Questions tagged [topological-sorting]

Questions related to the topological sort or topological ordering problem, on directed acyclic graphs (DAGs) or partial orders

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5
votes
2answers
145 views

Topological sorting of a DAG where special vertices have to come in even groups

Consider the following problem. The input is a directed acyclic graph (DAG) $G = (V, E)$, and a subset $V' \subseteq V$ of vertices, which we call special vertices. The question is to determine ...
3
votes
1answer
115 views

how to achieve a topological sort of an given sequence with minimum swaps

For example, given the constraints {$a<b,c<d$} and a sequence $[b,a,c,d]$. we just need swap $a$ with $b$ to get an topological sort, I want to ask how to find the sort solutions with minimum ...
7
votes
1answer
320 views

Computing topological sort while keeping edges “short”

Motivation: I want to compute a topological sort order in which the connected vertices are close to each other. Problem statement: Given a DAG $G(V,E)$ with $n$ vertices, compute a topological sort ...
1
vote
1answer
80 views

Constrained Topological Sorting with bounded number of chains

In general, constrained topological sorting is NP-hard. Now we add another constraint to it, such that take any k+1 nodes and there will be at least one pair ...
6
votes
1answer
883 views

Minimum cost topological ordering

We are given a $n$ vertex directed graph $G=(V,E)$ and also given a cost function $c:V\times [n]\to \mathbb{R}$. Consider a topological ordering of the vertices, $v_1,\ldots,v_n$, the cost of the ...
4
votes
0answers
181 views

NP-completeness of a specific topological sorting problem

Consider $(V, E)$ be a DAG, and $p_1, \dots, p_n$ be its topological sorting (i.e. such permutation $p$ of $V$ that $\forall(x, y) \in E.\ p^{-1}(x) < p^{-1}(y)$). Let's call the goodness of $p$ a ...
11
votes
1answer
750 views

Enumerating topological sorts of a vertex-labeled DAG

Let $G = (V, E)$ be a directed acyclic graph, and let $\lambda$ be a labeling function mapping each vertex $v \in V$ to a label $\lambda(v)$ in some finite alphabet $L$. Writing $n := |V|$, a ...
6
votes
1answer
503 views

Ordering of a DAG minimizing some definition of cost

Consider a DAG $(V,A)$ with a topological ordering $(v_1,v_2,\ldots,v_n)$. I define the cost of this ordering as the maximum over all $1\leq i\leq n$ of $|\{j\leq i \mid \exists k>i: (v_j,v_k)\in A\...
13
votes
2answers
3k views

Lexicographically minimal topological sort of a labeled DAG

Consider the problem where we are given as input a directed acyclic graph $G = (V, E)$, a labeling function $\lambda$ from $V$ to some set $L$ with a total order $<_L$ (e.g., the integers), and ...
3
votes
1answer
498 views

Topological sort with alternative choices of predecessors

I have a family of directed graphs over the same set of nodes $V$ defined as follows. Each node $v \in V$ has $k_v$ alternative choices for its set of predecessors. In other words, I am given a ...
29
votes
2answers
1k views

Why is “topological sorting” topological?

Why is "topological sorting" called "topological"? Is it just because it determines an order without altering any vertices or edges -- like a doughnut and coffee cup are topologically equivalent? Why ...
15
votes
3answers
948 views

Complexity of topological sort with constrained positions

I am given as input a DAG $G$ of $n$ vertices where each vertex $x$ is additionally labeled with some $S(x) \subseteq \{1, \ldots, n\}$. A topological sort of $G$ is a bijection $f$ from the vertices ...
9
votes
2answers
2k views

Testing/Identifying a Topological Sorting

You're given a set of $n$ Directed Acyclic Graphs $G_1, G_2, ..., G_n$ over the same set of $m$ vertices $V$. You're also given a permutation of the set of vertices $(v_1,v_2,...,v_m)$. What is the ...
20
votes
4answers
1k views

Positive topological ordering, take 3

Suppose we have an n by n matrix. Is it possible to reorder its rows and columns such that we get an upper-triangular matrix? This question is motivated by this problem: Positive topological ordering ...
46
votes
5answers
2k views

Positive topological ordering

Suppose I have a directed acyclic graph with real-number weights on its vertices. I want to find a topological ordering of the DAG in which, for every prefix of the topological ordering, the sum of ...