Minimum Classes Inequalities in BPPS

• MCIs are class-aggregation inequalities used in BPPS that enforce a minimum number of active bins per class based on the residual capacity (d - s_c).
• They strengthen the LP relaxation by ensuring class setup costs are counted for each bin required, restoring a classical worst-case guarantee of 1/2.
• Computational experiments show that incorporating MCIs reduces the integrality gap and improves solver performance compared to the natural formulation.

Minimum Classes Inequalities (MCIs) are class-aggregation inequalities introduced for the natural integer linear programming formulation of the Bin Packing Problem with Setups (BPPS). In BPPS, items are partitioned into classes, and packing any item of class $c$ into a bin incurs both a setup weight $s_c$, which consumes bin capacity, and a setup cost $f_c$, which enters the objective. The MCIs strengthen the LP relaxation by enforcing, for each class, the minimum number of bins in which that class must be active. Their purpose is to correct a specific pathology of the natural relaxation: without them, a class setup can be counted only once even when the total weight of that class necessarily forces activation in multiple bins. The strengthened relaxation restores a worst-case guarantee of $1/2$, matching the classical Bin Packing Problem, and materially improves solution quality and running time in computational experiments (Baldacci et al., 12 Sep 2025).

1. BPPS formulation in which MCIs arise

The BPPS extends classical bin packing by associating each class with a setup weight and a setup cost. A bin of capacity $d$ can contain items whose total item weight, plus the setup weights of all classes active in that bin, does not exceed $d$. The objective includes a bin cost $r$ for each used bin and a setup cost $f_c$ for each class activated in a bin (Baldacci et al., 12 Sep 2025).

The compact ILP formulation uses binary variables $x_{ib}$, $y_{cb}$, and $s_c$0, where $s_c$1 if item $s_c$2 is packed in bin $s_c$3, $s_c$4 if class $s_c$5 is active in bin $s_c$6, and $s_c$7 if bin $s_c$8 is used. With $s_c$9, $f_c$0, and $f_c$1 the items of class $f_c$2, the formulation is

$f_c$3

subject to

$f_c$4

$f_c$5

$f_c$6

Within this model, the linking constraints $f_c$7 guarantee that if an item of class $f_c$8 is packed in bin $f_c$9, then the class is active in that bin. The MCIs do not replace these constraints; they add a class-level lower bound that aggregates the packing consequences of all items of a class across the full bin set.

2. Deficiency of the natural LP relaxation

The paper shows that the LP relaxation of the natural formulation can be arbitrarily weak. Its closed-form optimum is obtained by spreading each item and each class setup fractionally across all bins: $1/2$0 The resulting LP value is

$1/2$1

The conceptual problem is that each class setup cost $1/2$2 is counted only once in the relaxation, because the linking structure implies only $1/2$3. In an integer solution, however, the items of class $1/2$4 may require that class to be activated in several bins once the setup weight $1/2$5 is reserved (Baldacci et al., 12 Sep 2025).

This mismatch is not merely qualitative. The paper proves an extreme worst case in which the ratio $1/2$6, where $1/2$7 is the optimal BPPS objective value. A plausible implication is that, without additional valid inequalities, the natural LP can fail to capture the defining combinatorics of repeated class activation and therefore provides a poor basis for either bounding or branch-and-bound.

3. Formal definition and interpretation of the MCIs

The Minimum Classes Inequalities are

$1/2$8

The right-hand side is the minimum number of bins that class $1/2$9 must occupy once the setup weight $d$0 has been reserved in every bin containing that class (Baldacci et al., 12 Sep 2025).

Their interpretation is direct. If class $d$1 is active in a bin, the residual capacity available to items of that class is $d$2. Consequently, the total weight $d$3 cannot be packed into fewer than

$d$4

class activations. The paper emphasizes that this bound can be computed in linear time in $d$5 (Baldacci et al., 12 Sep 2025).

The terminology “Minimum Classes” refers to this minimum class-wise packing count. The inequalities do not merely assert that a class must appear somewhere; they assert that it must appear in at least as many bins as required by its total weight under residual capacity $d$6. In that sense, they are aggregate lower bounds on class activation multiplicity rather than simple linking constraints.

The paper proves validity by summing capacity contributions over all bins where a given class is active. For every class $d$7,

$d$8

hence

$d$9

and integrality of the left-hand side yields the ceiling form. This proof clarifies that the MCIs are not heuristic corrections; they are valid inequalities satisfied by every feasible BPPS solution.

4. Strengthened relaxation and worst-case theory

Once MCIs are added, the LP relaxation acquires a new closed-form optimum: $d$0 with objective value

$d$1

Thus the relaxation counts each class’s setup cost and setup weight not once, but $d$2 times, where $d$3 is the minimum number of required activations for that class (Baldacci et al., 12 Sep 2025).

The central guarantee is

$d$4

The proof compares the strengthened LP bound with a feasible BPPS solution constructed through class-wise packing arguments and uses a generalized BPP lower bound lemma stating that

$d$5

where $d$6 is the optimum number of bins in the ordinary BPP (Baldacci et al., 12 Sep 2025).

