# Questions tagged [simplex]

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### Complexity of the simplex algorithm

What is the upper bound on the simplex algorithm for finding a solution to a Linear Program? How would I go about finding a proof for such a case? It seems as though the worst case is if each vertex ...
2k views

### Consequences of existence of a strongly polynomial algorithm for linear programming?

One of the holy grails of algorithm design is finding a strongly polynomial algorithm for linear programming, i.e., an algorithm whose runtime is bounded by a polynomial in the number of variables and ...
516 views

### The structure of pathological instances for simplex algorithms

As far as I understand, all know deterministic pivot rules for simplex algorithms have specific inputs on which the algorithm requires exponential time (or at least not polynomial) to find the optimum....
2k views

### Best book on Simplex Method implementation?

I'm interested in implementing SM for LP task, however I've heard about possible pitfalls: Cormen's book says that it is possible to have input data which will make naive implementation to behave in ...
2k views

### Justification for the Hungarian method (Kuhn-Munkres)

I wrote an implementation of the Kuhn-Munkres algorithm for the minimum-weight bipartite perfect matching problem based on lecture notes I found here and there on the web. It works really well, even ...
395 views

### Motivation for Developing Shortest Path Simplex Algorithms

I'm reading Efficient Shortest Path Simplex Algorithms by Donald Goldfarb, Jianxiu Hao and Shen-Roan Kai who considered "the specialization of the primal simplex algorithm to the problem of finding a ...
918 views

### The Average-case Complexity of Simplex Algorithm

I was wondering if there are any results on the average case complexity of the simplex algorithm. Let $A \in \mathbb{R}^{m \times n}$ be the matrix in the linear constraint. I know that Smale did ...
186 views

### LP-type vs. Approximation

I'm interested in an computational geometry problem that's sensibly expressed as an infinite dimensional 0-1 integer program. I'm not worried about finding an actual minimum for the objective ...
57 views

### Using Baire Category to analyze the efficiency of the Simplex Method

I read from the wiki page of the Simplex Algorithm that we can "use Baire category theory from general topology, and to show that (topologically) "most" matrices can be solved by the ...
248 views

### Generating points uniformly distributed over the SURFACE of a standard simplex

I would like to generate points that are uniformly distributed over the SURFACE of a standard $k$-simplex ($k$ dimensions, $k+1$ vertices). One way to efficiently generate points that are uniformly ...
559 views

### equivalent way(s) of expressing P=?NP problem in linear programming?

the paper "In defense of the Simplex Algorithm's worst-case behavior" Disser/Skutella  was recently cited on this tcs.se site by saeed on another interesting question. the paper introduces the idea ...
122 views

### Enumerate all allocations of points in a simplex

Consider the standard 2-simplex $\{(x,y)~|~x+y=1~;~ x,y\geq 0\}$. Given a set $M$ of $m$ points in this simplex, we allocate each point either to X or to Y by the following process: Fix two positive ...
93 views

### Is this proof of $LP$ being in $coNP$ correct?

I am referring to the natural decision version of the Linear Programming problem: given $A \in \mathbb{Q}^{m \times n}, \ b \in \mathbb{Q}^m, \ c \in \mathbb{Q}^n, \ \alpha \in \mathbb{Q}$, does there ...
### Optimizing along a cube $s=\{0,1\}^n$
I am doing an optimization on a n-dimensional cube. That means that every solution is a set of $0$ and $1$, hence $s=\{0,1\}^n$. Most optimization algorithms though need a differential to work. E.g. ...