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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Existence of some lattice path connecting all given lattice paths
My daily work concerns analysis on metric spaces and some time ago it turned out that the problem I am dealing with boils down to a certain combinatorial problem. I've checked a lot of examples and it ...
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Partitioning vectors from Z^k into bundles preserving their additive properties
Let $B_1, B_2, \dots, B_m$ be disjoint subsets of $\mathbb{Z}^k$ and $B$ denote their union.
Also suppose that $k$ upper bounds the $\ell^\infty$-norm of every vector in $B$.
A set $V \subseteq B$ of ...
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Modular inverse computation - avoiding Euclidean algorithm
Modular inverse is known to be computable by Extended Euclidean algorithm which is the reaping the rewards of computing the GCD of two numbers or proving two numbers are coprime.
If we already know ...
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On optimizing a multivariate quadratic function subject to certain conditions
The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...
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Will an integer program to deterministically factor integers help derandomize $\mathbb F_q[x]$ factoring?
There are many analogies between the objects $\mathbb F_q[x]$ and $\mathbb Z$.
Supposing there is a fixed (say $10^9$) dimension linear integer program (describable without any objective function) in ...
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Hemisphere containing the maximum number of points scattered on a sphere
Consider a set of points $x_1, \ldots,x_n$ on $\mathbb{S}^{k-1}$ (the unit sphere in $\mathbb{R}^k$). The goal is finding the hemisphere which contains the maximum number of $x_i$'s. Basically, we ...
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The matrix representation of an interval graph
It is well-known that many classes of graphs have matrix representations that can be written concisely. For example,
The set of all directed acyclic graphs consists of binary matrices $x_{ij} \in \{...
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Benefit of adding a trivial constraint to ILPs
let ILP be an integer linear program with constraints-matrix $\boldsymbol{\mathrm{M}}\in\mathbb{Z}^{m\times n}$ and cost vector $\boldsymbol{\mathrm{c}}\in\mathbb{Z}^n$,
${\boldsymbol{\mathrm{x}}^*}\...
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How quickly can this IQP or its MILP relaxation be solved
Let $A\in\{0,1\}^{(n,n)}$ be a $n$ by $n$ boolean matrix (in particular think of an adjacency matrix of a graph), and consider the following optimization problem:
$$\begin{align*}&&\max_{P\in\{...
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Partially relaxing integer programs while preserving unique integral solution
Consider the program
$$\exists x\in\mathbb Z$$
$$\exists y\in\mathbb Z^d$$
$$A[x,y]'\leq b$$
and assume exactly one $(x_0,y_0)\in\mathbb Z^{d+1}$ satisfies the program.
Under what conditions on $A,b$ ...
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Questions in number theory related to $NC$ and $P$-completeness
Given $a,b\in\mathbb N$ find $\operatorname{GCD}(a,b)$.
Given $a,b,c\in\mathbb N$ find $x,y\in\mathbb Z$ such that $ax+by=c$.
Euclidean algorithm solves both.
My question is if either 1 or 2 is in ...
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Only trivial solutions to system of linear diophantine equations possibly related to hamiltonian cycles in graphs
This might be related to counting hamiltonian cycles.
@Peter Taylor gave negative result about the one dimensional case, but we believe his attack is
not directly applicable to this question.
Given ...
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Integer linear constraint(s) for y= x1 XOR x2 [closed]
Is there any way to convert $y=x_1~ \text{XOR} ~x_2$ to linear constraints? It means we write some linear relations with:
if $x_1=x_2 =0$ or $x_1=x_2=1$ $\to$ $y=0$,
else, $y=1$?
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Only trivial solution to a pair of constrained linear diophantine equations
Given positive integer $n$, we are looking for a set
of $n$ positive integers $a_i$.
The following linear integer program must have only
the trivial integer solution of all ones.
$0 \le x_i \le \frac{...
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Fastest way to solve non-negative linear diophantine equations
Let $A$ be a matrix in $M_{n \times m}(\mathbb{Z}_{\ge 0})$ without zero column. Let $V$ be a vector in $\mathbb{Z}_{> 0}^m$.
Question: What is the fastest way to find all the solutions $X \in \...
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Integer programming using the Steinitz lemma
I am trying to implement an algorithm that I read on the paper entitled: "Proximity results and faster algorithms for integer programming using the Steinitz lemma", published by Friedrich ...
