Locally Optimal Cuts: Definitions, Methods & Complexity

• Locally optimal cuts are defined via distinct locality notions such as single-vertex flips, seed-biased clustering, and vertexwise objectives.
• The study unifies algorithmic frameworks that guarantee stability under local perturbations with rigorous complexity analyses including NP-hardness and smoothed behavior.
• Applications include Max-Cut, parameterized local search, and local graph clustering, illustrating practical trade-offs between local stability and global performance.

A locally optimal cut is a cut that is optimal only relative to a restricted notion of locality. In the literature, the restriction is not uniform. In local-search formulations of Max-Cut, a cut is locally optimal if no allowed local move—most commonly a single-vertex flip—improves the cut value. In local graph clustering, a cut is locally optimal when it is optimal for a seed-biased conductance objective near a reference set, and the algorithm is “strongly local” when its work depends primarily on the seed region or output size rather than the full graph. In local-guarantee formulations, locality refers instead to vertexwise quality measures such as the maximum disagreement incident on any node. The term is therefore objective-dependent, but in every case it replaces unconstrained global optimality by stability under local perturbations, local bias, or local constraints (Elsaesser et al., 2010, Garvardt et al., 2024, Veldt et al., 2016, Charikar et al., 2017).

1. Formal meanings of locality

Across the cited literature, “local” appears in at least four distinct senses: move-locality, neighborhood-locality, seed-locality, and vertex-local objectives. These senses are related but not interchangeable.

Setting	Locality notion	Representative criterion
Max-Cut local search	Single-vertex FLIP neighborhood	No improving flip exists (Elsaesser et al., 2010)
Parameterized local search	$k$-flip neighborhood	No improving recoloring changing at most $k$ vertices exists (Garvardt et al., 2024)
Local graph clustering	Seed-biased region near $R$	Minimize $\bar{\phi}_R(S)$ near the seed set (Veldt et al., 2016)
Local guarantees	Vertexwise objective	Minimize $\max_u \text{disagree}_{\mathcal S}(u)$ or maximize $\min_u \text{agree}_{\mathcal S}(u)$ (Charikar et al., 2017)

The distinction matters algorithmically. In one line of work, locality defines the admissible perturbations of a current cut; in another, it defines which parts of the graph may be explored; in a third, it defines the objective itself. A locally optimal cut is therefore not a single canonical object but a family of optimization targets indexed by the choice of local structure.

2. Stable cuts and single-vertex local optimality

For an edge-weighted graph $G=(V,E)$ with edge weights $w:E\to \mathbb N$, a cut is a partition $V=V_0\cup V_1$, and its weight is

$$w(V_0,V_1)=\sum_{\substack{uv\in E\ u\in V_0,\ v\in V_1}} w(uv).$$

A cut is stable if every vertex already has at least half of its incident weight crossing the cut. Writing

$k$0

stability means that for every $k$1 and every $k$2,

$k$3

Equivalently, no single vertex can switch sides and increase the total cut weight (Lampis, 2021).

The same notion appears in the FLIP-neighborhood formulation of local Max-Cut. There, a move changes exactly one node’s side of the partition. A node is called happy if flipping it does not increase the cut weight, and a partition is a local optimum if all nodes are happy. This is precisely the single-vertex local-search interpretation of a locally optimal cut (Elsaesser et al., 2010).

This stability notion is sufficiently rich to support secondary optimization problems over the set of local optima. The problem $k$4 asks for a stable cut of minimum weight, i.e. the worst local optimum of the Max-Cut game. The problem is NP-hard in general, weakly NP-hard on trees of diameter $k$5, weakly NP-hard on bipartite graphs of vertex cover $k$6, and strongly NP-hard and APX-hard on bipartite graphs of maximum degree $k$7 in the unweighted case. Exact algorithms exist on bounded treewidth and degree, including a pseudo-polynomial DP with running time

$k$8

and an FPT algorithm running in

$k$9

with matching ETH-based lower bounds up to the stated parameter dependence (Lampis, 2021).

