Cover–Relax–Search: A Heuristic for Constrained BQP

• The paper introduces CRS, a heuristic combining a graph cover, NLP relaxation, and restricted sub-BQP search to quickly produce feasible incumbents.
• CRS reduces quadratic nonlinearity by selectively fixing cover variables using confidence measures from the relaxation phase, enhancing search efficiency.
• Empirical results demonstrate CRS’s competitive primal-integral improvements over branch-and-bound, Undercover, and Relax–Search on various benchmark instances.

Cover–Relax–Search (CRS) denotes, in its primary and most specific current usage, a primal heuristic for constrained binary quadratic programs (BQPs) of the form

$$\begin{aligned} \min\; & x^\top H x + c^\top x \ \text{s.t. } & A x \le B, \ & x \in \{0,1\}^n, \end{aligned}$$

where $H$ is real symmetric and not necessarily positive semidefinite, so the objective may be nonconvex. The method combines a graph-theoretic cover of quadratic interactions, an NLP relaxation over $x \in [0,1]^n$, and a restricted sub-BQP search neighborhood. In this form, CRS is presented as a start heuristic for large BQPs, intended to produce feasible incumbents quickly and to improve the primal integral earlier than exact branch-and-bound alone (Huang et al., 9 Jan 2025).

1. Problem class and motivation

The target class is the constrained BQP, a family that includes classical problems such as quadratic assignment, stable set, and vertex coloring, and that also appears in finance, energy systems, logistics, and machine learning. Its difficulty comes from two sources that compound each other: the combinatorial explosion induced by $x \in \{0,1\}^n$, which yields $2^n$ possible assignments, and the nonlinearity of $x^\top H x$, which becomes nonconvex when $H$ is indefinite. The paper emphasizes that even identifying a feasible solution can be hard when the constraints $Ax \le B$ are tight and highly coupled, when the continuous relaxation has fractional points far from any feasible integer point, or when branch-and-bound for MINLP or MIQP explores substantial parts of the tree before reaching a feasible node (Huang et al., 9 Jan 2025).

Within this setting, primal heuristics serve a specific role. They are not intended to certify optimality; rather, they try to generate feasible solutions quickly, improve the incumbent during the search, reduce the optimality gap earlier, and improve the primal integral. For large BQPs, exact branch-and-bound alone may be too slow or may fail to produce strong incumbents early, which motivates heuristics that exploit the quadratic structure and the binary domain more directly (Huang et al., 9 Jan 2025).

2. Three-phase structure

CRS is organized around the phases named in its title: cover, relax, and search. The method begins from the Hessian graph $G=(V,E)$, whose vertices correspond to variables and whose edges connect pairs of variables that appear together in a quadratic term. A vertex cover $C\subseteq B$ of this graph identifies variables whose fixing intersects every quadratic interaction. This is the structural reduction step: fixing variables from $H$0 reduces the number of surviving quadratic terms in the subproblem (Huang et al., 9 Jan 2025).

The relaxation phase solves the nonlinear programming relaxation obtained by replacing $H$1 with $H$2. Let $H$3 denote the resulting fractional solution. CRS then evaluates the confidence of each cover variable through

$H$4

Large $H$5 indicates a relaxed value closer to $H$6 or $H$7, hence a more decisive rounding signal. If $H$8 is the fixing ratio, the algorithm sets

$H$9

and selects the subset $x \in [0,1]^n$0 of size $x \in [0,1]^n$1 with the largest $x \in [0,1]^n$2 values. These variables are fixed to their rounded relaxation values $x \in [0,1]^n$3 (Huang et al., 9 Jan 2025).

The search phase substitutes those fixed values into the original BQP and solves the resulting restricted subproblem over the remaining binary variables. The resulting subproblem is still a BQP, not necessarily an ILP, because CRS fixes only part of the cover. This is the central departure from Undercover. Undercover fixes all cover variables and removes all original nonlinear terms, yielding a purely linear ILP subproblem; CRS retains some nonlinearity in exchange for a larger neighborhood and, empirically, better early primal performance. The paper also compares CRS to a Relax–Search baseline that fixes the same number of variables using fractionality alone but does not restrict the candidates to the cover set, and therefore does not explicitly target nonlinearity reduction (Huang et al., 9 Jan 2025).

3. Mathematical formulation

The vertex cover used in the first phase is itself computed by solving

$x \in [0,1]^n$4

If all variables in $x \in [0,1]^n$5 were fixed, every quadratic term $x \in [0,1]^n$6 would contain at least one fixed variable. CRS does not enforce that stronger condition; it uses the cover more selectively as a source of candidate fixings (Huang et al., 9 Jan 2025).

