12
votes
Accepted
Number of simple paths between two vertices in a DAG
Every simple path is uniquely determined by the subset of vertices that it passes through: if you topologically order the DAG (arbitrarily) then a path through any subset of vertices must go through ...
- 51k
8
votes
Accepted
Shortest distance problem with length as functions of time
This is known as the "time-dependent shortest path" problem. Indeed research has been done for this problem; see for example the classical paper by Orda and Rom, and this recent SODA paper which ...
- 7,648
5
votes
Accepted
Shortest path property and monadic second order logic
I'm assuming you want a formula $\varphi(s,t,X)$ in $MSO_2$ on graphs, stating that the set $X$ is the shortest path from $s$ to $t$. Under this meaning of "express the shortest path", no ...
- 8,843
5
votes
Accepted
Axioms for Shortest Paths
I just stumbled across this old question while conducting a lit search, and I happen to have recently gotten answers in this paper that I might as well share. I hope the combination of thread ...
- 2,403
5
votes
Accepted
Are there any heuristics that works solely on graphs?
There are known pre-processing methods that rely solely on the graph representation itself (and not on any kind of geometric embedding) to establish good heuristics for A*. Perhaps the most well-known ...
- 131
4
votes
Constant Width Max Sum Product Multi-objective Shortest path problem
For K=2, PARTITION reduces to this problem, so it is NP-hard.
Take an instance of PARTITION: a list of nonnegative integers $x_1,\dots ,x_n$, and you ask if there is a subset $I\subseteq [1,n]$ such ...
- 8,843
4
votes
Finding k shortest Paths with Eppstein's Algorithm
Pseudocode for Eppstein's algorithm (and the authors' lazy version of it) are given in:
V.M. Jiménez, A. Marzal, A lazy version of Eppstein’s shortest paths algorithm, in: 2nd International Workshop ...
- 41
4
votes
How to solve the Shortest Hamiltonian Path problem on Sparse Graphs?
Indeed: Eppstein has shown that the TSP can be solved in time $O(1.26^n)$ if all vertices are of degree at most 3.
David Eppstein:
The Traveling Salesman Problem for Cubic Graphs.
J. Graph ...
- 5,772
4
votes
Accepted
Shortest path on a hypergraph with no leftovers
This answer doesn't answer the question about previous work, but it does show the problem is NP-complete.
Lemma 1. Finding a shortest $s$-$t$ hyperpath (as defined in the post) in a given hypergraph ...
- 10.7k
4
votes
Accepted
Find the shortest s-t trail(edge disjoint path) in a graph with negative weight edges
The longest path problem can be reduced to this problem. Let $G = (V,E)$ be an instance of longest $s,t$-path problem. For each vertex $v \in V$ create two vertices, $v_{in}$ and $v_{out}$, and a ...
- 1,766
4
votes
Accepted
Polynomial time algorihtms for two variants of the decision version of longest walk problem
I deleted my previous answer because there were some inaccuracies. Also I am going to assume that either, you are looking for the longest walk, with any nodes as endpoints, or you are looking for a ...
- 329
3
votes
Accepted
How to solve the Shortest Hamiltonian Path problem on Sparse Graphs?
Considering your response in the comments where you do not necessarily need a provably-better runtime:
Have a look at the three methods described in this tutorial:
https://www.hackerearth.com/...
- 1,316
3
votes
Bellman-Ford with Non-edge-decomposable Path Weights
No, Bellman-Ford won't work because the problem you described is NP-hard.
This is pretty easy to prove. I've been able to come up with several reductions using the same strategy. The general idea is ...
- 2,768
3
votes
Solving All-Pairs Shortest Paths using a distance matrix in sub-cubic time
The discussion in this paper, Section 3 beginning on page 7, might be useful to you. It focuses on reducing distance product witnesses to distance product, which is the same as distance vs path APSP ...
- 2,403
3
votes
Accepted
Finding shortest path while maximizing the number of overlapping edges
There is a polynomial-time algorithm for this problem in the case where $C \le 2$. There is also a polynomial-time algorithm when $C > 2$, assuming paths are not required to be simple. If you ...