3. Beyond FLIP: $R$0-optimality and parameterized local search

Single-vertex local optimality is only the $R$1 case of a broader neighborhood concept. In $R$2, the input includes a $R$3-coloring $R$4, and one asks whether there exists an improving coloring $R$5 with

$R$6

A coloring is $R$7-optimal if no improving $R$8-neighbor exists. For $R$9, this is a locally optimal cut under the $\bar{\phi}_R(S)$0-flip neighborhood; for $\bar{\phi}_R(S)$1, it is the classical FLIP notion (Garvardt et al., 2024).

A key structural fact is that every inclusion-minimal improving $\bar{\phi}_R(S)$2-flip has a connected flip set. This reduces the search space from arbitrary $\bar{\phi}_R(S)$3-subsets to connected subgraphs of size at most $\bar{\phi}_R(S)$4. Given such a candidate set $\bar{\phi}_R(S)$5, the best recoloring of exactly those vertices can be computed by dynamic programming in

$\bar{\phi}_R(S)$6

time, and the resulting overall algorithm runs in

$\bar{\phi}_R(S)$7

(Garvardt et al., 2024).

The parameterized viewpoint also exposes hardness. $\bar{\phi}_R(S)$8 is W[1]-hard when parameterized by $\bar{\phi}_R(S)$9, even on bipartite $\max_u \text{disagree}_{\mathcal S}(u)$0-degenerate graphs with unit weights. Moreover, even the permissive version admits no $\max_u \text{disagree}_{\mathcal S}(u)$1-time algorithm unless ETH fails. These lower bounds extend to $\max_u \text{disagree}_{\mathcal S}(u)$2 for fixed $\max_u \text{disagree}_{\mathcal S}(u)$3, as well as to related local-search problems such as $\max_u \text{disagree}_{\mathcal S}(u)$4, $\max_u \text{disagree}_{\mathcal S}(u)$5, and $\max_u \text{disagree}_{\mathcal S}(u)$6-Flip Max SAT (Garvardt et al., 2024).

4. Seed-biased locally optimal cuts in local graph clustering

A different use of locality arises in local graph clustering and graph-based learning. Here the input includes a seed set $\max_u \text{disagree}_{\mathcal S}(u)$7, and the objective is not the best cut anywhere in the graph but the best cut near $\max_u \text{disagree}_{\mathcal S}(u)$8. In the flow-based framework of SimpleLocal, the relevant locally biased objective is

$\max_u \text{disagree}_{\mathcal S}(u)$9

where

$\min_u \text{agree}_{\mathcal S}(u)$0

This objective prefers sets that retain much of the seed set while penalizing drift outside it (Veldt et al., 2016).

The algorithmic mechanism is an augmented graph. Starting from $\min_u \text{agree}_{\mathcal S}(u)$1, one adds a source $\min_u \text{agree}_{\mathcal S}(u)$2 and sink $\min_u \text{agree}_{\mathcal S}(u)$3, with capacities determined by $\min_u \text{agree}_{\mathcal S}(u)$4 and by

$\min_u \text{agree}_{\mathcal S}(u)$5

The modified augmented graph $\min_u \text{agree}_{\mathcal S}(u)$6 adds a locality parameter $\min_u \text{agree}_{\mathcal S}(u)$7 to increase sink-side penalties outside $\min_u \text{agree}_{\mathcal S}(u)$8. A central theorem shows that this modification is exactly equivalent to adding an $\min_u \text{agree}_{\mathcal S}(u)$9-type penalty: $G=(V,E)$0 with

$G=(V,E)$1

Thus locality is not merely an implementation feature; it is equivalent to a sparsity-inducing regularizer that discourages large excursions from the seed region (Veldt et al., 2016).

In this setting, strong locality means that the algorithm need not inspect the entire graph. SimpleLocal begins with

$G=(V,E)$2

and expands only around vertices whose sink edges have become saturated. Its 3StageFlow procedure repeatedly expands the local graph, runs a max-flow solver, and updates the residual graph; when the expansion set becomes empty and the flow is nonzero, the resulting flow is a maximum flow on the full modified augmented graph and the returned set is the corresponding minimum $G=(V,E)$3-$G=(V,E)$4 cut. The cut-quality guarantee is also seed-relative: if $G=(V,E)$5, then

$G=(V,E)$6

and more generally, if $G=(V,E)$7 overlaps the seed set sufficiently, then

$G=(V,E)$8

under the stated overlap condition (Veldt et al., 2016).

