Constrained Max-Cut

• Constrained Max-Cut is a variant of the classical problem that integrates restrictions such as cardinality, graph-based constraints, and budget limitations.
• It features diverse formulations including linear, quadratic, and semidefinite relaxations that enable dynamic programming and advanced rounding techniques.
• Approximation algorithms and LP hierarchies are employed to overcome NP-hard challenges, with extensions addressing fault tolerance and robust adversarial scenarios.

Constrained Max-Cut denotes a family of extensions of the classical Max-Cut problem in which the cut objective is optimized over a restricted feasible region or against an adversarial post-processing rule. In the literature represented here, the term covers several non-equivalent models: exact cardinality constraints on one or more vertex classes, graph-based feasibility constraints such as independence, domination, vertex cover, or connectivity, knapsack-type budgets on cut edges, matroid base constraints, robustness to vertex failures, and structural promises such as 3-colorability or the existence of a large independent set (Makarychev et al., 16 Jul 2025, Lee et al., 2015, Puerto et al., 2023, Censor-Hillel et al., 2021, Ghoshal et al., 11 Apr 2026). A standard weighted cut is written as

$$\delta_w(S)=\sum_{\substack{u\in S,\,v\in V\setminus S}} w(\{u,v\}),$$

and constrained variants retain this objective while modifying admissibility or evaluation of $S$ (Makarychev et al., 16 Jul 2025).

1. Scope of the subject

A direct cardinality-constrained version fixes a partition

$V=\biguplus_{i\in[c]} V_i$

and integers $k_1,\dots,k_c$, and asks for a set $S\subseteq V$ such that

$|S\cap V_i|=k_i\quad \forall i\in[c]$

maximizing $\delta_w(S)$. In this model, $c=1$ gives the single-cardinality problem Max-Cut$_k$, and the special case $c=1,\ |S|=n/2$ is Max-Bisection (Makarychev et al., 16 Jul 2025).

A second major usage is graph-constrained Max-Cut: given a graph $S$0 and a family $S$1 defined by graph structure, the task is

$S$2

The families treated explicitly include independent sets, vertex covers, dominating sets, and connected subsets (Lee et al., 2015).

A third model is budget-constrained Max-Cut, in which cutting an edge yields both profit $S$3 and cost $S$4, and the cut must satisfy a single knapsack bound $S$5. This is a resource-constrained or knapsack-constrained Max-Cut in edge space (Puerto et al., 2023).

Robustness yields yet another definition. In fault tolerant Max-Cut, an adversary deletes $S$6 vertices after the cut is chosen. For a cut $S$7, the adaptive objective is

$S$8

while against an oblivious adversary the objective is defined on a distribution $S$9 over cuts by

$V=\biguplus_{i\in[c]} V_i$0

The same Max-Cut instance is therefore evaluated through worst-case residual cut value rather than immediate cut value (Censor-Hillel et al., 2021).

The structural-promise viewpoint studies unrestricted cuts on restricted graph classes. The papers summarized here isolate two such promises: 3-colorability, and the existence of a large independent set measured by edge volume $V=\biguplus_{i\in[c]} V_i$1 rather than by $V=\biguplus_{i\in[c]} V_i$2 (Ghoshal et al., 11 Apr 2026).

2. Mathematical formulations and encodings

The budget-constrained formulation $V=\biguplus_{i\in[c]} V_i$3 uses edge variables $V=\biguplus_{i\in[c]} V_i$4, objective

$V=\biguplus_{i\in[c]} V_i$5

triangle inequalities

$V=\biguplus_{i\in[c]} V_i$6

and the budget inequality

$V=\biguplus_{i\in[c]} V_i$7

The triangle constraints are the classical cut-polytope inequalities reused from the LP relaxation of Max-Cut, while the budget constraint is a single knapsack restriction (Puerto et al., 2023).

For bounded-treewidth graphs with logical side conditions, the formulation can instead be logical and dynamic-programmatic. An MSO formula $V=\biguplus_{i\in[c]} V_i$8 specifies feasible parts of a $V=\biguplus_{i\in[c]} V_i$9-partition, and the constrained problem becomes

$k_1,\dots,k_c$0

The crucial structural fact is that, for a graph $k_1,\dots,k_c$1 of treewidth $k_1,\dots,k_c$2, the CSP induced by $k_1,\dots,k_c$3 admits a CSP extension $k_1,\dots,k_c$4 with

$k_1,\dots,k_c$5

which makes a bounded-state dynamic program available on a tree decomposition (Koutecký et al., 2018).

A different formulation absorbs constraints into the objective. A linear or quadratic $k_1,\dots,k_c$6 program

$k_1,\dots,k_c$7

can be transformed, after the change of variables to $k_1,\dots,k_c$8, into an unconstrained Max-Cut-type quadratic optimization by using the penalized polynomial

$k_1,\dots,k_c$9

and then homogenizing it to a quadratic form $S\subseteq V$0 on $S\subseteq V$1 variables. The associated graph has an edge $S\subseteq V$2 exactly when the corresponding coefficient $S\subseteq V$3 is nonzero; its sparsity is controlled by $S\subseteq V$4 (Lasserre, 2015).

