# Questions tagged [integer-programming]

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### Do there exist two equivalent objective functions one of which can be approximated but another one cannot?

I have two equivalent problems A and B, meaning that the optimal solution of one must be the optimal solution of another one. However, it seems that problem A can be approximated but B cannot. Below ...
1answer
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### How hard is this combinatorial optimisation problem?

Suppose we have multiple intervals $R_1,R_2,...,R_i$ of non-negative integers. These intervals may overlap and we use $R_h(\mathrm{median})$ to denote the median integer in the $h$-th interval $R_h$, ...
1answer
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### Characterization of integral polyhedra

A rational polyhedron $P \subseteq \mathbb{R}^n$ is an integral polyhedron if it is the convex hull of its integer points. That is, if $P = conv(P \cap \mathbb{Z}^n)$. Equivalently, $P$ is integral if ...
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### Existence of $\{0,1\}$-solution to a system of linear equations with coefficients in $\{0,1\}$

Crossposted at MathOverflow 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 $a_{ij}\in\{0,1\}$. $\mathbf{1}$ is ...
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### Optimum partitioning of vertices into mutually disjoint subsets in a weighted graph

tl;dr I'm trying to partition my students into groups with respect to their preferences, i.e. they can declare if they want to be with someone in a group or if they do not want to be with someone in a ...
0answers
47 views

### what is the LP gap of this particular non metric facility location problem in planar graphs?

Suppose I have a facility location problem with all service costs 0 (or infinity (in particular service costs are not metric)) such that the edges $ij$ for which facility $i$ can service client $j$ ...
0answers
92 views

### (Integer) Linear Program formulation of planarity?

Q: Is there an efficient (I)LP formulation of planarity? More specifically, I am looking for a set of constraints that are satisfied by exactly all planar graphs on $n$ vertices, in order to optimize ...
1answer
104 views

### Ensuring integral maximizer from integral linear program

An integral linear program is one that has a maximizer that is integral. Sometimes it's possible to prove that a particular LP has this property, for example by proving that it's constraint matrix is ...
0answers
92 views

### Is the edge cover polytope integral on graphs with self-loops?

It is well known that the edge cover polytope is integral on simple graphs. I am wondering whether this also holds for graphs with self-loops. Here is a Linear Relaxation of the edge cover polytope, ...
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114 views

### Is the matching polytope integral?

In this document https://courses.engr.illinois.edu/cs598csc/sp2010/Lectures/Lecture9.pdf they prove the integrality of the matching polytope using the integrality of the perfect matching polytope. The ...
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85 views

### Separation oracle for hitting all small cut on a graph?

We are given as input an undirected graph $G=(V,E)$, weights $w_e \ge 0$ for all $e\in E$ and an positive integer $k$. We aim to select a set of edges with the minimum weight, such that the cut set of ...
1answer
86 views

### Name of (and solution to) this generalization of linear assignment

I would like to know if the following problem is known and has any efficient solution. Given an $n\times n$ score matrix $S$. Find the best $a$ elements, in terms of their sum of scores, such that no ...
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### An integer programming on maximum number of triangles

Consider three distinct sets $A, B, C$ of $N$ elements respectively. For each pair of sets, say $A,B$, we introduce an variable for each combination of values $a,b$ as $x_{ab}$. Similarly, we ...
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### Polynomial cases of 0 1 quadratic programm with linear constraints

A pseudo boolean function f:{0,1}^n-> R is defined as f(x)= x^tQx +cx where Q is a symmetric matrix with null elements in the diagonal. Finding the minimum of this function is solvable in polynomial ...
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### Fixed parameter Integer Programming circuit depth complexity

It is well known Lenstra's and Kannan's algorithm achieves $n$-variable parameter $L$-bit integer programming solvability in $O(n^nL)$ time and $O(L)$ space. If implemented as an arithmetic circuit ...
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### Under what conditions does an Integer Programming problem run in polynomial time?

Given $AX\leq B$ where $A\in\Bbb Z^{m\times n}$,$B\in\Bbb Z^m$ finding $X\in\Bbb Z^n$ where $m\geq n$ is the integer programming problem. If $A$ is totally unimodular then the problem is solvable in ...
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