Minimum Bins Inequality: Theory & Applications

• Minimum Bins Inequality (MBI) is a principle that provides a global lower bound on the number of bins needed by accounting for both item weights and setup requirements in BPPS.
• It is derived by complementing the natural ILP formulation with Minimum Classes Inequalities (MCIs), thereby dramatically strengthening LP relaxations with reduced integrality gaps.
• In stochastic settings, MBI principles ensure that as-balanced-as-possible allocations minimize the expected final residue in ball-removal processes.

Searching arXiv for papers on “Minimum Bins Inequality” and closely related formulations. Search query: "Minimum Bins Inequality Bin Packing Problem with Setups arXiv" The Minimum Bins Inequality (MBI) denotes a lower-bounding principle about the minimum number of bins required by a constrained allocation or occupancy process, but the term is used in more than one arXiv-adjacent context. In the most formal and explicit sense, it is a valid inequality for the Bin Packing Problem with Setups (BPPS) that strengthens a compact ILP by imposing a global lower bound on the number of opened bins (Baldacci et al., 12 Sep 2025). In a second, closely related usage, the phrase refers to the question of whether a most balanced initial allocation minimizes the expected final residue in a stochastic ball-removal process; that question is answered affirmatively for the uniform-selection model by a coupling argument (Correa et al., 13 Feb 2026). Related balls-and-bins, occupancy, hashing, and binomial-minimum papers do not always use the name MBI, but they study quantitatively similar threshold, occupancy, or minimum-load phenomena (Schulte-Geers et al., 2022, Aamand et al., 2018, Zhu et al., 25 Feb 2025).

1. Named inequality in BPPS

In the BPPS literature, the MBI is introduced as a global strengthening of the natural compact formulation for the Bin Packing Problem with Setups (BPPS) (Baldacci et al., 12 Sep 2025). In this model one is given a set of items

$\mathcal I=\{1,2,\dots,n\},$

identical bins of capacity

$d\in \mathbb Z_{>0},$

item weights $w_i$, a partition of the items into classes

$$\mathcal P=\{\mathcal I_1,\mathcal I_2,\dots,\mathcal I_m\},$$

setup weights $s_c$, setup costs $f_c$, and a bin cost $r$. If at least one item of class $c$ is packed in a bin, then class $c$ is active in that bin, and the bin incurs a one-time capacity reduction of $s_c$ and a one-time cost of $d\in \mathbb Z_{>0},$0 for that class-bin pair (Baldacci et al., 12 Sep 2025).

The natural compact ILP uses binary variables $d\in \mathbb Z_{>0},$1 for item assignment, $d\in \mathbb Z_{>0},$2 for class activation, and $d\in \mathbb Z_{>0},$3 for bin usage. Its capacity constraints are

$d\in \mathbb Z_{>0},$4

The MBI is designed to strengthen the global implication of these per-bin capacity constraints on the total number of bins (Baldacci et al., 12 Sep 2025).

The exact inequality is

$d\in \mathbb Z_{>0},$5

where

$d\in \mathbb Z_{>0},$6

The right-hand side is denoted

$d\in \mathbb Z_{>0},$7

The paper presents the MBI as a single inequality for a BPPS instance, not as a family indexed by subsets or bins, and it is described as a compact, closed-form valid inequality computable in linear time (Baldacci et al., 12 Sep 2025).

2. Derivation, validity, and relation to MCIs

The motivation for the MBI is the weakness of the LP relaxation of the natural formulation. The LP relaxation $d\in \mathbb Z_{>0},$8 is obtained by replacing the binary restrictions with

$d\in \mathbb Z_{>0},$9

and its optimal value is denoted $w_i$0. The paper proves that an optimal LP solution is the completely symmetric fractional point

$w_i$1

with objective value

$w_i$2

The paper states that this relaxation can be arbitrarily weak, because setup mass can be shared fractionally among bins even though in any integral solution a setup must be paid each time a class appears in a bin (Baldacci et al., 12 Sep 2025).

To repair the classwise undercounting, the paper first introduces the Minimum Classes Inequalities (MCIs): $w_i$3 These enforce that class $w_i$4 must be active in at least $w_i$5 bins, because once class $w_i$6 is active, only $w_i$7 capacity remains for its items. With MCIs, the LP optimal value becomes

$w_i$8

and the paper states that this restores the classical $w_i$9 worst-case guarantee: $$\mathcal P=\{\mathcal I_1,\mathcal I_2,\dots,\mathcal I_m\},$$0 However, MCIs remain classwise constraints on the $$\mathcal P=\{\mathcal I_1,\mathcal I_2,\dots,\mathcal I_m\},$$1-variables and do not directly impose a lower bound on the total number of opened bins $$\mathcal P=\{\mathcal I_1,\mathcal I_2,\dots,\mathcal I_m\},$$2. That remaining gap is exactly what the MBI fills (Baldacci et al., 12 Sep 2025).