For matroid-constrained Max-Cut, the relevant convex formulation is over the base polytope $S\subseteq V$5 of the matroid. The LP used in the cited work is

$S\subseteq V$6

subject to

$S\subseteq V$7

This formulation separates the matroid feasibility carried by $S\subseteq V$8 from the cut surrogate carried by $S\subseteq V$9 (Makarychev et al., 16 Jul 2025).

3. Complexity and hardness landscape

Budget-constrained Max-Cut is NP-Complete. The paper states this as Proposition 3.3 and motivates it by combining the hardness of classical Max-Cut with the hardness of Binary Knapsack (Puerto et al., 2023).

For multiple exact cardinality constraints, the complexity picture depends sharply on the number of parts. When $|S\cap V_i|=k_i\quad \forall i\in[c]$0 is unrestricted, it is NP-hard to decide whether there exists a feasible solution $|S\cap V_i|=k_i\quad \forall i\in[c]$1 satisfying all cardinality constraints and cutting all edges: $|S\cap V_i|=k_i\quad \forall i\in[c]$2 The proof is by reduction from Perfect 3D Matching and shows that even extreme-value feasibility becomes hard once many constraints are imposed (Makarychev et al., 16 Jul 2025).

Fault tolerant Max-Cut inherits the classical $|S\cap V_i|=k_i\quad \forall i\in[c]$3 barrier. Assuming the Unique Games Conjecture and $|S\cap V_i|=k_i\quad \forall i\in[c]$4, there is no polynomial-time $|S\cap V_i|=k_i\quad \forall i\in[c]$5-approximation for fault tolerant Max-Cut, for either adaptive or oblivious adversaries and any constant $|S\cap V_i|=k_i\quad \forall i\in[c]$6. The same work proves hardness by reducing classical Max-Cut to a one-fault robust instance in which a single added high-degree vertex becomes the critical failure point (Censor-Hillel et al., 2021).

Structural promises do not automatically weaken hardness. Max-Cut remains $|S\cap V_i|=k_i\quad \forall i\in[c]$7-hard on 3-colorable weighted graphs, and this hardness persists even when the 3-coloring is provided as part of the input (Ghoshal et al., 11 Apr 2026). The same paper shows a threshold phenomenon for hidden independent sets. With

$|S\cap V_i|=k_i\quad \forall i\in[c]$8

it is UGC-hard to beat $|S\cap V_i|=k_i\quad \forall i\in[c]$9 on graphs that contain an independent set $\delta_w(S)$0 with

$\delta_w(S)$1

while for graphs containing an independent set of volume $\delta_w(S)$2 there is an efficient $\delta_w(S)$3-approximation algorithm (Ghoshal et al., 11 Apr 2026).

A more geometric constrained variant is rank-$\delta_w(S)$4 Max-Cut, denoted Max-Cut$\delta_w(S)$5, in which vertices are assigned unit vectors in $\delta_w(S)$6 and the objective is

$\delta_w(S)$7

Here classical Max-Cut is Max-Cut$\delta_w(S)$8. There is a PTAS reduction from Max-Cut$\delta_w(S)$9 to Max-Cut$c=1$0, and Max-Cut$c=1$1 is APX-complete for every constant $c=1$2 (Piddock, 9 Oct 2025).

4. Approximation algorithms

For exact multiple cardinality constraints with constant $c=1$3, the strongest guarantee in the provided corpus is a $c=1$4-approximation. The algorithm constructs an approximate kernel, reducing each part $c=1$5 to $c=1$6 high-degree vertices plus a super-vertex, and then applies Lasserre-based correlation rounding on the kernel. Its running time is

$c=1$7

and it generalizes the earlier $c=1$8 guarantee known only for the dense single-cardinality regime $c=1$9 (Makarychev et al., 16 Jul 2025).

The same paper gives a deterministic $_k$0-approximation for Max-Cut under an arbitrary matroid base constraint. The method solves the base-polytope LP and then applies pipage rounding to the quadratic form

$_k$1

using the inequalities

$_k$2

to compare the LP optimum with the rounded integral cut (Makarychev et al., 16 Jul 2025).

Graph-constrained Max-Cut on bounded-treewidth graphs admits a generic $_k$3-approximation whenever the graph constraint supports an exact dynamic program for linear objectives on a tree decomposition. The explicit constraint families handled in this framework are independent set, vertex cover, dominating set, and connectivity. Using decomposition theorems, the same paper obtains $_k$4-approximations on $_k$5-minor-free graphs for independent set, vertex cover, and dominating set constraints, and on bounded-genus graphs for connected Max-Cut (Lee et al., 2015).

The logic-based generalization reaches the same factor. For any fixed $_k$6, any MSO-definable constraint on a bounded-treewidth graph yields a $_k$7-approximation for Max-$_k$8-Cut, and in the $_k$9 case this specializes to Max-Cut under arbitrary graph-MSO constraints on one side of the cut (Koutecký et al., 2018).

