Quantum Classical Branch-and-Price

• Quantum Classical Branch-and-Price (QCBP) is a hybrid optimization framework that integrates quantum subroutines with classical branch-and-price, retaining decomposition methods while delegating bottleneck tasks to quantum processors.
• It leverages diverse quantum models such as QAOA, QUBO, and adiabatic algorithms to solve subproblems like set-partitioning, pricing, and cut separation across various applications including Tail Assignment, EV charging, and Vehicle Routing.
• Empirical studies show that QCBP can reduce runtime and improve solution quality for large-scale integer programs, though its effectiveness is highly dependent on problem structure, penalty calibrations, and hardware capabilities.

Quantum Classical Branch-and-Price (QCBP) denotes a family of hybrid optimization architectures that augment classical Branch-and-Price, or branch-price-and-cut, with quantum, quantum-annealing-inspired, or quantum-speedup subroutines for selected bottlenecks. In the current literature, those bottlenecks include solving restricted master problems as $0$–$1$ set-partitioning models, solving pricing subproblems, solving cut-separation subproblems, and accelerating branch-and-bound tree exploration itself. The unifying design principle is not a single fixed algorithmic template, but the retention of a classical decomposition-and-search backbone—column generation, branching, bounding, pruning, and incumbent management—while delegating structurally compatible subproblems to Ising-, QUBO-, or oracle-based quantum machinery (Svensson et al., 2021, Wagner et al., 2024, Montanaro, 2019, Vercellino et al., 26 Aug 2025, Sun et al., 22 Mar 2026).

1. Classical substrate and problem structure

The classical core of QCBP is Branch-and-Price (B&P), which combines Column Generation (CG) and Branch-and-Bound (B&B). In its canonical set-partitioning form, the master problem uses columns $j \in J$ representing feasible routes or patterns, rows $i \in I$ representing items to cover, and a binary incidence matrix $A=(a_{ij})$ with $a_{ij}\in\{0,1\}$. The master integer program is

$\min \sum_{j \in J} c_j x_j$

subject to exact coverage

$\sum_{j \in J} a_{ij}x_j = 1 \quad \forall i\in I,$

with $x_j \in \{0,1\}$. Its LP relaxation replaces integrality by $x_j \ge 0$. Solving the restricted master problem (RMP) yields dual variables $1$0, and pricing generates columns with negative reduced cost

$1$1

CG iterates between solving the current RMP LP and solving pricing until no column with $1$2 remains. If the LP-optimal master solution is fractional, branching recursively partitions the feasible RMP solution space; pruning occurs by bound, infeasibility, or integrality (Svensson et al., 2021).

This substrate persists across application domains, but the master and pricing semantics vary. Tail Assignment uses a set-partitioning master over aircraft routes, with pricing formulated as an RCSPP under connection times, maintenance windows, curfews, and preassignments (Svensson et al., 2021). Vehicle Routing uses an exponentially sized set-cover or set-partitioning master over feasible capacity-respecting routes, together with rounded capacity cuts in a branch-price-and-cut setting (Wagner et al., 2024). Intra-day EV charging scheduling is modeled as a Partition Coloring Problem (PCP), where columns are independent sets of a conflict graph and the RMP minimizes makespan $1$3 (Sun et al., 22 Mar 2026). Vertex Coloring uses the standard set-partitioning master over all independent sets of a graph, with one column per color class (Vercellino et al., 26 Aug 2025).

A recurring misconception is that QCBP replaces Branch-and-Price by a quantum optimizer. The literature instead treats the classical decomposition as indispensable: dual extraction, lower-bound computation, incumbent management, and tree control remain classical in all practical variants currently described (Svensson et al., 2021, Wagner et al., 2024).

2. Algebraic formulations and quantum encodings

In the Tail Assignment formulation, the quantum subroutine targets the RMP as a $1$4–$1$5 set-partitioning model. The paper encodes Set Partitioning or Exact Cover into a penalty Hamiltonian

$1$6

with

$1$7

After expanding the quadratic penalties and applying the spin transformation $1$8 with $1$9, the model becomes an Ising Hamiltonian

$j \in J$0

where the local fields $j \in J$1 encode cost and cover counts, and the couplers $j \in J$2 penalize overlap of columns covering the same row. This mapping is central to the QAOA-based QCBP formulation, and it makes the penalty-balancing parameter $j \in J$3 an algorithmically critical quantity (Svensson et al., 2021).

