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Results tagged with optimization
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user 2205
Questions about problems that entail selecting the best element from some set of available alternatives, and methods to solve them.
3
votes
Formulating a linear program s.t. only extreme point solutions are found
Perturb the vector of the objective function slightly. If you got multiple optimal solutions than this vector is orthogonal to a facet $F$ of the polytope that defines the feasible area of the LP. Not …
- 12.3k
14
votes
Accepted
Given a set of sets, find the smallest set(s) containing at least one element from each set
The problem is the well-known NP-complete problem Hitting Set. It is closely related to Set-Cover. The NP-completeness proof can found in the classic book of Garey and Johnson.
If you want to approxi …
- 12.3k
7
votes
Accepted
Randomized Rounding of Solutions to Linear Programs
Of course, if you round, you have to verify that rounding preserves feasibility.
Let us for example consider the relaxed VERTEX-COVER LP formulation.
$$
\begin{array}{lll}
\text{min} & \sum_{v\in V}c …
- 12.3k
7
votes
Accepted
Find maximum distance between elements given constraints on some
What you describe is a linear program. You can use the following formulation:
Let $x_i$ be the $i$th element. The variables are $d_1,\ldots, d_{n-1}$, where $d_i$ denotes difference between $x_{i+1}$ …
- 12.3k
3
votes
Finding the required value of an algebric expression
This is classical Integer Linear Programming. You have the following problem:
$$\begin{align}
Ax+Bx+Cx & \to \max \\
\text{s.t} \quad x,y,z &\ge 0\\
Ax+Bx+Cx &\le T \\
x,y,z &\in \mathbb{Z}\\
\end{al …
- 12.3k
3
votes
Given an optimal solution to the LP, show how it can be used to construct a directed cycle w...
I traced the LP back to this article. Here you can find the answer to your question presented in Theorem 2.
It might be helpful to look at the dual problem of your LP. This is analyzed in the above p …
- 12.3k
0
votes
Accepted
ordered uniform distribution
You can now turn your decision problem into an optimization problem. For example with the technique called parametric search. …
- 12.3k
15
votes
How many cookies in the cookie box? -- Tiling stars
Let me answer your question partially for the hexagram case.
You can make the following tiling
$\hskip1.2in$
By this you will cover 12/14=6/7 of the plane (count the triangles in the dashed quadril …
- 12.3k