The $d$7 bound is also tight. The paper gives a single-class family with

$d$8

and growing item count. In that family, each bin can hold at most one item, the integer optimum uses one bin per item, and

$d$9

This establishes that MCIs restore, but do not exceed, the classical $r$0 worst-case behavior known for bin-packing relaxations (Baldacci et al., 12 Sep 2025).

The comparison with classical BPP is explicit. If $r$1 for all classes, BPPS reduces to the ordinary Bin Packing Problem. The paper’s interpretation is that the natural BPPS relaxation can be much weaker than the ordinary BPP relaxation because it ignores repeated class activations, whereas MCIs restore an analogous classical-style worst-case guarantee of $r$2.

5. Relation to the Minimum Bins Inequality and computational evidence

The paper introduces a second strengthening device, the Minimum Bins Inequality (MBI),

$r$3

The MBI is global, whereas the MCIs are class-specific. The paper is explicit that the two are complementary: MCIs force each class to be activated in enough bins, while the MBI forces the total number of opened bins to be large enough overall (Baldacci et al., 12 Sep 2025).

With both MCIs and the MBI, the LP optimum becomes

$r$4

and the objective value is

$r$5

The paper reports that this combined relaxation is computationally stronger than MCIs alone and still has a worst-case ratio approaching $r$6 (Baldacci et al., 12 Sep 2025).

The experimental evidence is based on a benchmark of 480 BPPS instances. For the 286 instances with certified optimal solutions, the average integrality gap drops from $r$7 for the baseline LP to $r$8 with MCIs, and to $r$9 with MCIs+MBI. For $f_c$0, the average integrality gap drops from $f_c$1 to $f_c$2 with MCIs, and then to $f_c$3 with MCIs+MBI. On the full set of 480 instances, the baseline formulation solves 162 instances to optimality, the version with MCIs solves 220, the version with MCIs+MBI solves 253, and the version with MCIs+MBI plus a stronger upper bound on the number of bins solves 268. Average optimality gaps on unsolved instances drop from $f_c$4 to $f_c$5, then $f_c$6, then $f_c$7 (Baldacci et al., 12 Sep 2025).

An important methodological point is that generic solver technology does not reproduce this effect. In 465 out of 480 instances, the root-node bound produced by standard CPLEX cuts coincides with the raw LP relaxation bound of the baseline formulation. This indicates that the observed improvement comes from the problem-specific structure encoded by MCIs and the MBI rather than from generic cutting-plane routines.

6. Scope of the term and related inequality families

In the current arXiv record represented here, the exact term “Minimum Classes Inequalities” is used explicitly for BPPS and its LP strengthening (Baldacci et al., 12 Sep 2025). Several nearby literatures contain structurally related inequality systems, but they do not use the term in the same way.

In multiple-hypothesis testing, “General Classes of Lower Bounds on the Probability of Error in Multiple Hypothesis Testing” derives two classes of lower bounds using Hölder’s inequality and reverse Hölder’s inequality. The paper does not explicitly use the term “Minimum Classes Inequalities (MCIs),” although its posterior-based expressions

$f_c$8

are described as “minimum-class-like” in structure (Routtenberg et al., 2010). This suggests an analogy in mathematical form, not a terminological identity.

In polyhedral studies of order relations, “Primary Facets Of Order Polytopes” likewise does not use the term MCI explicitly. Instead, it studies primary facet-defining inequalities with coefficients in $f_c$9, including nonnegativity, 2-cycle, transitivity, 3-cycle, and fence inequalities. The paper states that this primary-inequality program is “the paper’s analog of the MCI philosophy,” especially because it seeks simple structured inequalities capturing minimal forbidden combinatorial patterns (Doignon et al., 2015). Here again, the connection is conceptual rather than nominal.

In matroid prophet inequalities, “Non-Adaptive Prophet Inequalities for Minor-Closed Classes of Matroids” also does not explicitly mention MCIs. The relevant inequality language is instead the standard matroid polytope

$x_{ib}$0

together with structural results on minor-closed classes, representability, and decomposition. The paper explicitly notes that it does not develop an MCI-specific forbidden-minor or axiom system (Pashkovich et al., 2023).

A separate source of ambiguity is biomedical usage. In “Comprehensively stratifying MCIs into distinct risk subtypes based on brain imaging genetics fusion learning,” the abbreviation MCI denotes mild cognitive impairment, and “MCIs” refers to MCI subjects rather than to inequalities (Shang et al., 25 Aug 2025). In technical writing, disambiguation is therefore necessary: in combinatorial optimization, MCIs refer to Minimum Classes Inequalities in BPPS; in Alzheimer’s disease research, MCI denotes a clinical condition.

Taken together, these usages indicate that Minimum Classes Inequalities, in the strict sense, are a BPPS-specific strengthening device, while related domains contain either analogous simple inequality families or an unrelated acronym.