For EV charging scheduling, the pricing problem is a maximum-weight independent set with partition constraints on a conflict graph $j \in J$4. Dual variables $j \in J$5 and $j \in J$6 induce vertex contributions

$j \in J$7

and a candidate column $j \in J$8 has reduced cost

$j \in J$9

The corresponding QUBO drops charger-use variables and uses a penalized objective

$i \in I$0

where $i \in I$1 carries the dual-weighted objective, $i \in I$2 enforces one interval per vehicle, and $i \in I$3 penalizes conflicts on the same charger. Feasibility is restored after the QUBO solve by greedy repair: for a conflict $i \in I$4, the vertex with larger $i \in I$5 is retained, and for each partition only the max-$i \in I$6 vertex is kept (Sun et al., 22 Mar 2026).

For Vehicle Routing, both pricing and separation are mapped to QUBOs. Pricing is formulated as a capacitated price-collecting TSP (CPCTSP) with visit-at-time variables $i \in I$7, customer-incidence variables $i \in I$8, and binary capacity-encoding variables $i \in I$9. The reduced-cost information is folded into modified edge costs

$A=(a_{ij})$0

and feasibility is enforced by a penalty parameter $A=(a_{ij})$1 satisfying

$A=(a_{ij})$2

Rounded-capacity-cut separation is likewise encoded as a QUBO with subset variables $A=(a_{ij})$3 and route-incidence variables $A=(a_{ij})$4; for that formulation, $A=(a_{ij})$5 suffices and the experiments use $A=(a_{ij})$6 (Wagner et al., 2024).

For Vertex Coloring, pricing is a Maximum-Weight Independent Set (MWIS) problem with reduced cost

$A=(a_{ij})$7

so negative reduced cost is equivalent to $A=(a_{ij})$8. The quantum implementation uses a Rydberg-atom Hamiltonian

$A=(a_{ij})$9

in which the Rydberg blockade enforces independence and site-dependent detunings encode the dual weights $a_{ij}\in\{0,1\}$0 (Vercellino et al., 26 Aug 2025).

These encodings show that “QCBP” is not tied to one physics model. The literature spans Ising spin-glass mappings, generic QUBOs, and analog neutral-atom Hamiltonians. This suggests that the defining criterion is architectural placement within Branch-and-Price rather than a specific quantum ansatz.

3. Placement of the quantum subroutine

The locus of quantum intervention varies substantially across QCBP instantiations.

Setting	Quantum target	Classical remainder
Tail Assignment	Integer solutions of the RMP via QAOA	CG, RCSPP pricing, branching, postprocessing (Svensson et al., 2021)
EV charging scheduling	Pricing as MIS QUBO via BSB or SimCIM	RMP by Gurobi, branching, repair, tree search (Sun et al., 22 Mar 2026)
Vehicle Routing	Pricing and cut separation as QUBOs	RMP, exact fallback, branch-price-and-cut control (Wagner et al., 2024)
Vertex Coloring	MWIS pricing via QAA on neutral-atom QPU	RMP, branching on maximal ISs, primal heuristic (Vercellino et al., 26 Aug 2025)
Oracle-based theoretical lift	Global tree search and counting	Classical B&P semantics encoded in $a_{ij}\in\{0,1\}$1, $a_{ij}\in\{0,1\}$2, $a_{ij}\in\{0,1\}$3 (Montanaro, 2019)

In Tail Assignment, the quantum component is inserted as a primal heuristic at selected CG iterations. Given a promising RMP instance, the current master is encoded into a Hamiltonian and QAOA is run to sample integer RMP solutions. Feasible high-quality samples can tighten the incumbent upper bound, improve pruning by bound, and guide branching or fixing decisions. Pricing remains classical because, in that application, it is an RCSPP with intricate operational constraints (Svensson et al., 2021).

In EV charging scheduling, QCBP places the nonclassical component in pricing rather than in the master. The RMP is solved by Gurobi, while pricing is cast as a MIS QUBO and handled by MindQuantum’s QAIA solvers, specifically ballistic simulated branching (BSB) and simulated coherent Ising machine (SimCIM). Branching constraints are propagated back into pricing by forbidding or forcing vertices and updating the conflict graph and partitions (Sun et al., 22 Mar 2026).