The derivation is direct. Since each item must be packed, all item weights contribute to required capacity. Since MCIs imply that class $$\mathcal P=\{\mathcal I_1,\mathcal I_2,\dots,\mathcal I_m\},$$3 must be activated in at least $$\mathcal P=\{\mathcal I_1,\mathcal I_2,\dots,\mathcal I_m\},$$4 bins, setup weight $$\mathcal P=\{\mathcal I_1,\mathcal I_2,\dots,\mathcal I_m\},$$5 must be paid at least $$\mathcal P=\{\mathcal I_1,\mathcal I_2,\dots,\mathcal I_m\},$$6 times in total. Therefore any feasible integral solution must consume at least

$$\mathcal P=\{\mathcal I_1,\mathcal I_2,\dots,\mathcal I_m\},$$7

units of capacity, and since each used bin contributes at most $$\mathcal P=\{\mathcal I_1,\mathcal I_2,\dots,\mathcal I_m\},$$8, the number of used bins must satisfy the MBI (Baldacci et al., 12 Sep 2025).

The formal proof in the paper derives the inequality algebraically from the relaxation with MCIs. It uses

$$\mathcal P=\{\mathcal I_1,\mathcal I_2,\dots,\mathcal I_m\},$$9

and a lower bound on the total objective to obtain

$s_c$0

Since $s_c$1 and the $s_c$2 are integer in every feasible solution, the ceiling form follows (Baldacci et al., 12 Sep 2025).

3. Structural role in the BPPS formulation

The paper emphasizes that MCIs and the MBI are complementary rather than interchangeable (Baldacci et al., 12 Sep 2025). The MCIs act at the class level,

$s_c$3

while the MBI acts at the global bin-count level,

$s_c$4

The MBI constrains only the aggregate number of bins, whereas the MCIs ensure that the contribution of each class is explicitly enforced. The paper does not claim that one dominates the other, and mathematically they live in different variable spaces (Baldacci et al., 12 Sep 2025).

Conceptually, the MBI is a BPPS analogue of the classical bin-packing weight lower bound

$s_c$5

The BPPS adaptation replaces total item weight with

$s_c$6

which includes the minimum setup burden implied by the MCIs. This can be read as a setup-aware global volume bound (Baldacci et al., 12 Sep 2025).

The paper also derives a paired upper bound on the number of bins used in an optimal solution. For each class $s_c$7, let $s_c$8 be an upper bound on the minimum number of bins of capacity $s_c$9 needed to pack all items of class $f_c$0. Then

$f_c$1

is a valid upper bound on the number of bins used in any optimal BPPS solution. This upper bound is used to replace the trivial choice $f_c$2 in the compact model, and the paper notes that the number of variables and constraints scales linearly with $f_c$3: variables $f_c$4, constraints $f_c$5 (Baldacci et al., 12 Sep 2025).

The addition of the MBI does not improve the asymptotic worst-case guarantee beyond the $f_c$6 already obtained with MCIs. The paper states that there exists a sequence of instances such that

$f_c$7

This shows that the MBI is structurally important without changing the established worst-case ratio (Baldacci et al., 12 Sep 2025).

4. Stochastic balancing interpretation of MBI

A second usage of the term appears in the stochastic balls-and-bins removal problem studied in "About a Ball Removal Process on Bins" (Correa et al., 13 Feb 2026). There the process starts from an occupancy vector

$f_c$8

and repeatedly removes a ball from a non-empty bin chosen uniformly at random, stopping as soon as only one non-empty bin remains. If $f_c$9 denotes the number of balls in the last non-empty bin at the stopping time $r$0, the objective is

$r$1

and the optimization problem is

$r$2

The paper states Will Ma’s conjecture as: the minimizing allocation is one that distributes the balls as evenly as possible, i.e. occupancies differ by at most $r$3 (Correa et al., 13 Feb 2026).

The main theorem proves that conjecture. It states that

$r$4

is attained only by assignments satisfying

$r$5

Equivalently, among all allocations of $r$6 balls into $r$7 bins, the as-balanced-as-possible allocation uniquely minimizes the expected number of balls remaining, up to permutation symmetry (Correa et al., 13 Feb 2026).

A key structural point is that the objective function is not Schur-concave. The result is instead proved through a local balancing or smoothing inequality: whenever one bin exceeds another by at least $r$8, moving one ball from the larger bin to the smaller bin strictly improves the objective. The relevant statement is

$r$9

For $c$0, the paper gives the recursion

$c$1

with boundary values

$c$2

and proves

$c$3

The general-$c$4 proof uses a coupling on the common sample space of selected bins, together with a stopping time $c$5 and a partition of sample paths into event classes that are compared one by one (Correa et al., 13 Feb 2026).

This usage differs from the BPPS inequality in formal status. In the BPPS paper, MBI is a named linear inequality. In the ball-removal paper, the term refers to a balancing principle or conjecture about the minimization of the expected final residue. This suggests that current usage treats MBI as a broader label for minimum-bin-count or minimum-residue phenomena driven by balancing.

5. Related occupancy and minimum-load results

Several nearby papers do not use the exact label “Minimum Bins Inequality,” but they establish MBI-type results for occupancy thresholds, occupied-bin counts, or minimum-load surrogates.

