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# Questions tagged [graph-algorithms]

Algorithms on graphs, excluding heuristics.

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### Approximation algorithms for Metric TSP

It is known that metric TSP can be approximated within $1.5$ and cannot be approximated better than $123\over 122$ in polynomial time. Is anything known about finding approximation solutions in ...
1k views

### Examples where the uniqueness of the solution makes it easier to find

The complexity class $\mathsf{UP}$ consists of those $\mathsf{NP}$-problems that can be decided by a polynomial time nondeterministic Turing machine which has at most one accepting computational path. ...
2k views

### Are any of the state of the art Maximum Flow algorithms practical?

For the maximum flow problem, there seem to be a number of very sophisticated algorithms, with at least one developed as recently as last year. Orlin's Max flows in O(mn) time or better gives an ...
5k views

### Hard-looking algorithmic problems made easy by theorems

I am looking for nice examples, where the following phenomenon occurs: (1) An algorithmic problem looks hard, if you want to solve it working from the definitions and using standard results only. (2) ...
2k views

### What is simplest polynomial algorithm for PLANARITY?

There are several algorithms that decide in polynomial time whether a graph can be drawn in the plane or not, even many with a linear running time. However, I could not find a very simple algorithm ...
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 ...
705 views

### How to find the cycles which, together, involve the biggest number of non-shared edges in a directed graph?

I am not a computer science theorist, but think this real world problem belongs here. The problem My company have several units accross the country. We offered to employees the possibility to work ...
477 views

### Complexity of “is a graph a product”

This question arises out of pure curiosity (it came up while thinking about unshuffling a string, but I'm not sure if it's actually related) so I hope it's appropriate. There are various graph ...
941 views

### Reverse Graph Spectra Problem?

Usually one constructs a graph and then asks questions about the adjacency matrix's (or some close relative like the Laplacian) eigenvalue decomposition (also called the spectra of a graph). But what ...
621 views

### The complexity of determining if a fixed graph is a minor of another

The result by Robertson and Seymour demonstrates an $O(n^3)$ algorithm for testing whether a fixed graph $G$ is a minor of $H$. I have two and a half questions on this topic: 1) It appears that there ...
6k views

### What is the maximum number of stable marriages for an instance of the Stable Marriage Problem?

Stable Marriage Problem: http://en.wikipedia.org/wiki/Stable_marriage_problem I am aware that for an instance of a SMP, many other stable marriages are possible apart from the one returned by the ...
745 views

### Minimum Flip Connectivity Problem

I formulated the following problem today while playing with my GPS. Here it is : Let $G(V,E)$ be a directed graph such that if $e=(u,v) \in E$ then $(v,u) \notin E$, i.e., $G$ is an orientation of ...
1k views

### What is the best exact algorithm to compute the core of a graph?

A graph H is a core if any homomorphism from H to itself is a bijection. A subgraph H of G is a core of G if H is a core and there is a homomorphism from G to H. http://en.wikipedia.org/wiki/Core_%...
966 views

### Counting Isomorphism Types of Graphs

Polya's counting theorem leads to an algorithm for counting (precisely) the number of isomorphism types of graphs with $n$ vertices in $\exp (\sqrt n )$ steps. From Polya theorem you get a formula ...
899 views

### Optimization problems with good characterization, but no polynomial-time algorithm

Consider optimization problems of the following form. Let $f(x)$ be a polynomial-time computable function that maps a string $x$ into a rational number. The optimization problem is this: what is the ...
3k views

### Approximation algorithms for Maximum Independent Set on special classes of graphs

We know that Maximum Independent Set (MIS) is hard to approximate within a factor of $n^{1-\epsilon}$ for any $\epsilon > 0$ unless P = NP. What are some special classes of graphs for which better ...
522 views

### The randomized query complexity of the conjoined trees problem

An important 2003 paper by Childs et al. introduced the "conjoined trees problem": a problem admitting an exponential quantum speedup that's unlike just about any other such problem that we know of. ...
2k views

### Problems that are easy on unweighted graphs, but hard for weighted graphs

Many algorithmic graph problems can be solved in polynomial time both on unweighted and weighted graphs. Some examples are shortest path, min spanning tree, longest path (in directed acyclic graphs), ...
2k views

### Finding twin vertices in graphs

Let $G=(V,E)$ be a graph. For a vertex $x\in V$, define $N(x)$ to be the (open) neighbourhood of $x$ in $G$. That is, $N(x)=\{y\in V \,\vert\, \{x,y\}\in E\}$. Define two vertices $u,v$ in $G$ to be ...
9k views

### Vertex Cover applications in the real world

What applications does the Vertex Cover Problem have in the real world? Which industry or research projects use actually implemented software that is based on theoretical results for the Vertex Cover ...
542 views

