Configuration linear program: Difference between revisions

Revision as of 16:50, 28 January 2023

Revision as of 20:49, 9 February 2023

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First, they construct the [[dual linear program]] of the fractional LP:<blockquote><math>\text{maximize}~~\mathbf{n}\cdot \mathbf{y}~~~\text{s.t.}~~ A^T \mathbf{y} \leq \mathbf{1}~~~\text{and}~~ \mathbf{y}\geq 0</math>.</blockquote>It has ''S'' variables ''y''₁,...,''y_S'', and ''C'' constraints: for each configuration ''c'', there is a constraint <math>A^c\cdot y\leq 1</math>, where <math>A^c</math> is the column of '''''A''''' representing the configuration ''c''. 3It has the following economic interpretation.<ref name=":12" /> 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|''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]].