This seed-biased view is closely related to the earlier cut-improvement problem addressed by LocalImprove. There the input is a set $G=(V,E)$9, and one seeks a set similar to $w:E\to \mathbb N$0 but with smaller conductance. LocalImprove was introduced as the first cut-improvement algorithm that is local in the sense that its running time depends on the size of $w:E\to \mathbb N$1 rather than on the size of the whole graph. It essentially matches the guarantee of the global Andersen–Lang approach and yields an $w:E\to \mathbb N$2 approximation as long as $w:E\to \mathbb N$3. The same abstract contrasts it with earlier local random-walk methods, which guaranteed only $w:E\to \mathbb N$4, and with the improvement $w:E\to \mathbb N$5 of Zhu, Lattanzi, and Mirrokni (Orecchia et al., 2013).

Random-walk locality remains an important baseline. For the small sparsest cut problem, truncated random walks yield local bicriteria algorithms: if a set $w:E\to \mathbb N$6 with $w:E\to \mathbb N$7 and $w:E\to \mathbb N$8 exists, then one can find a set $w:E\to \mathbb N$9 with either

$V=V_0\cup V_1$0

or

$V=V_0\cup V_1$1

and the truncated implementation runs in time almost linear in the explored volume (Kwok et al., 2012). In this branch of the literature, a locally optimal cut is therefore a conductance-optimal or conductance-improved set found near seeds or within small explored regions.

5. Local objectives and per-vertex guarantees

A third interpretation of locality concerns the objective rather than the search region or neighborhood. In correlation clustering, one may measure quality at each vertex instead of globally over all edges. If $V=V_0\cup V_1$2 is a clustering and $V=V_0\cup V_1$3, the local disagreement incident on $V=V_0\cup V_1$4 is

$V=V_0\cup V_1$5

and the local agreement is

$V=V_0\cup V_1$6

This yields the objectives

$V=V_0\cup V_1$7

Here the “local” cut is one with good worst-case behavior at each vertex, not one that is stable under local moves (Charikar et al., 2017).

The prototypical graph-cut specialization is Min Max $V=V_0\cup V_1$8-$V=V_0\cup V_1$9 Cut: find an $$w(V_0,V_1)=\sum_{\substack{uv\in E\ u\in V_0,\ v\in V_1}} w(uv).$$0-$$w(V_0,V_1)=\sum_{\substack{uv\in E\ u\in V_0,\ v\in V_1}} w(uv).$$1 cut minimizing the largest total cut weight incident on any node. This objective differs sharply from ordinary minimum cut, which minimizes total cut capacity. The paper gives an $$w(V_0,V_1)=\sum_{\substack{uv\in E\ u\in V_0,\ v\in V_1}} w(uv).$$2-approximation for minimizing the maximum total weight of disagreement edges incident on a node, a deterministic $$w(V_0,V_1)=\sum_{\substack{uv\in E\ u\in V_0,\ v\in V_1}} w(uv).$$3-approximation for minimizing local disagreements in complete graphs, and a $$w(V_0,V_1)=\sum_{\substack{uv\in E\ u\in V_0,\ v\in V_1}} w(uv).$$4-approximation for maximizing the minimum total weight of agreement edges incident on a node (Charikar et al., 2017).

This line of work also shows why globally natural relaxations are insufficient for local objectives. The natural LP and SDP relaxations for local min-disagreements and local max-agreements have integrality gap $$w(V_0,V_1)=\sum_{\substack{uv\in E\ u\in V_0,\ v\in V_1}} w(uv).$$5. A plausible implication is that locally optimal cuts in the per-vertex sense require techniques that are structurally different from standard global cut rounding, even when the underlying cut family includes classical problems such as Min $$w(V_0,V_1)=\sum_{\substack{uv\in E\ u\in V_0,\ v\in V_1}} w(uv).$$6-$$w(V_0,V_1)=\sum_{\substack{uv\in E\ u\in V_0,\ v\in V_1}} w(uv).$$7 Cut, Multiway Cut, and Multicut (Charikar et al., 2017).