In Vehicle Routing, the same hybrid logic is extended to branch-price-and-cut. Pricing returns routes with negative reduced cost, and separation returns violated rounded capacity cuts. A distinctive design choice is to exploit the randomness of the quantum heuristic by accepting all distinct solutions below a cost threshold, not only the best one. This multi-solution strategy is used both for route generation and cut generation (Wagner et al., 2024).

In the Vertex Coloring implementation, the quantum device is used only for MWIS pricing. Branching is performed exclusively on maximal independent sets discovered during pricing, and a classical primal heuristic builds feasible colorings from the accumulated column pool without extra ILP or QPU calls (Vercellino et al., 26 Aug 2025).

The theoretical formulation based on Montanaro’s branch-and-bound speedup differs from the heuristic variants. There, the branch-and-price node state is abstracted as a tree with a branch oracle and a monotone cost oracle, and quantum search and tree-size estimation are used to accelerate global exploration of the truncated tree (Montanaro, 2019).

4. Representative problem classes and algorithmic workflows

In Tail Assignment, feasible aircraft routes form the columns and flights are the rows. The master selects a set of legal routes that covers each flight exactly once at minimum cost, while pricing generates new legal routes with negative reduced cost using dual information from the master. The QCBP workflow keeps the column-generation machinery, pricing via RCSPP, dual stabilization, and branching or fixing heuristics classical, and uses QAOA only to solve the current RMP as a $a_{ij}\in\{0,1\}$4–$a_{ij}\in\{0,1\}$5 set-partitioning model. Preprocessing reduces RMP size, and postprocessing filters infeasible samples and optionally performs local improvements (Svensson et al., 2021).

In intra-day EV charging scheduling, each vehicle induces a partition of feasible charging intervals, vertices represent intervals, and conflict edges encode either intra-vehicle exclusivity or inter-vehicle temporal overlap. The RMP minimizes the latest completion time $a_{ij}\in\{0,1\}$6 over a dynamically growing family of independent-set columns. Column generation constructs dual-weighted vertex scores $a_{ij}\in\{0,1\}$7, pricing solves the associated MIS QUBO, and repaired negative-reduced-cost independent sets are added back to the RMP. Branching includes subset-based branching, which forces or forbids an interval for a selected vehicle, and pairwise branching, which either adds a conflict edge or merges a co-occurring pair into a super-vertex (Sun et al., 22 Mar 2026).

In Vehicle Routing, the classical master contains route variables $a_{ij}\in\{0,1\}$8 and may use either set-cover or set-partitioning semantics. Pricing seeks a route with minimum reduced cost, and cut separation seeks violated rounded capacity cuts. The QCBP formulation maps the pricing problem to a CPCTSP QUBO and the separation problem to a fractional-capacity-cut oracle QUBO. Each node of the branch-and-bound tree alternates between RMP solution, quantum pricing, quantum separation, and exact fallback if no improving column or cut is found. The accepted pricing outputs are

$a_{ij}\in\{0,1\}$9

which are filtered for feasibility, deduplicated, and added in batch (Wagner et al., 2024).

In Vertex Coloring, the master problem chooses independent sets as color classes. Pricing is MWIS under dual weights, implemented by quantum adiabatic algorithms on neutral-atom registers. The workflow initializes the RMP with singletons or other feasible independent sets, solves the RMP LP, invokes QAA pricing, and adds all columns with negative reduced cost. If quantum pricing yields none, a classical MWIS solve with GLPK certifies that no improving column exists. Branching then fixes a selected maximal independent set to one color, removes its vertices and incident edges, and recurses on the reduced subgraph. The lower bound at a node of depth $\min \sum_{j \in J} c_j x_j$0 is

$\min \sum_{j \in J} c_j x_j$1

and pruning occurs if this bound is at least the current upper bound or if a duplicate subgraph has already been processed (Vercellino et al., 26 Aug 2025).