### Is there a problem that is easy for cubic graph but hard for graphs with maximum degree 3?

Cubic graphs are graphs where every vertex has degree 3. They have been extensively studied and I'm aware that several NP-hard problems remain NP-hard even restricted to subclasses of cubic graphs, ...
1k views

### Generating a tower defense maze, aka Finding the K most vital nodes (“nodewise interdiction”) in an unweighted grid-graph

In a tower defense game, you have an NxM grid with a start, a finish, and a number of walls. Enemies take the shortest path from start to finish without passing through any walls (they aren't usually ...
2k views

### Graphs in which every minimal separator is an independent set

Background: Let $u, v$ be two vertices of an undirected graph $G=(V,E)$. A vertex set $S\subseteq V$ is a $u,v$-separator if $u$ and $v$ belong to different connected components of $G-S$. If no proper ...
2k views

### Program for computing Tree decomposition of a graph

Does anybody know of an open-source program for computing Tree decomposition of graphs for a fixed "k"(width)? I know that the problem of finding Tree-Decomposition is NP-Hard for variable "k", but my ...
907 views

### Finding a 5-cycle in a sparse graph efficiently.

(crossposted from MathOverflow) Hi, I was reading this thread: https://mathoverflow.net/questions/16393/finding-a-cycle-of-fixed-length I want to find a 5-cycle in a graph. Actually, what I really ...
798 views

### Max-Cut algorithm that shouldn't work, unclear why

OK, this might seem like a homework question and, in a sense, it is. As a homework assignment in an undergraduate algorithms class, I gave the following classic: Given an undirected graph $G=(V,E)$, ...
1k views

### Deterministic Parallel algorithm for perfect matching in general graphs?

In complexity class $\mathsf{P}$, there are some problems conjectured NOT to be in the class $\mathsf{NC}$, i.e. problems with deterministic parallel algorithms. Maximum Flow problem is one example. ...
468 views

### Is it necessary to call matrix multiplication $n$ times to find a claw

A claw is a $K_{1,3}$. A trivial algorithm will detect a claw in $O(n^4)$ time. It can be done in $O(n^{\omega+1})$, where $\omega$ is the exponent of fast matrix multiplication, as follows: take the ...
420 views

### Reducing space usage of st-connectivity with multiple passes?

Suppose a graph $G$ with $n$ vertices is presented as a stream of $m$ edges, but multiple passes are allowed over the stream. Monika Rauch Henzinger, Prabhakar Raghavan, and Sridar Rajagopalan ...
3k views

### Recognizing line graphs of hypergraphs

The line graph of a hypergraph $H$ is the (simple) graph $G$ having edges of $H$ as vertices with two edges of $H$ are adjacent in $G$ if they have nonempty intersection. A hypergraph is an $r$-...
825 views

### How fast can we compute the set inclusion poset of a set family?

Given a set family $\mathcal{F}$ of subsets of a universe $U$. Let $S_1,S_2 \in \mathcal F$ and we want to answer is $S_1 \subseteq S_2$. I am looking for a data-structure that will allow me to ...
572 views

### Complexity of finding the smallest well-covered completion

This is related to an earlier question on which graphs have the property that all maximal independent sets are maximum — such graphs turn out to be known as the well-covered graphs. Any graph $G$ is ...
929 views

### Is feedback vertex set problem is solvable in polynomial time for 3-degree bounded graphs?

Feedback Vertex Set is NP-complete for general graphs. It is known to be NP-complete for degree-8 bounded graphs due to a reduction from vertex cover. The Wikipedia article says that it is poly-time ...
1k views

### Computing the Cheeger constant: feasible for which classes?

Computing the Cheeger constant of a graph, also known as the isoperimetric constant (because it is essentially a minimum area/volume ratio), is known to be NP-complete. Generally it is approximated. ...
2k views

### Data structure for shortest paths

Let $G$ be an unweighted undirected graph with $n$ vertices and $m$ edges. Is it possible to preprocess $G$ and produce a data structure of size $m \cdot \mathrm{polylog}(n)$ so that it can answer ...
756 views

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### Connectivity of graphs by edge and vertex removal

Let us say that a graph $G$ is $(a,b)$-connected if the removal of any $a$ vertices and any $b$ edges from $G$ leaves always a connected graph. For example, a $k$-connected graph, according to the ...
1k views

### DAG reachability with O(n log n) space and O(log n)-time queries?

For a directed acyclic graph ${\langle}V,E{\rangle}$, is there a data structure that allows for reachability queries without requiring quadratic space or linear time? Ideally I seek an algorithm ...
731 views

### Complexity of a switch network problem

A switch network (the name is invented) is made with three types of nodes: one Start node one End node one or more Switch nodes The switch node has 3 exits: Left, Up, Right; has two states L and R ...
### Approximation for counting the number of simple $s$-$t$ paths in a general graph
I have been told that there are some good polynomial time algorithms for approximating the number of simple paths in an directed graph from given starting vertex $s$ to given ending vertex $t$. Does ...