6. Continuous reformulations and iterative local-optimum methods

Local optimality can also be encoded through continuous reformulations. For MaxCut, the SI method uses

$$w(V_0,V_1)=\sum_{\substack{uv\in E\ u\in V_0,\ v\in V_1}} w(uv).$$8

together with the exact equivalence

$$w(V_0,V_1)=\sum_{\substack{uv\in E\ u\in V_0,\ v\in V_1}} w(uv).$$9

The algorithm updates $k$00 by solving analytic subproblems derived from a subgradient linearization of $k$01, and the sequence of objective values is monotone increasing. The local-optimum set is

$k$02

where $k$03 flips the sign of coordinate $k$04. The method converges in finite steps to points in this set, so the resulting cut is locally optimal under the paper’s sign-flip neighborhood (Shao et al., 2018).

An analogous program exists for the anti-Cheeger cut. There the discrete objective

$k$05

is reformulated as

$k$06

with

$k$07

The CIA1 method again uses a Dinkelbach-type iteration plus carefully chosen boundary subgradients. Its main convergence theorem states that after finitely many steps the objective values stabilize and the iterates become local maximizers in the neighborhood defined by coordinate sign changes. The paper also introduces CIA2, which alternates anti-Cheeger iterations with MaxCut iterations to escape local optima (Shao et al., 2021).

These continuous formulations show that locally optimal cuts need not be treated solely as combinatorial endpoints of local search. They can also arise as finite-step local maximizers of exact continuous surrogates whose neighborhood structure is encoded by sign flips or coordinatewise transformations. This suggests a methodological bridge between discrete local search and nonsmooth continuous optimization (Shao et al., 2018, Shao et al., 2021).

7. Complexity landscape and smoothed behavior

The complexity of finding locally optimal cuts depends strongly on the locality model. For FLIP-local Max-Cut, the worst-case picture is sharp: computing a local optimum on graphs of maximum degree five is PLS-complete, and the threshold degree for PLS-completeness is therefore either four or five, unless $k$08. By contrast, in cubic graphs every FLIP local search takes $k$09 improving steps, and on graphs of degree $k$10 every FLIP local search has probably polynomial smoothed complexity under Gaussian perturbations (Elsaesser et al., 2010).

On complete graphs, smoothed analysis gives polynomial bounds for arbitrary implementations of FLIP. Under independent edge weights with densities bounded by $k$11, one result proves that for every $k$12, with high probability any implementation of FLIP terminates in

$k$13

steps (Angel et al., 2016). A later analysis improves this substantially, showing smoothed complexity

$k$14

for Max-Cut in complete graphs, polynomial smoothed complexity

$k$15

for Max-$k$16-Cut in complete graphs, and quasi-polynomial smoothed complexity for Max-$k$17-Cut in arbitrary graphs for every fixed $k$18 (Bibak et al., 2018).

The same section of the literature clarifies an important asymmetry. Existence of a local optimum is trivial for finite local-search spaces, but optimization over local optima may remain hard. $k$19 exemplifies this: stable cuts are easy to define and always exist, yet selecting the minimum-weight stable cut is NP-hard and only becomes fixed-parameter tractable when treewidth and maximum degree are controlled jointly (Lampis, 2021). A plausible implication is that “finding a locally optimal cut” and “optimizing within the set of locally optimal cuts” are computationally different tasks, even when they share the same stability definition.

Taken together, these results portray locally optimal cuts as a broad organizing concept rather than a single theorem schema. In some settings they are Nash-type equilibria under one-vertex moves; in others they are seed-biased minimum cuts of augmented graphs; in still others they are solutions with explicit per-vertex quality guarantees. The common principle is locality of comparison, but the mathematical object depends on whether locality is imposed on moves, explored subgraphs, seeds, or objectives.