The theoretical oracle-based formulation lifts this workflow to a more abstract level. A QCBP node consists of the base constraint system, a sequence of branching decisions, and a current master LP model whose columns are an implicit subset of an exponentially large set. The cost oracle solves the LP relaxation via column generation to optimality, the branch oracle returns the children induced by the branching rule, and a marking oracle identifies feasible leaves within a cost threshold. Search and Count are then orchestrated as in quantum branch-and-bound (Montanaro, 2019).

5. Empirical behavior, scaling, and performance diagnostics

The Tail Assignment study simulates ideal QAOA with depths up to $\min \sum_{j \in J} c_j x_j$2 and qubit counts $\min \sum_{j \in J} c_j x_j$3. For Exact Cover, the required depth $\min \sum_{j \in J} c_j x_j$4 decreases as the number of feasible solutions increases for a fixed success probability of finding a feasible solution. For Set Partitioning, if the Hamiltonian is poorly balanced, the required depth for a fixed success probability of finding the optimal solution can increase with the number of feasible solutions, and in the worst case the depth requirement can be exponential in the problem size. Proper balancing of objective versus constraints significantly improves success probabilities and reduces required depth for most instances: $\min \sum_{j \in J} c_j x_j$5 of $\min \sum_{j \in J} c_j x_j$6 instances with more than one feasible solution benefited. Instances with a single feasible solution were best treated as Exact Cover. The paper also identifies the smallest nonzero energy gap relative to the maximum eigenvalue as a useful diagnostic, because increasing this ratio correlates with lower $\min \sum_{j \in J} c_j x_j$7 requirements (Svensson et al., 2021).

The EV charging study uses synthetic instances on a Windows 10 machine with Intel Core i7-12700H and Python 3.10.18, with horizon $\min \sum_{j \in J} c_j x_j$8, fixed charging duration $\min \sum_{j \in J} c_j x_j$9, charger count $\sum_{j \in J} a_{ij}x_j = 1 \quad \forall i\in I,$0, and a time limit of $\sum_{j \in J} a_{ij}x_j = 1 \quad \forall i\in I,$1 s. On small and medium instances with $\sum_{j \in J} a_{ij}x_j = 1 \quad \forall i\in I,$2, all methods prove optimality with sub-second to tens-of-seconds runtimes. On large and hard instances with $\sum_{j \in J} a_{ij}x_j = 1 \quad \forall i\in I,$3, the pure Gurobi-based baseline often hits the $\sum_{j \in J} a_{ij}x_j = 1 \quad \forall i\in I,$4 s limit with non-zero gaps, whereas the QAIA-based variants close the gap and prove optimality within the same budget. The example instance v100c10k10s1 is reported explicitly: the Gurobi baseline terminates at $\sum_{j \in J} a_{ij}x_j = 1 \quad \forall i\in I,$5 s with gap $\sum_{j \in J} a_{ij}x_j = 1 \quad \forall i\in I,$6, BSB reaches gap $\sum_{j \in J} a_{ij}x_j = 1 \quad \forall i\in I,$7 in $\sum_{j \in J} a_{ij}x_j = 1 \quad \forall i\in I,$8 s, and SimCIM in $\sum_{j \in J} a_{ij}x_j = 1 \quad \forall i\in I,$9 s (Sun et al., 22 Mar 2026).

The Vehicle Routing study compares quantum annealing (QA), simulated annealing (SA), and classical exact subroutines. In pricing, SA reduces the number of exact pricing calls by an average of $x_j \in \{0,1\}$0 relative to pure IP, while QA reduces them by $x_j \in \{0,1\}$1. QA’s average per-pricing runtime is lower than SA on all but the smallest instance, but pure IP pricing is still fastest in the small-size regime. In separation, QA reduces integrality gaps more than CVRPSEP on two instances, while SA matches or exceeds QA on all instances. A central systems-level observation is that QPU time is about $x_j \in \{0,1\}$2 of total wall-clock time, so classical overhead dominates runtime (Wagner et al., 2024).

