# Questions tagged [integer-programming]

Integer programming regards optimization problems, where one seeks to find integer values for a set of unknowns, that optimizes the objective function. A common subset of this type of problems are integer linear programming problems, where all inequalities, equalities and the objective function are linear in the unknowns.

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### What is the probability of interpolating the Tutte polynomial of a planar graph from the values at the two hyperbolas?

The Tutte polynomial is a bivariate polynomial with positive integer coefficient which is a graph invariant and can be defined recursively. Evaluating it is $\#P$-complete even when restricted to (...
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### Lenstra's integer programming algorithm: Finding a lattice point “near the center”

I have already posted this question on the mathematics forum, but I suspect the question needs more detailed knowledge than most users have; please excuse the duplicate post. Any help is greatly ...
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### Reference request: how to find the k'th best solution to the 1-0 knapsack problem?

How do I find the k'th best solution to the 1-0 knapsack problem without finding the Top-k solutions? Is there any mathematical research that deals with the k'th best solutions to the mixed integer ...
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### About the partial expectation polynomials in the Interlacing-I paper and perfect matchings

I am thinking of the polynomials $f_{s_1,s_2,..,s_k}$ as in the definition 4.3 in this paper http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf In the use of these ...
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I am seeking some reference on already existing work for the following problem. Given an $n$-dimensional square matrix $A=DP$ where $D$ is a diagonal and $P$ is a permutation matrix (think of Gaussian ...
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### 0,1 solution to system of linear integer equations

I have the following problem: $A x = b$ where $A, b$ - $m \times n$-matrix and $m$-vector of nonnegative integers (respectively). $x \in \{0,1\}^n$ - vector of binary variables, which need to be ...
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### What does Lenstra's MILP do?

Honestly I do not understand why Lenstra's MILP is in $P$ if the number of integer dimensions is fixed. Here is what Lenstra says in 'http://people.csail.mit.edu/rrw/presentations/Lenstra81.pdf' in ...
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### Smallest integer lattice point by box measure in a polytope?

Given an integer lattice $\mathcal L\subseteq\mathbb Z^n$ represented by basis $\mathcal B$ and an integer linear program $Ax\leq b$ where $x\in\mathbb Z^n$ is unknown with $A\in\mathbb Z^{m\times n}$ ...
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### Distribution of maximum minor of a random matrix with one special column

Given $m,n,\ell\in\Bbb N$ and $\beta\in(0,1)$ consider the uniformly picked random matrix $A\in\Bbb Z^{n\times (n+1)}$ with $0\neq|\mathsf{det}(A^\circ)|\leq m^{\frac 1\ell}$ where $A^\circ$ is the ...
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### Strong Duality of Mixed Integer Linear Program

The problem at hand is to optimize a mixed-integer linear program closely related to the maximum flow problem. I would like to reformulate the problem with its dual and I'm concerned with the ...
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