Questions tagged [graph-algorithms]

Algorithms on graphs, excluding heuristics.

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3
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0answers
9 views

Complexity of finding an edge set yielding specified vertex degrees

I'm trying to figure out if the following two problems are known in general to be in P or NP-complete: Q1: Given a graph $G=(V,E)$ and integers $d_i,\,1\leq\,i\leq|V|$, does there exist a subset $E'...
7
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89 views

Algebraic methods for testing planarity

Mac Lane's planarity criterion states that a graph is planar if and only if it has a cycle basis such that each edge is contained in at most two cycles, we call such a basis a 2-basis. A 2-basis of a ...
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18 views

Efficient Graph Affinity Matrix Computation

I need to compute an affinity matrix for an unweighted undirected graph of related musical artists for the purposes of spectral clustering. Now, the most obvious affinity measure to use is shortest ...
6
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1answer
174 views

Complexity of finding the largest induced subgraph with all even degrees

What is the complexity of the following problem? Instance: Simple, undirected graph $G$, and a positive integer $k$. Question: Does $G$ have an induced subgraph on at least $k$ vertices, such that ...
9
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1answer
673 views

Relationship between two graph optimization problems

Let $Q$ be a polynomial time computable graph property of simple, undirected graphs. Consider the following two optimization problems on any input graph: P1. Find a largest induced subgraph of the ...
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36 views

Scheduling and routing

Given: $k > 1 $ sales execs, each specializing in one of 4 lines of business (LOBs), where each exec works (sales and travel) at 7.5 hours / day $n > 1$ client sites. Constraints: Each ...
9
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122 views

What are some examples of algorithmic applications of noncommutative rational identity testing?

The problem of polynomial identity testing (PIT) is known to be in $\mathsf{RP}$, but not known to be in $\mathsf{P}$. The related problem of noncommutative rational identity testing (NCIT) is known ...
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14 views

Capacitated Vehicle Routing Problem with multiple pickups at some sites but constrained to max 1 pickup per vehicle per site

I am looking for literature or code for a variant of capacitated vehicle routing problem. The variance is that some sites have multiple items for pickup, but there is a constraint such that even if a ...
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114 views

Is APSP verification easier than APSP?

In APSP, the input is an $n$-node directed weighted graph $G$, and the output is an $n \times n$ matrix holding pairwise shortest path distances between nodes in $G$. Define "APSP-Verification" as ...
2
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1answer
87 views

Diameter vs. Tractability

I would like to find examples of (unweighted) optimization problems $\Pi$ that satisfy 1) or 2): 1) $\Pi$ is NP-hard but polytime solvable in the class of graphs with diameter $\le k$, for all $k\ge ...
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1answer
87 views

Finding simple fixed length paths in directed graphs

Is there an efficient algorithm to enumerate unique simple fixed-length paths (of size $k$) in directed graphs? What would be its time complexity?
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1answer
68 views

Dividing a complete graph into two cliques with maximal sum of edge weights

Problem: Considering a complete weighted graph $G$ with $n$ vertices, where $n\in2\mathbb Z$ is an even number, remove edges in such a way that you end up with two cliques of graph $G$, each having $\...
9
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2answers
273 views

How long does it take to find a short cycle in a random graph?

Let $G \sim G(n, n^{-1/2})$ be a random graph on $\approx n^{3/2}$ edges. With very high probability, $G$ has many $4$-cycles. Our goal is to output any one of these $4$-cycles as quickly as ...
10
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2answers
180 views

“Relatives” of the shortest path problem

Consider a connected undirected graph with non-negative edge weights, and two distinguished vertices $s,t$. Below are some path problems that are all of the following form: find an $s-t$ path, such ...
3
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121 views

Enumerating Minimal (a,b) vertex separators in a DAG

A vertex subset $S \subseteq V$ is an $(a,b)$ separator for nonadjacent vertices $a$ and $b$ if the removal of $S$ from a graph $G$ separates $a$ and $b$ into distinct connected components. $S$ is a ...
2
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1answer
107 views

Relationship between $O(\log n)$ (bounded) treewidth and H-minor-free

What is the relationship between graphs which have $O(\log n)$ treewidth and $\mathcal{H}$-minor-free graphs? Are graphs which have $O(\log n)$ treewidth $\mathcal{H}$-minor-free? I know that graphs ...
6
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1answer
202 views

What is the fastest known algorithm for computing a 1.99-approximation of Vertex Cover?

It is known that computing $(\sqrt 2 -\epsilon)$-approximation for VC is NP-hard and that UGC implies that even a $(2 -\epsilon)$-approximation is hard. There is also a parameterized algorithm for ...
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79 views

Shortest s-t Path with a covering constraint

Instance: an undirected graph $G=(V,E)$ with edge-weights $w:E\to{\mathbb{R}}$; a source $s\in V$ and a sink $t\in V$; a ground set $X=\{x_1, ..., x_k\}$, and for every $v\in V$ a corresponding ...
8
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1answer
224 views

3-coloring planar graphs in $O\left(3^{n^.5}\right)$?

