Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [integrality-gap]

The integrality gap is the maximum ratio between the values of the optima of the integer program (IP) and of its LP relaxation.

The integrality gap is the maximum ratio between the values of the optima of the integer program (IP) and of its LP relaxation.

8 questions
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Proving that a polyhedron has integer vertices

Prove that the polyhedron $$ P = \left\{ (x_1, \ldots , x_m, y) \in \mathbb R_{+}^{m+1}: y \leq 1,x_i \leq y \text{ for } i = 1,\dots, m \right\} $$ has integer vertices. It seems obvious to me but don't know how to prove it. Any thoughts ?
Bassem
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Proof of binary solution of a linear program with specific structure

When solving instances of the following linear program (LP), I always get an integral (actually binary) solution. Is it just a coincidence or is it possible to prove that there always exists a binary solution? (Sorry if the definitions are hard to…
hdb
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Is there a theorem that states that Integer Linear Problems with a Totally Unimodular constraint matrix are solvable in polynomial time?

Is there a theorem that states that Integer Linear Problems with a Totally Unimodular constraint matrix are solvable in polynomial time? If the answer is positive, is it also valid for Mixed-Integer Linear Problems? And in which books can I find…
Mathgineer94
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How to prove LP integral solution when total unimodularity fails?

I have a problem that I managed to write as a binary integer linear program. As a natural first step, I relaxed the integrality constraint to solve a regular LP. To my surprise, the solutions where all integral (either $0$ or $1$). I then ran some…
AspiringMat
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Adapting a proof to show that a polyhedron has integer extreme points

Prove that the polyhedron $$P =\{(x_1, \ldots , x_m, x_{m+1}): 0\leq x_m \leq 1, 0 \leq x_i \leq x_{m+1} \text{ for } i = 1, \ldots , m \}$$ has integer extreme points. My Attempt: I have a similar proof in my notes to prove that the polyhedron…
UniStuffz
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Relationship between Optimal Solutions of a Linear Program and its Integer Counterpart with Binary Constraint Matrix

Consider a matrix $\mathbf{A} \in \{0, 1\}^{K \times N}$ and a vector $\mathbf{b} \in \mathbb{Z}_{>0}^{K \times 1}$ (i.e., elements of $\mathbf{A}$ are either $0$ or $1$, and elements of $\mathbf{b}$ are positive integers). We define two…
Andrew Follett
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Integral solution of a linear program

Consider the following linear program (LP) $$ \begin{align*} &\text{maximize }& &c^\prime x\\ &\text{subject to }& &Ax \leq b\\ & && 0\leq x_i \leq 1 \quad \forall i \end{align*} $$ We know that $b$ is integral, $A_{i,j} \in \{-1,0,1\}$. However,…
hdb
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Integrality Gap and Linear Relaxation vs. Binary Problem

For the following problem, can we say that its linear relaxation is equivalent to the binary problem? Problem 1 ($y_j$ and $u_j$ are $0-1$ parameters.): Given that $u_j=0$ the problem becomes (as $z_{ij}$ becomes zero): Problem (2) Given that…
madetolast
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