In "Balls and Bins -- Simple Concentration Bounds," the central quantity is

$c$6

for a nonuniform occupancy model in which $c$7 balls are thrown independently according to a distribution $c$8. The paper’s core inequality is

$c$9

where $c$0 is the maximum load after $c$1 throws. It interprets this through a phase change around $c$2: if $c$3, a $c$4-loaded bin is unlikely; if $c$5, a $c$6-loaded bin is likely (Schulte-Geers et al., 2022).

The same paper derives threshold statements for the waiting time

$c$7

and expectation bounds

$c$8

These are not MBI in name, but they provide a minimum-throw-scale characterization for the first $c$9-loaded bin (Schulte-Geers et al., 2022).

In "Non-Empty Bins with Simple Tabulation Hashing," the relevant variable is the number of occupied bins

$s_c$0

Under fully random hashing the baseline expectation is

$s_c$1

For simple tabulation hashing, the paper proves

$s_c$2

and the high-probability lower bound

$s_c$3

It also gives lower-tail bounds around $s_c$4, but with an unavoidable extra term $s_c$5, so the lower tail is not purely subgaussian in general (Aamand et al., 2018).

In "Tight Bounds on the Binomial CDF, and the Minimum of i.i.d Binomials, in terms of KL-Divergence," the object is

$s_c$6

For $s_c$7, the paper proves

$s_c$8

and then shows that, with explicit finite-sample correction

$s_c$9

the minimum satisfies

$d\in \mathbb Z_{>0},$00

with probability $d\in \mathbb Z_{>0},$01, except in the regime where the theorem implies $d\in \mathbb Z_{>0},$02 with high probability (Zhu et al., 25 Feb 2025).

These papers show that, even when the exact label MBI is absent, a common mathematical pattern recurs: thresholding or bounding a minimal occupancy-related quantity by translating local collision or activation constraints into a global condition.

6. Computational significance, limitations, and non-uniform variants

The strongest computational evidence for the MBI as a named inequality comes from BPPS. The experiments compare four variants:

1. $d\in \mathbb Z_{>0},$03,
2. $d\in \mathbb Z_{>0},$04,
3. $d\in \mathbb Z_{>0},$05,
4. $d\in \mathbb Z_{>0},$06.

Over 480 instances, the paper reports:

• $d\in \mathbb Z_{>0},$07: 162 solved to optimality, average gap $d\in \mathbb Z_{>0},$08;
• $d\in \mathbb Z_{>0},$09: 220 solved, average gap $d\in \mathbb Z_{>0},$10;
• $d\in \mathbb Z_{>0},$11: 253 solved, average gap $d\in \mathbb Z_{>0},$12;
• $d\in \mathbb Z_{>0},$13: 268 solved, average gap $d\in \mathbb Z_{>0},$14.

On the subset of 286 instances solved optimally by at least one formulation, the overall average integrality gap is reported as:

• $d\in \mathbb Z_{>0},$15: $d\in \mathbb Z_{>0},$16,
• $d\in \mathbb Z_{>0},$17: $d\in \mathbb Z_{>0},$18,
• $d\in \mathbb Z_{>0},$19: $d\in \mathbb Z_{>0},$20.

The paper also notes that $d\in \mathbb Z_{>0},$21, so the LP improvement from $d\in \mathbb Z_{>0},$22 to $d\in \mathbb Z_{>0},$23 is attributable specifically to the MBI, whereas the upper bound $d\in \mathbb Z_{>0},$24 reduces model size rather than the LP value (Baldacci et al., 12 Sep 2025).

In the stochastic ball-removal setting, the main limitation concerns non-uniform bin-selection rules. The paper explicitly studies a non-uniform two-bin extension with recursion

$d\in \mathbb Z_{>0},$25

and derives

$d\in \mathbb Z_{>0},$26

Using $d\in \mathbb Z_{>0},$27, $d\in \mathbb Z_{>0},$28, and

$d\in \mathbb Z_{>0},$29

the paper shows that the proportional allocation $d\in \mathbb Z_{>0},$30 is worse than $d\in \mathbb Z_{>0},$31: $d\in \mathbb Z_{>0},$32 Thus

$d\in \mathbb Z_{>0},$33

so probability-proportional balance is not optimal. The paper does not provide a full characterization of optimal allocations in the non-uniform setting and ends with the conjecture that the optimal allocation may still stay within $d\in \mathbb Z_{>0},$34 in $d\in \mathbb Z_{>0},$35 distance of the balanced allocation (Correa et al., 13 Feb 2026).

Taken together, these results distinguish three levels of usage. First, MBI is an exact named valid inequality in BPPS. Second, it denotes a solved balancing principle in the uniform ball-removal model. Third, it functions as a useful umbrella for occupancy-threshold and minimum-load results in hashing and balls-and-bins analyses. A plausible implication is that the term is stabilizing around a common methodological theme: converting unavoidable local burden—item weight, setup multiplicity, collision mass, or one-bin lower-tail probability—into a rigorous global lower bound on the number of bins, occupied bins, or surviving mass.