G'$, a (1+$\epsilon$)-approximate minimum spanning forest packing guarantees the existence of a tree $T$ in which the cut crosses at most $2\beta$ edges (El-Hayek et al., 15 Dec 2025, Gawrychowski et al., 2020).

Color-coding coverage: The constructed coloring families $F_1$ and $F_2$ ensure:

• For every $F_r$ (crossing) and $F_b$ (interior) edge subset of requisite size, a coloring exists isolating $F_r$ (as red) and $F_b$ (as blue).
• Similarly, $F_g$ (internal connectors) and $F_y$ (cut-edges) can be colored green and yellow, ensuring the traversal in $H$ reconstructs exactly $S$.

Invariants: At every invocation, all connected sets containing $v$ and satisfying the volume, cut-size, and approximation requirements are returned—no missed cuts, no false positives (El-Hayek et al., 15 Dec 2025).

4. Complexity and Implementation Analysis

Component	Complexity (per operation)	Source
Forest packing ($\mathcal{F}$)	$\widetilde O(s^2\log^2 m) = n^{o(1)}$ amortized	(El-Hayek et al., 15 Dec 2025)
Coloring families	$|F_1| = 2^{O(\beta\log\nu)}\log m$; $|F_2| = 2^{O(\beta\log s)}\log m$	(El-Hayek et al., 15 Dec 2025)
Query (per $v$)	$k\,|F_1|\,|F_2|\,O(\nu)=n^{o(1)}$ (for $s,\nu=n^{o(1)},\,\beta=O(1)$)	(El-Hayek et al., 15 Dec 2025)
Edge updates	$n^{o(1)}$ amortized per update	(El-Hayek et al., 15 Dec 2025)

The algorithm achieves $n^{o(1)}$ amortized time per operation for cut sizes up to $s=2^{\Theta(\log^{3/4-c} n)}$. The exhaustive enumeration of coloring combinations, leveraged with structural sparsification via cluster/expander decompositions, ensures overall subpolynomial amortized update and query time in the dynamic setting.

5. Determinism Compared to Prior Work

Earlier fully-dynamic or local minimum cut algorithms relied either on randomized tree-packing (Karger) or random walk–based local separations. The prior randomized LocalKCut subroutines required repeated random walk/forest growth from $v$, succeeding only with high probability after $\operatorname{poly}(s)$ attempts (El-Hayek et al., 15 Dec 2025). Deterministic improvements:

• Derandomized forest packing: Every step of the forest-packing is performed greedily, not via probabilistic choices.
• Deterministic color coding: Explicit coloring-set constructions, as in Chitnis et al., replace probabilistic hash/isolation methods.
• Combinatorial coverage: All $(T,\chi_1,\chi_2)$ configurations are exhaustively checked, exploiting the fact that the number is only $n^{o(1)}$ for the relevant parameter ranges.

This transition removes Las Vegas and Monte Carlo behaviors, yielding predictable and verifiable per-query and per-update costs, critical for worst-case guarantees (El-Hayek et al., 15 Dec 2025, Kawarabayashi et al., 2014, Henzinger et al., 2017).

6. Illustrative Example and Broader Context

Consider a 6-node expander $C$ attached by 10 boundary edges to a larger graph, with $v \in C$. For a cut $S\subset C$ of size $s=4$ and volume $\nu=8$:

1. The forest packing contains spanning trees of $C$; some $T_j$ has $\leq 8$ crossing and $\leq8$ interior edges.
2. $F_1$ and $F_2$ colorings enable coloring of all such edge subsets aligning with $S$.
3. A truncated BFS from $v$ in $H$ (blue and green edges only) reconstructs $S$ exactly, and the volume/cut-size filters exclude supersets.
4. All steps involve only deterministic enumerations and checking; the guarantee is that all valid $S$ are output.

This local deterministic construction can be incorporated recursively into dynamic cluster decompositions and expander-based frameworks, yielding the overall subpolynomial exact minimum cut algorithm (El-Hayek et al., 15 Dec 2025).

7. Relation to Other Deterministic Local and Global Min-Cut Algorithms

Deterministic local minimum cut algorithms are a critical ingredient in the most advanced deterministic global min-cut algorithms:

• Fully-dynamic and local-to-global frameworks: The approach of recursive decomposition and “local” certificate/cut search via deterministic procedures is present both in the near-linear time global cut methods of Kawarabayashi–Thorup (Kawarabayashi et al., 2014) and in flow-based partitioning approaches (Henzinger et al., 2017), where deterministic local routines replace random walk/diffusion Monte Carlo steps.
• Bounded-degree and LOCAL model: In the bounded-degree, bounded-capacity setting, deterministic constant-time local algorithms construct $(1+\epsilon)$-approximate minimum fractional (and via randomized rounding, integral) cuts, emphasizing the universality of local deterministic methods (Csóka et al., 2010).
• 2-respecting and structure-aware reductions: In global min-cut routines relying on tree-respecting reductions (such as those improving on Karger’s random contraction), deterministic 2D range-search and centroid enumeration replace prior random path sampling, achieving deterministic $O(m\log n + n\log^2 n)$ for the 2-respect subproblem of min-cut (Gawrychowski et al., 2020).

These approaches collectively shift the landscape from randomized Las Vegas/Monté Carlo paradigms to deterministic, structure-exploiting, and efficiently enumerable local routines, making deterministic dynamic and near-linear min-cut feasible in both theory and practice.

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References:

• El-Hayek, Henzinger & Li, "Deterministic and Exact Fully-dynamic Minimum Cut of Superpolylogarithmic Size in Subpolynomial Time" (El-Hayek et al., 15 Dec 2025)
• Henzinger, Rao & Wang, "Local Flow Partitioning for Faster Edge Connectivity" (Henzinger et al., 2017)
• Kawarabayashi & Thorup, "Deterministic Edge Connectivity in Near-Linear Time" (Kawarabayashi et al., 2014)
• Gawrychowski, Mozes & Weimann, "A Note on a Recent Algorithm for Minimum Cut" (Gawrychowski et al., 2020)
• Csóka, Pongrácz & Lippner, "Local algorithms for the maximum flow and minimum cut in bounded-degree networks" (Csóka et al., 2010)