The NLP relaxation solved in the second phase is

$x \in [0,1]^n$7

Because $x \in [0,1]^n$8 may be indefinite, this relaxation is still potentially nonconvex. The method does not require a global optimum of the relaxation; it uses the computed relaxation point as guidance for choosing which cover variables to fix (Huang et al., 9 Jan 2025).

After fixing $x \in [0,1]^n$9, CRS solves a reduced sub-BQP over $x \in \{0,1\}^n$0,

$x \in \{0,1\}^n$1

where $x \in \{0,1\}^n$2 arise from substitution. Feasibility is not imposed by repair or projection; it is enforced by solving this subproblem as an exact constrained optimization problem, so the returned solution satisfies the original constraints once the fixed values are reinserted. No formal guarantees are given that CRS will always find a feasible solution within time, converge to an optimal solution, or dominate other heuristics in final objective value; its claims are empirical and are especially framed in terms of primal integral (Huang et al., 9 Jan 2025).

4. Implementation and parameterization

The implementation described in the paper is external to SCIP but uses SCIP 8.0.1 for all three optimization tasks: solving the vertex-cover ILP, solving the NLP relaxation, and solving the restricted sub-BQP via MIQP or MINLP branch-and-bound. CRS is presented as a start heuristic: it does not require an incumbent and is intended to run before or early in a branch-and-bound process (Huang et al., 9 Jan 2025).

The experimental configuration fixes a total time budget of $x \in \{0,1\}^n$3 s per run. Cover computation is limited to $x \in \{0,1\}^n$4 s, the NLP relaxation to $x \in \{0,1\}^n$5 s, and the sub-BQP receives the remaining time, approximately $x \in \{0,1\}^n$6 s. Variable selection inside the restricted subproblem is left to SCIP’s default branch-and-bound settings; CRS only determines the presolve-style fixing pattern before handing the subproblem to the solver (Huang et al., 9 Jan 2025).

The fixing ratio $x \in \{0,1\}^n$7 is the principal tuning parameter. The paper explores $x \in \{0,1\}^n$8 and tunes it per instance group to minimize the final primal bound after $x \in \{0,1\}^n$9 s. The reported choices are $2^n$0 for UBQP-, QMKP-, and kQKP-small instances with $2^n$1 variables, $2^n$2 for the corresponding large instances with $2^n$3 variables, and $2^n$4 for the MIPcc25 BQPs. For fair comparison, Relax–Search is always configured to fix the same number of variables as CRS, though drawn from all binary variables rather than from the cover (Huang et al., 9 Jan 2025).

5. Benchmarks and empirical behavior

The evaluation uses four benchmark families: UBQP, QMKP, kQKP, and a subset of BQPs from the MIP Workshop 2025 Competition. For each of UBQP, QMKP, and kQKP, the experiments use $2^n$5 small instances with $2^n$6 binary variables and $2^n$7 large instances with $2^n$8 binary variables, with quadratic density approximately $2^n$9 and $x^\top H x$0. The MIPcc25 subset contains $x^\top H x$1 large BQP instances with more than $x^\top H x$2 binary variables, genuine quadratic cross terms, and an average of about $x^\top H x$3 binary variables. The baselines are SCIP branch-and-bound with default settings, Undercover, and Relax–Search. The metrics are time to first feasible solution, number of instances with at least one feasible solution, and primal integral at $x^\top H x$4 s (Huang et al., 9 Jan 2025).

On the synthetic benchmarks, CRS finds a feasible solution for all instances. Its start time is larger than SCIP branch-and-bound because cover computation and relaxation consume time before the restricted subproblem is solved, but the reported primal-integral improvements are much stronger. On the MIPcc25 instances, all methods find feasible solutions on all $x^\top H x$5 instances. The paper also notes an important nuance: CRS is often, but not always, best in final primal bound at $x^\top H x$6 s; Relax–Search can occasionally obtain slightly better final objective values, but typically with slower early improvement (Huang et al., 9 Jan 2025).

Benchmark	vs BnB PI reduction (%)	vs Relax–Search / Undercover PI reduction (%)
UBQP-Small	99.78	24.86 / 73.92
UBQP-Large	99.93	64.28 / 64.32
QMKP-Small	95.19	55.23 / 73.28
QMKP-Large	97.69	56.89 / 89.61
kQKP-Small	95.09	58.61 / 81.41
kQKP-Large	97.63	49.66 / 83.21
MIPcc25	41.89	89.88 / 18.78

These figures characterize the main empirical regime claimed for CRS. It is particularly strong on the large synthetic BQPs, and it remains positive on the more heterogeneous MIPcc25 set, though the gains over Undercover are more modest there. The paper additionally reports that Undercover sometimes fails to find feasible solutions within the time limit on the more constrained knapsack-type problems QMKP and kQKP, whereas CRS does not have this issue in the reported experiments (Huang et al., 9 Jan 2025).