- 12.1k
3
votes
Accepted
What exactly is Lawler's modification to Yen's algorithm and how does it work?
Yen's original paper 1, from 1971, only establishes an upper bound of $O(Kn^4)$ operations (see Table 1).
Lawler's original paper 2, from 1972, improves the time complexity upper bound to $O(Kn^3)$.
...
- 4,461
3
votes
Accepted
Minimum Union-Sum Cost Path
EDIT (Jan 2019): Lemma 2 as currently stated below is wrong. (Indeed, given any instance, adding a single edge with a single type of very large cost will not change the instance but will yield $N(I)=1$...
- 10.7k
3
votes
Cooperative Pathfinding to minimize global costs
This may be related to network formation games - games in which players try to find a path in a network, and collaborating along a path reduces the cost for both players (i.e. the cost is shared)
This ...
- 5,571
3
votes
Shortest distance problem with length as functions of time
Are you aware of the "shortest nondecreasing paths" problem? It was defined to model situations such as these. Although it's a bit less expressive compared to your formulation, there are fast algs for ...
- 27.5k
3
votes
Accepted
Shortest path with affine updates and fixed dimension
Lemma 1. The problem is strongly NP-hard for $n=2$, even in directed acyclic graphs (DAGs).
[EDIT: strong NP-hardness depends on the encoding. See the comments at the end.]
Proof sketch. The proof is ...
- 10.7k
3
votes
Accepted
Shortest path with permutations and fixed dimension
Summary:
A dimension restriction is necessary. Lemma 2 below observes that if arbitrary dimension is allowed the problem (even restricted to permutation matrices) is at least as hard as Graph ...
- 10.7k
2
votes
What is the proof that visibility graphs can be used to compute the shortest path?
I think the first proof was given in Der-Tsai's 1978 thesis on pages 111-113.
With the above result it is immediate to realize that the shortest
path problem with line segments as obstacles can ...
- 151
2
votes
Accepted
Max Sum Product Multi-objective Shortest path problem
It seems NP-complete even with weights in $\{0,1\}$.
I reduce from the MINSAT problem: given a SAT instance, find an assignment that minimizes the number of satisfied clauses.
More precisely, an ...
- 8,843
2
votes
Dynamic programming and shortest path problem
Here's a less formal answer that I hope nonetheless addresses the spirit of the question.
Many standard dynamic-programming algorithms are easily seen to be equivalent to shortest-path (or longest-...
- 10.7k
2
votes
Accepted
Proof of SPFA's worst-case complexity?
Here's the algorithm (from the wikipedia page) then a proof of the time bound:
...
- 10.7k
2
votes
Shortest path with permutations and fixed dimension
The usual rule is to ask only one question per post. I'll answer the first, about $A=\text{Id}$. For that special case, the shortest path problem is easy to solve in polynomial time. I'll describe ...
- 12.1k
1
vote
Algorithm for Shortest Path in a DAG with Multiple Transportation Modes and Associated Setup Costs
This problem appears to be an NP problem, and here's a proof for it. Considering the presence of fixed setup costs in the problem, it brings to mind the uncapacitated facility location problem. ...
- 21
1
vote
Accepted
Multi agent path following with collision avoidance with pre-determined path
In general, your problem is NP-hard, naturally (if you have arbitrarily many agents and an arbitrary arena, of course).
Combining paths will naturally cause collisions (otherwise we wouldn't need ...
- 5,571
1
vote
When is extra vertex required in arbitrage detection using Bellman Ford?
Yes there is no reason to add the extra vertex $v_0$ since the graph is fully connected and every vertex is reachable from every other vertex (the table specified in the problem seems to be a complete ...
- 152
1
vote
Finding Cheapest n-Path
I am assuming you are given a weighted directed acyclic graph with source $s$ and destination $t$ and you want to find the shortest path from $s$ to $t$ with length exactly $n$ , this can be done ...
- 94
Only top scored, non community-wiki answers of a minimum length are eligible