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Maximally sparse integer solutions
Suppose I have a system of $n$ inhomogeneous linear equations in $m$ variables, where $n$ and $m$ are of the order of a few hundred, and $m$ is significantly larger than $n$. All the coefficients are ...
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Constructing an integer with small residues for two distinct primes in polynomial time
Given two primes $p,q\in[T,2T]$, how many integers $m$ of size $O(T^{3/2+\epsilon})$ are there such that the residues $m\bmod p$ and $m\bmod q$ are both $O(polylog(T))$? Looking for an answer
Is it ...
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Correct way to conduct equilibrium scaling of linear/integer/MIP program
I would like to scale my linear/integer program and also mixed-integer program using the equilibrium scaling method. I have worked on two research papers and one research book. However, they did the ...
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Reliability of ILP approach to number-theoretic optimization
This question is inspired by the recent answer, where @RobPratt proposed to use integer linear programming (ILP) for solving a number-theoretic optimization problem. I will consider a very similar ...
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Modified quadratic assignment problem
Let $Y,Z$ be $n\times k$ matrices and assume all columns have been standardized such that their means are zero and variances 1. I seek an $n\times n$ permutation matrix $P$ such that
$$\left\Vert Y^{T}...
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Formulating a problem as a mixed-integer conic program
I have the following integer optimisation problem, and I wonder whether it can be reformulated as a conic program that can be solved with, e.g., Mosek. Suppose the $n$-dimensional vectors $a, b$ and $...
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Adding valid cuts for integer feasibility problem under Benders decomposition framework?
Traditional infeasibility cut is derived under the assumption that the feasibility problem is LP instead of ILP and relies on the dual form of the LP.
Is there a systematic way of adding valid cuts ...
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Subtour-gluing constraints for ILP formulation of TSPs
If one doesn't want to introduce additional variables to the ILP of a TSP instance, one has to add exponentially many so-called subtour-elimination constraints; in practical calculations subtour-...
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Sum of all integer binary solutions of a TUM linear system
I have the following problem: $A x = b$ where $A$ is a $m \times n$ total unimodular matrix (TUM) with entries in $\{0,1\}$ and $b$ is a $m$-vector of strictly positive integers. Let $\mathcal X$ be ...
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Knapsack like problem with nonnegative weight constraint
I am dealing with a knapsack-like problem with one difference from the conventional problem: the “weights” can be positive or negative and the constraint is $\sum w_i x_i \ge 0$ instead of $\sum w_i ...
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Integer programming for bin covering problem
I encounter an integer programming problem like this:
Suppose a student needs to take exams in n courses {math, physics, literature, etc}. To pass the exam in course i, the student needs to spend an ...
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Is this variant of knapsack problem strongly NP-hard?
Suppose we have a sequence of containers each of which contains multiple items. Each item $I_i$ is associated with an nonnegative weight $w_i$, a nonnegative value $v_i$, and $I_i(C)$ denotes the ID ...
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Polyhedron coordinate bound
Given a polyhedron
$$Ax\leq b$$
where we assume $A\in\mathbb Q^{m\times n}$ and $b\in\mathbb Q^{m}$ and it takes $L$ bits to represent the inequalities what is a good bound on the quantity $\|y\|_\...
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Modular counting of integral points under sparse non-negativity
Given a polyhedron
$$Ax\geq b$$
where every entry of $A,b$ are non-negative and $A\in\{0,1\}^{m\times n}$ and there are $O(1)$ (say $\leq8$) non-negative entries per row of $A$ is it possible to ...
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Knapsack problem with capacity constraint
The traditional knapsack problem is that: given a sequence of $i$ items with positive weights $w_1,w_2,...,w_i$, positive values $v_1,v_2,...,v_i$, and a bag with capacity $B$, we want to insert items ...
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Existence of $\{0,1\}$-solution to a system of linear equations with coefficients in $\{0,1\}$
Crossposted at Theoretical Computer Science SE
A problem I study reduces to a system of linear equations $A\mathbf{x}=\mathbf{1}$ where $A$ is an $m\times n$ matrix with each entry $a_{ij}\in\{0,1\}$....
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Knapsack problem with value range constraint
The traditional knapsack problem is that: given a sequence of $i$ items with positive weights $w_1,w_2,...,w_i$, positive values $v_1,v_2,...,v_i$, and a bag with capacity $B$, we want to insert items ...