6. Interpretation, limitations, and common confusions

A common misunderstanding is to equate CRS with full linearization. That is not its design. The method intentionally fixes only a fraction of the cover variables. The limiting case $x^\top H x$7 reproduces Undercover’s full-cover fixing strategy and yields an ILP subproblem with no original nonlinear terms, whereas $x^\top H x$8 leaves some quadratic terms in place and thus preserves a larger neighborhood (Huang et al., 9 Jan 2025).

Another potential misunderstanding is to treat the cover phase as computationally negligible in theory because it is “only” preprocessing. The paper states that the vertex-cover problem used in the cover phase is itself NP-hard, although in practice its time is small relative to the rest of CRS, and the implementation caps it at $x^\top H x$9 s. The overall method remains purely heuristic: there is no guarantee of optimality, no guarantee that it will always return a feasible solution within the budget, and no theorem establishing dominance over alternative heuristics in final objective value (Huang et al., 9 Jan 2025).

The paper also identifies several practical limitations. Performance depends on the quality of the NLP relaxation; if the relaxation is poor or highly fractional, the $H$0 scores may be misleading. The fixing ratio $H$1 matters and varies by problem class. The time allocation between cover, relaxation, and restricted search also affects the trade-off. Finally, the authors explicitly hypothesize that subproblems with more remaining nonlinearity, as in Relax–Search, may sometimes permit lower attainable objective values at the end of the run because they are less restricted, even though CRS usually has much better primal-integral behavior (Huang et al., 9 Jan 2025).

7. Broader uses of the expression

The expression “Cover–Relax–Search” is also used in several adjacent literatures as a methodological pattern rather than as the name of the BQP heuristic. In parameterized local search for Vertex Cover, the phrase denotes a transition from a covering problem to a relaxed local-search requirement and then to an FPT search procedure. There, strict $H$2-exchange local search for Vertex Cover is $H$3-hard even on $H$4-subdivided graphs, whereas permissive local search is FPT on $H$5-separable graphs, with running time $H$6 for $H$7; the paper explicitly describes this phenomenon as a “Cover–Relax–Search” effect (Gaspers et al., 2012).

In heuristic planning, the phrase labels a different architecture: complete $H$8 provides a baseline heuristic that covers all states, relaxed search in the $H$9-regression space computes partial $Ax \le B$0 information for $Ax \le B$1, and a final regression search uses the combined heuristic table. In that setting, the comparison between TP4 and HSP$Ax \le B$2 shows that relaxed search can improve admissible heuristics in some IPC-2004 domains but not all, especially when expanding small states is cheap and small states tend to have small successor states (Haslum, 2011).

In stochastic search theory, the same triad appears as a design perspective for exhaustive exploration. The cover phase aims to visit many new sites, the relax or reset phase relocates the searcher to avoid overexploring depleted regions, and the search phase repeats this cycle. For Markovian, non-compact random walks in large domains, the full cover time satisfies

$Ax \le B$3

and the parameter values that minimize the global mean first-passage time $Ax \le B$4 also minimize mean cover time. This is illustrated for persistent random walks, Lévy walks, intermittent strategies, and random walks on Erdős–Rényi networks (Chupeau et al., 2015).

In statistical mechanics of the minimum vertex cover problem on Erdős–Rényi random graphs, the cover–relax–search pattern is realized through a penalty formulation. The relaxed Hamiltonian

$Ax \le B$5

permits infeasible states with uncovered edges, weighted by a penalty $Ax \le B$6. The analysis shows that $Ax \le B$7 enforces feasibility in the zero-temperature ground state, $Ax \le B$8 permits infeasible ground states, the cover and penalty ratios exhibit devil’s-staircase-like discontinuities at rational $Ax \le B$9, and constraint relaxation expands the RS region while lowering the critical temperature (Dote et al., 2023).

In approximation algorithms for Partial Cover, the triad takes the form cover formulation, Lagrangian relaxation of the global profit constraint, and search over the multiplier $G=(V,E)$0. The relaxation produces a prize-collecting cover problem. For black-box use of an $G=(V,E)$1-LMP prize-collecting algorithm, the paper proves a $G=(V,E)$2 barrier. For Partial Totally Balanced Cover, however, a non-black-box analysis based on Kolen’s primal–dual algorithm, threshold search in $G=(V,E)$3, and structured merging of two neighboring solutions yields the bound

$G=(V,E)$4

for any integer $G=(V,E)$5 (0712.3936).

Taken together, these works suggest that “Cover–Relax–Search” has acquired a broader technical meaning alongside its specific BQP sense: begin from a covering or exhaustive-search objective, relax a hard structural requirement, and then search within the relaxed formulation or between relaxed solutions to recover good feasible solutions or improved algorithmic behavior.