--- title: Constrained Max-Cut url: https://www.emergentmind.com/topics/constrained-max-cut type: topic --- # Constrained Max-Cut 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 [2507.12607] [1511.08152] [2308.07063] [2105.01138] [2604.10318]. 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 \(S\) [2507.12607]. ## 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 [2507.12607]. A second major usage is graph-constrained Max-Cut: given a graph \(G=(V,E)\) and a family \(\mathcal{C}_G\subseteq 2^V\) defined by graph structure, the task is \[ \max_{S\in\mathcal{C}_G} c(\delta(S)). \] The families treated explicitly include independent sets, vertex covers, dominating sets, and connected subsets [1511.08152]. A third model is budget-constrained Max-Cut, in which cutting an edge yields both profit \(w_{ij}\) and cost \(c_{ij}\), and the cut must satisfy a single knapsack bound \(\sum c_{ij}x_{ij}\le T\). This is a resource-constrained or knapsack-constrained Max-Cut in edge space [2308.07063]. Robustness yields yet another definition. In fault tolerant Max-Cut, an adversary deletes \(k\) vertices after the cut is chosen. For a cut \(S\subseteq V\), the adaptive objective is \[ \varphi(S,k,G)=\min_{F\in \binom{V}{k}} C_{S-F,G-F}, \] while against an oblivious adversary the objective is defined on a distribution \(\mathcal D\) over cuts by \[ \mu(\mathcal D,k,G)=\min_{F\in\binom{V}{k}}\mathbb E_{S\sim \mathcal D}[C_{S-F,G-F}] . \] The same Max-Cut instance is therefore evaluated through worst-case residual cut value rather than immediate cut value [2105.01138]. 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 \(w(I,I^c)\) rather than by \(|I|\) [2604.10318]. ## 2. Mathematical formulations and encodings The budget-constrained formulation \( \text{(b-maxC)} \) uses edge variables \(x_{ij}\in\{0,1\}\), objective \[ \max \sum_{(i,j)\in E} w_{ij}x_{ij}, \] triangle inequalities \[ x_{ik}+x_{jk}\ge x_{ij},\qquad x_{ij}+x_{ik}+x_{jk}\le 2\qquad (i0\). 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 [2105.01138]. Structural promises do not automatically weaken hardness. Max-Cut remains \(\alpha_{GW}\)-hard on 3-colorable weighted graphs, and this hardness persists even when the 3-coloring is provided as part of the input [2604.10318]. The same paper shows a threshold phenomenon for hidden independent sets. With \[ \alpha^*=\frac{2\rho^*}{\rho^*-1}\approx 0.81597, \] it is UGC-hard to beat \(\alpha_{GW}\) on graphs that contain an independent set \(I\) with \[ w(I,I^c)\ge \alpha^*-\eta, \] while for graphs containing an independent set of volume \(> \alpha^*\) there is an efficient \(>\alpha_{GW}\)-approximation algorithm [2604.10318]. A more geometric constrained variant is rank-\(k\) Max-Cut, denoted Max-Cut\(_k\), in which vertices are assigned unit vectors in \(S_{k-1}\subset \mathbb R^k\) and the objective is \[ \text{Max-Cut}_k(G)= \max_{\vec x\in (S_{k-1})^n} \sum_{ij\in E}\frac12(1-\vec x_i\cdot \vec x_j). \] Here classical Max-Cut is Max-Cut\(_1\). There is a PTAS reduction from Max-Cut\(_k\) to Max-Cut\(_{k+1}\), and Max-Cut\(_k\) is APX-complete for every constant \(k\) [2510.07995]. ## 4. Approximation algorithms For exact multiple cardinality constraints with constant \(c\), the strongest guarantee in the provided corpus is a \((0.858-\varepsilon)\)-approximation. The algorithm constructs an approximate kernel, reducing each part \(V_i\) to \(O(k_i/\varepsilon)\) high-degree vertices plus a super-vertex, and then applies Lasserre-based correlation rounding on the kernel. Its running time is \[ O\!\left(\min\{k/\varepsilon,n\}^{\mathrm{poly}(c/\varepsilon)}+\mathrm{poly}(n)\right), \qquad k=\sum_i k_i, \] and it generalizes the earlier \(0.858-\varepsilon\) guarantee known only for the dense single-cardinality regime \(\min\{k,n-k\}=\Omega(n)\) [2507.12607]. The same paper gives a deterministic \(1/2\)-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 \[ F(x)=\sum_{\{u,v\}\in E} w_e\,(x_u+x_v-2x_ux_v), \] using the inequalities \[ x_u+x_v-2x_ux_v \le \min\{x_u+x_v,\,2-x_u-x_v\} \le 2(x_u+x_v-2x_ux_v) \] to compare the LP optimum with the rounded integral cut [2507.12607]. Graph-constrained Max-Cut on bounded-treewidth graphs admits a generic \(1/2\)-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 \((\tfrac12-\varepsilon)\)-approximations on \(H\)-minor-free graphs for independent set, vertex cover, and dominating set constraints, and on bounded-genus