# Questions tagged [partial-order]

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### How to compute all inequivalent (under Aut(P)) nonnegative integer weight assignments (with fixed sum) to the vertices of a finite poset P?

Let $P$ be a poset on $n$ points, $\text{Aut}(P)$ its automorphism group, and $a_1,a_2,\dots,a_k$ the lengths of the orbits under $\text{Aut}(P)$. Goal: An algorithm to generate a member from each ...
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### Finding all minimal upper bounds in a partially ordered set

I have a partially ordered set of numbers, represented as a vector<set<int>> (e.g. if $2 \preceq 4$ in this order, then ...
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### Efficient cardinality of set overlap relation

Assume that we have a set S of sets s. Every pair (s,s') in ...
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### How do we know that $F^{n + 1}(\overrightarrow{\emptyset}) = F(F^n(\overrightarrow{\emptyset}))$?

I am currently studying the textbook Principles of Program Analysis by Flemming Nielson, Hanne R. Nielson, and Chris Hankin. Chapter 1.3 Data Flow Analysis says the following: The least solution. The ...
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### Showing that $F$ is a monotone function

I am currently studying the textbook Principles of Program Analysis by Flemming Nielson, Hanne R. Nielson, and Chris Hankin. Chapter 1.3 Data Flow Analysis says the following: The least solution. The ...
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### Methods for generating DAG with small Minimum Path Cover

On a directed acyclic graph $G=(V,E)$ the Minimum Path Cover (MPC) is the minimum number of paths that can be constructed on the DAG such that all vertices are covered by at least one path. If one was ...
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### Components of subset partial order

Given a collection C of sets, there are a number of proposed algorithms for building the subset partial order, e.g. this paper. But is there any work on algorithms ...
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### Finding connected components without building the graph first

What are good algorithms for finding connected components in a graph defined by a set of elements X, where each x in ...
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### Why is there so little literature on partial order production?

Please excuse or improve the poor title of this question. My question is rather undirected, but I guess I am trying to find out if I might be missing a keyword for my problem. So there is plenty of ...
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### Optimally find one of the total orderings for a poset based on some metadata about the elements

Given a finite, partially ordered set with the following two properties: Every element in the set has one of two types: "A" or "B". The type does not define the total ordering of the set and is ...
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### What are all linear extensions of the product order of $\{1, \dots, M\} \times \{1, \dots, N\}$?

Note: I have read somewhere that finding all linear extensions of a partial order is in general a #P-complete problem (which apparently means difficult, and thus no closed form expression), but just ...
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### Is this algorithm for partial ordering of sets complete and sound?

I need to build a partial order tree of sets for analysis. Where the order is defined as A <= B <=> for all x in A, y in B, x <= y. I realized that if ...
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### Algorithm to establish a global ranking given individual rankings

I am looking for an algorithm(s) that can compute a global ranking (partial ordering) given individual rankings, in some kind of principled manner. I want to establish a partial-ordering of some ...
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### effective, efficient algorithms on antichains

In a partially ordered set L, an antichain is a subset A of L such that no two elements of A are comparable. Antichains are commonly used to represent upward-closed subsets of L, that is, sets S such ...
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### Counting number of permutations respecting partial order

Suppose that we have an array $A$ of $n$ elements with some partial order known, e.g. for example as a $n\times n$ matrix containing $c_{ij} \in \{-1, 0, 1\}$ where $0$ represents unknown and $-1, 1$ ...
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### How to convert a dependency graph to series-parallel representation?

I'm given a finite partial order, in the form of a dependency graph between items, and I'd like to have it in series-parallel form (Wikipedia). So formally, given a finite partial order $\le$ on a ...
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### Checking if the mimimum is unique

We have a finite poset and its subset $S$. We can enumerate elements of $S$ using an iterator. I need to check if there are more than one minimal elements of $S$ (regarding the above poset). The ...
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### Two related partial order relations

We will consider a set whose elements I call "precedences". Precedences are related by two relations: "is subclass of" and "higher than", which are specified by some pairs (given on algorithm input) ...
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### Path optimization in a DAG: maximizing number of least cost arcs

I've got the following problem. I've a graph $G=(V,E)$ as in the picture and I have to calculate the optimal path from $R$ to $S$. The optimal path has to maximize the number of least cost arcs. In ...
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### Algorithm for partial order width [duplicate]

I want to compute the partial order width i.e. the size of the maximum antichain in a given partial order. By Dilworth's Theorem this is the same as minimal chains required to decompose the graph, ...
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### Complete Partial Order of Partial Functions with Different Outputs

Since a partial function can be seen as a set of tuples, there is a trivial CPO defined by the subset relation on partial functions of the same (co-)domain. However, this is not really useful. What I'...
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### What is the name of the property where $f(A) \supseteq f(B)$ when $A\supseteq B$?

Suppose I have a function $f$ on sets. What is the property of $f$ called when, for all sets $x$, $y$: $f(x)$ is a superset of $f(y)$ when $x$ is a superset of $y$ i.e. \forall x,y : x\...
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### What kind of order is binary tree ancestry?

Let isAncestor be a relation on binary tree nodes such that isAncestor x y means y can be ...
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### Search in a partial ordering defined by tuples of numbers

This is a graph theory and partial ordering problem. Consider a set of triples {(di,ai,ci)}i=1...N, which specify edges between two nodes A and B, d denotes a departure time, a an arrival time and c a ...
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### What does Tarski's Fixed-Point theorem give us that that Y-Combinator does't

I'm taking a graduate course on the theory of functional programming, based on Paul Taylor's "Practical Foundations of Mathematics." I understand the statement of Tarski's theorem about how for any \$\...