🧩 Constraint Solving POTD:Bin Packing

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Category: Packing & Cutting | Date: 2026-04-23

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Problem Statement

You have n items, each with a size s_i ∈ (0, 1], and an unlimited supply of bins each with capacity 1. Pack all items into as few bins as possible.

Small Concrete Instance

Item	Size
A	0.4
B	0.7
C	0.3
D	0.5
E	0.6
F	0.2

Optimal solution (2 bins): {B, C, F} = 1.2 → not valid! Actually:

• Bin 1: B (0.7) + C (0.3) = 1.0 ✓
• Bin 2: D (0.5) + A (0.4) = 0.9 ✓
• Bin 3: E (0.6) + F (0.2) = 0.8 ✓

Optimal: 3 bins (lower bound: ⌈total size⌉ = ⌈2.7⌉ = 3 ✓).

Formal Input/Output

• Input: n items with sizes s_1, ..., s_n ∈ (0,1]
• Output: An assignment bin[i] ∈ {1..k} minimizing k such that ∑_{i: bin[i]=j} s_i ≤ 1 for all bins j

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Why It Matters

Cloud resource allocation: Virtual machines with varying CPU/memory demands must be packed onto physical hosts to minimize active server count — directly saving energy and cost.

Cutting stock & manufacturing: Raw material rolls or sheets must be cut to fulfill orders while minimizing waste (a continuous generalization). Appears in paper, steel, and textile production daily.

Compiler register allocation: Variables with overlapping lifetimes compete for a fixed set of registers — a graph-coloring variant of bin packing. Getting it wrong forces costly memory spills.

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Modeling Approaches

Approach 1 — Mixed Integer Programming (MIP)

Decision variables:

• y_j ∈ {0,1} — bin j is used
• x_{ij} ∈ {0,1} — item i is placed in bin j

Model:

minimize  ∑_j y_j
subject to:
  ∑_j x_{ij} = 1          ∀i          (each item assigned once)
  ∑_i s_i · x_{ij} ≤ y_j  ∀j          (capacity + bin activation)
  y_j ≥ y_{j+1}            ∀j          (symmetry breaking: use bins in order)
  x_{ij}, y_j ∈ {0,1}

Trade-offs: Compact formulation; LP relaxation gives a useful lower bound. Symmetry among bins causes massive branch-and-bound blowup without the ordering constraint. Scales poorly beyond ~100 items without column generation.

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Approach 2 — Constraint Programming (CP)

Decision variables:

• bin[i] ∈ {1..n} — bin assigned to item i
• load[j] — total size in bin j (auxiliary)
• k — number of bins used (objective)

Model (MiniZinc-style):

var 1..n: k;
array[1..n] of var 1..n: bin;
array[1..n] of var 0..100: load;  % sizes scaled to integers

constraint bin_packing_load(load, bin, sizes);
constraint k = max(bin);
constraint forall(j in 1..n)(load[j] <= capacity);

% Symmetry breaking
constraint value_precede_chain(1..n, bin);

solve minimize k;

The bin_packing_load global constraint propagates both the capacity bounds and the cardinality of each bin simultaneously — far stronger than decomposing into individual sum constraints.

Trade-offs: Global constraints enable powerful propagation. CP excels at finding feasible packings quickly; proving optimality often requires combining with a MIP lower bound (hybrid approach).

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Example Model (OR-Tools CP-SAT, Python)

from ortools.sat.python import cp_model

def solve_bin_packing(sizes, capacity):
    model = cp_model.CpModel()
    n = len(sizes)
    
    # Upper bound: use one bin per item
    bin_of = [model.NewIntVar(0, n-1, f'bin_{i}') for i in range(n)]
    num_bins = model.NewIntVar(1, n, 'num_bins')
    
    # num_bins = max(bin_of) + 1
    model.AddMaxEquality(num_bins - 1, bin_of)
    
    # Capacity constraints via AddLinearConstraint per bin (or bin_packing)
    for j in range(n):
        model.Add(
            sum(sizes[i] * (bin_of[i] == j) for i in range(n)) <= capacity
        )  # Note: use intermediate bool vars in practice
    
    # Symmetry: item i goes to bin <= i
    for i in range(n):
        model.Add(bin_of[i] <= i)
    
    model.Minimize(num_bins)
    solver = cp_model.CpSolver()
    solver.Solve(model)
    return solver.ObjectiveValue()

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Key Techniques

1. The bin_packing Global Constraint

Most CP solvers provide a dedicated bin_packing or bin_packing_load global. It enforces both capacity and item count per bin simultaneously, enabling bounds consistency across all bins in a single pass — much stronger than n separate sum constraints.

2. Lower Bound: First-Fit Decreasing (FFD) Gap

Sort items by decreasing size and apply the First Fit Decreasing heuristic to get a tight upper bound. Pair it with the LP relaxation lower bound (total size / capacity). The gap between them guides branch-and-bound: small gap → prune aggressively; large gap → focus search.

3. Symmetry Breaking with Value Precedence

Bins are interchangeable — swapping items between two equally-loaded bins gives an equivalent solution. Value precedence (value_precede_chain) forces items to fill bins in order: bin j+1 is only used if bin j is already occupied. This alone can reduce the search space by a factor of k!.

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Challenge Corner

Extension: Suppose each item also has a color and bins can hold at most 2 distinct colors. How would you extend the CP model to enforce this? What global constraint (or decomposition) would you use, and does the symmetry-breaking argument still hold?

As a bonus: can you construct a worst-case instance where First Fit Decreasing uses 11/9 · OPT + 6/9 bins — and verify the bound experimentally?

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References

1. Martello & Toth (1990). Knapsack Problems: Algorithms and Computer Implementations. Wiley. — Foundational reference for bin packing algorithms and bounds.
2. Coffman, Garey & Johnson (1997). "Approximation algorithms for bin packing: A survey." In Approximation Algorithms for NP-Hard Problems, PWS Publishing. — Comprehensive survey of FFD, BFD, and their guarantees.
3. Shaw (2004). "A constraint for bin packing." CP 2004, LNCS 3258. — Introduces the bin_packing global constraint with efficient propagation.
4. Google OR-Tools Bin Packing example — [(developers.google.com/redacted)(developers.google.com/redacted) — Hands-on tutorial with CP-SAT and MIP solvers.

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• expires on Apr 30, 2026, 12:56 PM UTC

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