Fault tolerant Max-Cut admits two distinct guarantees, reflecting the adversary model. For any constant number of failures $c=1,\ |S|=n/2$0, there is a polynomial-time $c=1,\ |S|=n/2$1-approximation against an adaptive adversary on unweighted graphs, and a $c=1,\ |S|=n/2$2-approximation against an oblivious adversary on weighted graphs. The adaptive algorithm combines a heavy-vertex reduction with simultaneous Max-Cut, while the oblivious algorithm uses LP duality, the ellipsoid method, and a derandomized Goemans–Williamson oracle (Censor-Hillel et al., 2021).

5. Relaxation hierarchies, LP power, and structured instances

Subexponential linear programming becomes relevant once the constraint system is viewed through local pseudodistributions. A degree-$c=1,\ |S|=n/2$3 Sherali–Adams relaxation of Max-Cut has size $c=1,\ |S|=n/2$4 and yields a worst-case approximation ratio at least

$c=1,\ |S|=n/2$5

This gives a converse to earlier extension-complexity lower bounds near the $c=1,\ |S|=n/2$6 barrier and separates Sherali–Adams from Lovász–Schrijver for approximate Max-Cut (Hopkins et al., 2019).

On graphs of low threshold rank, the same hierarchy is much stronger. If $c=1,\ |S|=n/2$7, then degree

$c=1,\ |S|=n/2$8

gives an $c=1,\ |S|=n/2$9-additive approximation to the Max-Cut value, and the paper highlights the example that $S$00 levels approximate Max-Cut within $S$01 on graphs whose normalized adjacency matrices have $S$02 eigenvalues larger than $S$03 (Hopkins et al., 2019).

That work does not treat balanced Max-Cut, bisection, or explicit side constraints directly. It nevertheless states that Sherali–Adams is defined for general $S$04-CSPs and for linear constraints in $S$05 variables, and the accompanying discussion notes that balance constraints $S$06, cardinality constraints, and degree constraints can be encoded as polynomial inequalities in the relaxation. It also identifies the unconstrained component-flipping step as the main obstruction to immediate extensions to globally balanced variants. This suggests a route to subexponential LP-based algorithms for constrained Max-Cut, but not a complete transfer theorem (Hopkins et al., 2019).

A different relaxation perspective comes from reducing constrained $S$07 optimization to Max-Cut itself. The reformulation

$S$08

derived from the penalized and homogenized form $S$09 allows the Shor/Goemans–Williamson semidefinite relaxation

$S$10

to serve as a lower bound for the original constrained problem. The reported comparisons state that this Max-Cut-based SDP is much better than the LP bound in most small $S$11 knapsack examples, slightly better than the first Lasserre relaxation in most tested cases, always better than the convex quadratic relaxation considered for quadratic $S$12 problems, and nearly identical to the first copositive relaxation while using a semidefinite matrix of half the dimension (Lasserre, 2015).

6. Special cases, applications, and open directions

Some constrained variants admit strong simplifications on special graph classes. For budget-constrained Max-Cut on a tree, the problem is trivial: the optimum is the edge of maximum weight whose cost is below the budget. For separable graphs with articulation points, the optimum budget-constrained Max-Cut is the maximum of the budget-constrained Max-Cut values of the blocks. The same paper also notes that the usual max-flow/min-cut duality breaks under budget constraints (Puerto et al., 2023).

Budget-constrained Max-Cut has also been studied from the exact-algorithmic side. The cited work develops a branch-and-bound method driven by an adapted Stoer–Wagner procedure, BudgetedCut, and reports that on instances solved by both approaches the IP formulation required about $S$13 seconds on average while the combinatorial algorithm required about $S$14 seconds. It further reports $S$15 instances solved by the algorithm but not by CPLEX within $S$16 seconds, with average algorithm time $S$17 seconds and average $S$18, and $S$19 instances unsolved by both within the time limit with average $S$20 (Puerto et al., 2023).

The application range is correspondingly broad. Budget-constrained cuts are connected in the paper to attack strategies on networks with a budget constraint and to scheduling applications, while graph-constrained and logic-constrained formulations cover clustering, connected partitioning, and sparse-graph partition problems (Puerto et al., 2023, Lee et al., 2015, Koutecký et al., 2018).

Several open directions remain explicit in the cited literature. For multiple cardinality constraints, the $S$21-approximation currently requires $S$22, while the unrestricted-$S$23 regime is supported only by the general $S$24-approximation under matroid constraints and the NP-hardness of deciding whether all edges can be cut (Makarychev et al., 16 Jul 2025). For fault tolerant Max-Cut, extending the adaptive result to weighted graphs and closing the gap from $S$25 to $S$26 are highlighted as open (Censor-Hillel et al., 2021). For graph-MSO constraints, improving the $S$27 factor and replacing the current XP dependence on treewidth by an FPT dependence are identified as natural next problems (Koutecký et al., 2018). For LP hierarchies, the subexponential Sherali–Adams work explicitly asks whether subexponentially-sized LPs offer a nontrivial approximation for every $S$28-CSP, and whether the phenomenon extends beyond constraint satisfaction problems (Hopkins et al., 2019).