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What's the meaning of this inequality in the lot-sizing and scheduling problem
I learned about the MILP models proposed by Pochet and Wolsey. Here are the formulations of one of these models(MILP3).
So the decision variables and the primary formulation are as following:
Based ...
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What is the computational complexity of the calculation of $ \Psi(x) $?
What is the computational complexity of the calculation of $ \Psi(x) $ described below:
Let $\left\{ f_i : \{0,1,\dots,m\} \to \mathbb{R} \right\}_{i=1}^n$. For each $x \in \{0,1,\dots,m\}$ we ...
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$\mathsf{NP}$ complete version of Skolem arithmetic
Definable subsets of $\mathbb N$ in the language of Presburger arithmetic are exactly the eventually periodic sets and quantifier free part corresponds to Integer Programming with linear inequalities.
...
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Allowing an "OR" option between equations in a linear program
I am looking for a way to express an "or" option in a system of linear inequalities for a linear program I am working on.
I will explain what I mean precisely: Lets say I have a set of ...
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Complete graph invariant based on integer programming?
Roughly speaking, we are trying to find complete graph invariant
as the lexicographically first matrix from the permutations
of the adjacency matrix.
Let $G$ be graph, possibly directed graph, of ...
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Solutions to matrix equations in the non-negative integers
For an integer matrix $S$, and an integer vector $y$, I'm looking for solutions to $xS = y$ where the entries in $x$ are in the non-negative integers.
I've been doing this with Sage's mixed integer ...
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How to solve MILP problem on several linear subspaces
I have a set of close mixed-integer programming problems. More exactly, all the problems share the same set of (binary and continuous) variables, the same set of linear inequality constraints, and the ...
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Does Barvinok's algorithm apply to convex integer program?
Barvinok provided a counting algorithm to count number of integer solutions to integer linear program that runs in polynomial time if the number of integer variables is fixed.
If we have convex ...
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Structural properties of polytopes for mainstream integer or linear programs
Are there any papers/textbooks/monographs that describe distinguishing properties of the polytopes that arise when solving the linear relaxation of well-known integer programs? For example, it is ...
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Constructivity of two problems on a standard simplex?
Maximizing a hyperplane $\sum_i a_ix_i$ where $a_i\in\mathbb R$ and each $a_i$ are fixed and non-negative and $x_i$ are variables over a standard simplex $\sum_i x_i\leq 1$ with $0\leq x_i$ always ...
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Maximizing the length of a sequence under constraints
Fix $\{w_n\}_n$ a sequence of positive real numbers, fix positive integers $N,K$, and fix $\eta>1$. I'm looking for a sequence of integers $\{k_n\}_n$ optimizing the following problem:
$$
\begin{...
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Pros and cons of using integer programming alone or combined integer and global optimization?
First, I am not sure if this is the right question to ask in this forum. But I have been looking for answers for a long time and I have been also asking my university's "engineering" professors but I ...
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Deciding whether a system of linear integer inequalities has infinitely many solutions
I have a quick question that I am struggling to find a solution to:
Given a system of linear integer inequalities $A\textbf{x} \leq \textbf{b}$, where $A\in \mathbb{Z}^{m\times n}$ and $\textbf{b}\...
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Maximize sum of supermodular functions over nested sets
Let $R$ be a function that maps a set and a positive integer to a real positive number. We have that for any positive integer $t$ and $S \subseteq \{1, \ldots, t\}$, $R(S, t)$ satisfies:
For all $t &...
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Integer linear programming (ILP) formulation of connectivity of induced subgraph
Can anyone assist me to find out what should be the ILP formulation of a case when I try to label vertices by say $0$, $1$ and $2$ and want the subgraph of graph $(V,E)$ made by same vertex set but ...
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How do you refer to the feasible set of solutions to a mixed-integer program?
I frequently want to refer to the feasible set of solutions to a mixed integer programming instance. Is there a name for a subset of $\mathbb{R}^n\times\{0,1\}^m$ of the form $\{(x,a)| Ax + Ba\leq b\}$...
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Gröbner basis via integer programming
I have studied some papers related to solving integer programs via Gröbner bases. I wonder if the other way is possible or not — i.e., given any ideal, can we find the Gröbner basis by translating ...