The Vertex Coloring experiments report an emulation set of $x_j \in \{0,1\}$3 instances with $x_j \in \{0,1\}$4, including $x_j \in \{0,1\}$5 unit-disk and $x_j \in \{0,1\}$6 non-unit-disk graphs. QCBP solves $x_j \in \{0,1\}$7 optimally overall, compared with $x_j \in \{0,1\}$8 for HCG and $x_j \in \{0,1\}$9 for BBQ-mIS. The breakdown is $x_j \ge 0$0 on unit-disk instances and $x_j \ge 0$1 on non-unit-disk instances. Optimality gaps are usually $x_j \ge 0$2, with worst gap $x_j \ge 0$3; by contrast, HCG reaches gaps up to $x_j \ge 0$4 and BBQ-mIS up to $x_j \ge 0$5. QCBP typically uses fewer than $x_j \ge 0$6 shots per instance and often generates at most $x_j \ge 0$7 nodes while exploring at most $x_j \ge 0$8 (Vercellino et al., 26 Aug 2025).

These results do not support a uniform performance narrative. Some QCBP variants already outperform classical baselines on designated large-instance regimes, while others remain behind classical exact or heuristic competitors once orchestration overhead is counted. The empirical record is therefore application-specific and hardware-specific rather than universal.

6. Exactness, theory, limitations, and future directions

Exactness in QCBP is conditional on where the quantum component is placed and whether heuristic outputs are eventually certified. In the EV charging framework, branch-and-price remains exact if the master provides valid lower bounds and column generation continues until no negative-reduced-cost column exists; if pricing is heuristic, periodic exact pricing checks or a final exact pricing call restore formal correctness (Sun et al., 22 Mar 2026). The Vehicle Routing formulation makes the same point for both pricing and separation: heuristic QUBO solves can be interleaved with exact fallback, so optimality is still proved by the classical branch-and-bound framework (Wagner et al., 2024). By contrast, the Tail Assignment variant uses QAOA as a primal heuristic for integer RMP solutions, so its immediate role is incumbent improvement rather than exact certification (Svensson et al., 2021).

The strongest complexity statement in the literature comes from the oracle-based quantum branch-and-bound formulation. Let $x_j \ge 0$9 be the full search tree of depth $1$00, let $1$01 be the minimal cost of a valid solution, and let $1$02 be the size of the tree truncated at $1$03. Theorem 1 states that the quantum algorithm uses

$1$04

oracle calls and, except with failure probability at most $1$05, returns a minimal-cost solution if one exists, else “no solution.” This yields a near-quadratic speedup in the truncated tree size under the stated oracle assumptions. For the Sherrington–Kirkpatrick model, the paper reports a quantum runtime $1$06 on most instances, compared with Grover’s $1$07 and a classical branch-and-bound bound of $1$08 with high probability (Montanaro, 2019).

The practical limitations are equally explicit. Tail Assignment notes that typical RMPs have $1$09–$1$10 columns, dense couplings imply many entangling gates, Hamiltonian construction for dense RMPs is $1$11, and success degrades under noise because the reported QAOA simulations are ideal and shot-free (Svensson et al., 2021). Vehicle Routing emphasizes sparse hardware connectivity, minor embedding overhead, chain errors, feasibility filtering, and the fact that dense QUBOs are limited by current annealer embedding capacity (Wagner et al., 2024). Vertex Coloring adds neutral-atom embedding constraints such as minimum spacing, maximum adjacency distance, and register diameter, and notes that non-unit-disk graphs may require approximate embeddings that effectively replace the target graph by a supergraph, potentially missing some columns (Vercellino et al., 26 Aug 2025). EV charging highlights penalty calibration, hyperparameter sensitivity in BSB and SimCIM, and the possibility that heuristic pricing may miss a negative-reduced-cost column unless exact fallback is used (Sun et al., 22 Mar 2026).

Several future directions recur across the literature. These include extending QCBP to other large-scale ILPs with column-generation structure such as crew pairing, crew rostering, vehicle routing, and facility location; using alternative quantum subroutines such as quantum annealing, VQE, Grover-based search, or amplitude amplification; designing constraint-preserving mixers instead of penalty-only encodings; improving dual stabilization and warm-start schedules; and exploiting batch sampling more systematically to add many columns or cuts per iteration (Svensson et al., 2021, Wagner et al., 2024, Sun et al., 22 Mar 2026). A plausible implication is that QCBP is best viewed not as a single algorithm with settled performance guarantees, but as a modular research program centered on one question: which subproblems inside Branch-and-Price are small enough, structured enough, and influential enough that a quantum or quantum-inspired coprocessor can improve the overall decomposition without destroying its bound structure or correctness semantics.