graphs for connected Max-Cut [1511.08152]. The logic-based generalization reaches the same factor. For any fixed \(k\), any MSO-definable constraint on a bounded-treewidth graph yields a \(1/2\)-approximation for Max-\(k\)-Cut, and in the \(k=2\) case this specializes to Max-Cut under arbitrary graph-MSO constraints on one side of the cut [1803.05718]. Fault tolerant Max-Cut admits two distinct guarantees, reflecting the adversary model. For any constant number of failures \(k\), there is a polynomial-time \((0.878-\epsilon)\)-approximation against an adaptive adversary on unweighted graphs, and a \(GW\)-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 [2105.01138]. ## 5. Relaxation hierarchies, LP power, and structured instances Subexponential linear programming becomes relevant once the constraint system is viewed through local pseudodistributions. A degree-\(n^\alpha\) Sherali–Adams relaxation of Max-Cut has size \(\exp(\tilde O(n^\alpha))\) and yields a worst-case approximation ratio at least \[ \frac12+\epsilon_\alpha, \qquad \epsilon_\alpha=\exp(-O(1/\alpha^3)). \] This gives a converse to earlier extension-complexity lower bounds near the \(1/2\) barrier and separates Sherali–Adams from Lovász–Schrijver for approximate Max-Cut [1911.10304]. On graphs of low threshold rank, the same hierarchy is much stronger. If \(r=\mathrm{rank}_\tau(G)\), then degree \[ k\ge \frac{r}{\epsilon^{O(\log(n/r)/\log(1/\tau))}} \] gives an \(\epsilon\)-additive approximation to the Max-Cut value, and the paper highlights the example that \(O(r)\) levels approximate Max-Cut within \(0.01\) on graphs whose normalized adjacency matrices have \(r\) eigenvalues larger than \(n^{-0.01}\) [1911.10304]. That work does not treat balanced Max-Cut, bisection, or explicit side constraints directly. It nevertheless states that Sherali–Adams is defined for general \(2\)-CSPs and for linear constraints in \(0/1\) variables, and the accompanying discussion notes that balance constraints \(\sum_i x_i=0\), 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 [1911.10304]. A different relaxation perspective comes from reducing constrained \(0/1\) optimization to Max-Cut itself. The reformulation \[ \min_{y\in\{-1,1\}^{n+1}} y^TQy \] derived from the penalized and homogenized form \(Q(x,x_0)\) allows the Shor/Goemans–Williamson semidefinite relaxation \[ \min \langle Q,X\rangle \quad\text{s.t.}\quad X\succeq 0,\ X_{ii}=1 \] 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 \(0/1\) knapsack examples, slightly better than the first Lasserre relaxation in most tested cases, always better than the convex quadratic relaxation considered for quadratic \(0/1\) problems, and nearly identical to the first copositive relaxation while using a semidefinite matrix of half the dimension [1505.06840]. ## 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 [2308.07063]. 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 \(207\) seconds on average while the combinatorial algorithm required about \(65\) seconds. It further reports \(906\) instances solved by the algorithm but not by CPLEX within \(3600\) seconds, with average algorithm time \(88.62\) seconds and average \(\%GAP=-14.27\%\), and \(71\) instances unsolved by both within the time limit with average \(\%GAP=-71.57\%\) [2308.07063]. 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 [2308.07063] [1511.08152] [1803.05718]. Several open directions remain explicit in the cited literature. For multiple cardinality constraints, the \((0.858-\varepsilon)\)-approximation currently requires \(c=O(1)\), while the unrestricted-\(c\) regime is supported only by the general \(1/2\)-approximation under matroid constraints and the NP-hardness of deciding whether all edges can be cut [2507.12607]. For fault tolerant Max-Cut, extending the adaptive result to weighted graphs and closing the gap from \(0.878-\epsilon\) to \(\alpha_{GW}\) are highlighted as open [2105.01138]. For graph-MSO constraints, improving the \(1/2\) factor and replacing the current XP dependence on treewidth by an FPT dependence are identified as natural next problems [1803.05718]. For LP hierarchies, the subexponential Sherali–Adams work explicitly asks whether subexponentially-sized LPs offer a nontrivial approximation for every \(2\)-CSP, and whether the phenomenon extends beyond constraint satisfaction problems [1911.10304]. Source: https://www.emergentmind.com/topics/constrained-max-cut