Configuration linear program

In the bin packing problem, there are n items with different sizes. The goal is to pack the items into a minimum number of bins, where each bin can contain at most B. A feasible configuration is a set of sizes with a sum of at most B.

• Example:[7] suppose the item sizes are 3,3,3,3,3,4,4,4,4,4, and B=12. Then the possible configurations are: 3333; 333; 33, 334; 3, 34, 344; 4, 44, 444. If we had only three items of size 3, then we could not use the 3333 configuration.

Denote by S the set of different sizes (and their number). Denote by C the set of different configurations (and their number). For each size s in S and configuration c in C, denote:

• n_s - the number of items of size s.
• a_s,c - the number of occurrences of size s in configuration c.
• x_c - a variable denoting the number of bins with configuration c.

Then, the configuration LP of bin-packing is:

${\displaystyle {\text{minimize}}~~~\sum _{c\in C}x_{c}~~~{\text{subject to}}}$

${\displaystyle \sum _{c\in C}a_{s,c}x_{c}\geq n_{s}}$ for all s in S (- all n_s items of size s are packed).

${\displaystyle x_{c}\in \{0,\ldots ,n\}}$ for all c in C (- there are at most n bins overall, so at most n of each individual configuration).

The configuration LP is an integer linear program, so in general it is NP-hard. Moreover, even the problem itself is generally very large: it has C variables and S constraints. If the smallest item size is eB (for some fraction e in (0,1)), then there can be up to 1/e items in each bin, so the number of configurations C ~ S^1/e, which can be very large if e is small (if e is considered a constant, then the integer LP can be solved by exhaustive search: there are at most S^1/e configurations, and for each configuration there are at most n possible values, so there are at most ${\displaystyle n^{S^{1/e}}}$ combinations to check. For each combination, we have to check S constraints, so the run-time is ${\displaystyle S\cdot n^{S^{1/e}}}$, which is polynomial in n when S, e are constant).[7]

However, this ILP serves as a basis for several approximation algorithms. The main idea of these algorithms is to reduce the original instance into a new instance in which S is small and e is large, so C is relatively small. Then, the ILP can be solved either by complete search (if S, C are sufficiently small), or by relaxing it into a fractional LP.

The fractional configuration LP of bin-packing It is the linear programming relaxation of the above ILP. It replaces the last constraint ${\displaystyle x_{c}\in \{0,\ldots ,n\}}$ with the constraint ${\displaystyle x_{c}\geq 0}$. In other words, each configuration can be used a fractional number of times. The relaxation was first presented by Gilmore and Gomory,[2] and it is often called the Gilmore-Gomory linear program.[8]

• Example: suppose there are 31 items of size 3 and 7 items of size 4, and the bin-size is 10. The configurations are: 4, 44, 34, 334, 3, 33, 333. The constraints are [0,0,1,2,1,2,3]*x=31 and [1,2,1,1,0,0,0]*x=7. An optimal solution to the fractional LP is [0,0,0,7,0,0,17/3] That is: there are 7 bins of configuration 334 and 17/3 bins of configuration 333. Note that only two different configurations are needed.

In short, the fractional LP can be written as follows:

${\displaystyle {\text{minimize}}~~\mathbf {1} \cdot \mathbf {x} ~~~{\text{s.t.}}~~A\mathbf {x} \geq \mathbf {n} ~~~{\text{and}}~~\mathbf {x} \geq 0}$

Where 1 is the vector (1,...,1) of size C, A is an S-by-C matrix in which each column represents a single configuration, and n is the vector (n₁,...,n_S).

A linear program with no integrality constraints can be solved in time polynomial in the number of variables and constraints. The problem is that the number of variables in the fractional configuration LP is equal to the number of possible configurations, which might be huge. Karmarkar and Karp[9] present an algorithm that overcomes this problem.

First, they construct the dual linear program of the fractional LP:

${\displaystyle {\text{maximize}}~~\mathbf {n} \cdot \mathbf {y} ~~~{\text{s.t.}}~~A^{T}\mathbf {y} \leq \mathbf {1} ~~~{\text{and}}~~\mathbf {y} \geq 0}$.

It has S variables y₁,...,y_S, and C constraints: for each configuration c, there is a constraint ${\displaystyle A^{c}\cdot y\leq 1}$, where ${\displaystyle A^{c}}$ is the column of A representing the configuration c. 3It has the following economic interpretation.[9] For each size s, we should determine a nonnegative price y_s. Our profit is the total price of all items. We want to maximize the profit n y subject to the constraints that the total price of items in each configuration is at most 1.

Second, they apply a variant of the ellipsoid method, which does not need to list all the constraints - it just needs a separation oracle. A separation oracle is an algorithm that, given a vector y, either asserts that it is feasible, or finds a constraint that it violates. The separation oracle for the dual LP can be implemented by solving the knapsack problem with sizes s and values y: if the optimal solution of the knapsack problem has a total value at most 1, then y is feasible; if it is larger than 1, than y is not feasible, and the optimal solution of the knapsack problem identifies a configuration for which the constraint is violated.

Third, they show that, with an approximate solution to the knapsack problem, one can get an approximate solution to the dual LP, and from this, an approximate solution to the primal LP; see Karmarkar-Karp bin packing algorithms.

All in all, for any tolerance factor h, finds a basic feasible solution of cost at most LOPT(I) + h, and runs in time:

${\displaystyle O\left(S^{8}\log {S}\log ^{2}({\frac {Sn}{eh}})+{\frac {S^{4}n\log {S}}{h}}\log({\frac {Sn}{eh}})\right)}$,

where S is the number of different sizes, n is the number of different items, and the size of the smallest item is eB. In particular, if e ≥ 1/n and h=1, the algorithm finds a solution with at most LOPT+1 bins in time: ${\displaystyle O\left(S^{8}\log {S}\log ^{2}{n}+S^{4}n\log {S}\log {n}\right)}$. A randomized variant of this algorithm runs in expected time:

${\displaystyle O\left(S^{7}\log {S}\log ^{2}({\frac {Sn}{eh}})+{\frac {S^{4}n\log {S}}{h}}\log({\frac {Sn}{eh}})\right)}$.

Karmarkar and Karp further developed a way to round the fractional LP into an approximate solution to the integral LP; see Karmarkar-Karp bin packing algorithms. Their proof shows that the additive integrality gap of this LP is in O(log²(n)). Later, Hoberg and Rothvoss[10] improved their result and proved that the integrality gap is in O(log(n)). The best known lower bound on the integrality gap is a constant Ω(1). Finding the exact integrality gap is an open problem.

The integrality gap of the configuration-LP for unrelated-machines scheduling is 2.[5]