I was wondering if the task of searching for planar 3-colorings is known to be of complexity $O\left(c^{\sqrt{n}}\right)$ or lower? This feels like it would be an intuitive consequence based from ...
2
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0answers
72 views

Algorithm for computing the smallest subset of nodes to remove from a graph to make it a tree

I have encountered an interesting problem that I couldn't find any references to solve: Determine the smallest subset of nodes that need to be removed from an undirected graph to make it a tree. ...
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0answers
44 views

Matching of two weighted graphs allowing one-to-many mapping

I am looking for a heuristic for a graph matching problem as follows. Given two graphs: $A$ (consisting of nodes $a_i$) and $B$ (consisting of nodes $b_i$). Typically the size of $B$ is larger than ...
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50 views

Directed Acyclic Graph partition into minimum subgraphs with a constraint

I have this problem, not sure there is a name for it, wherein a Directed Acyclic Graph has different colored nodes. The idea is to partition it into minimum number of subgraphs with the following 2 ...
2
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76 views

Common techniques for the acyclic orientation problem under some special constraint?

An acyclic orientation of an undirected graph is an assignment of a direction to each edge(an orientation) that does not form any directed cycle and therefore generates a directed acyclic graph(DAG). ...
5
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262 views

Does Depth-First-Search admit a quasilinear time algorithm in mutitape Turing Machine model?

Depth-First-Search (DFS) has a quasilinear (i.e.,$\widetilde{O}(m+n)$) time algorithm in random access model (RAM). I am curious about whether DFS still admits a $\widetilde{O}(m+n)$ time algorithm in ...
10
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1answer
215 views

Efficient graph isomorphism for similar graph queries

Given the graph G1, G2 and G3, we want to perform isomorphism test F between G1 and G2 as well as G1 and G3. If G2 and G3 are very similar such that G3 is formed by deleting one node and inserting one ...
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20 views

Connectivity with ordered adjacency list

In the adjacency list model, a graph is described through lists that contain the neighbors of any node $i \in [n]$. A query is of the form "What is the $k$-th neighbor of node $i$?". BFS allows to ...
2
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1answer
69 views

Computing the existence of a path in a code execution graph

I have a need for an algorithm which I can express as a reachability problem in a graph. Note that I'd appreciate any advices with respect to better wording this question. Also please tell me if this ...
0
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0answers
68 views

Unknown gaps in computation models

I'm looking for computatuon models where it is known that there are problems that we can solve in time T1 and T2. where T1 is smaller then T2 and it is unknown if there are problems where their ...
0
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1answer
87 views

finding maximum weight subgraph

My graph is as follows: I need to find a maximum weight subgraph. The problem is as follows: There are n Vectex clusters, and in every Vextex cluster, there are some vertexes. For two vertexes in ...
3
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53 views

Algorithms for Maximum weight connected subgraph in planar graphs

I wonder what is known about the two following maximisation problems. Maximum weight connected subgraph : Input : A graph $G$, with weights $w_v\in \mathbb{R}$ for each vertex $v \in V(G)$ Output :...
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1answer
51 views

Min Cut with Vertices

I have an undirected graph G with a set of vertices and edges. Each vertex has a weight w. Let's assume we have all vertices connected with some paths. I'm looking ...
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56 views

Graph automorphism with prescribed values

Consider a graph $G$ with vertices labeled $1,...,n$ and edge weights $w_{ij}$. Recall an automorphism of G is a permutation $\sigma$ of the vertex labels such that $w_{\sigma(i),\sigma(j)}=w_{ij}$ ...
2
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1answer
164 views

How many samples are needed to reconstruct a path?

Consider an input set of vertices $V$ and vertices $s,t\in V$. The goal is to learn some unknown shortest path from $s$ to $t$; the set of edges of the graph is hidden at first and there may be ...
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19 views

The set of weight functions for which the assignment problem has non-trivial solutions

The standard assignment problem is specified with a square matrix ${\bf W}$ of weights (values, costs): $$ V_{\cal P} = \sum_i w(i, b(i)) = \sum_{(i, j) \in {\cal P}} w_{ij}, $$ where $\cal P$ is a ...
0
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1answer
69 views

Finding a Hamiltonian cycle from perfect matching of a bipartite graph

A disjoint vertex cycle cover of G can be found by a perfect matching on the bipartite graph, H, constructed from the original graph, G, by forming two parts G (L) and its copy G(R) with original ...
4
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3answers
292 views

Is counting simple cycles in $P$ for graphs of bounded tree width?

Motivation: Determining if a graph has a Hamiltonian cycle is $NP$-hard in general. However, determining if there is a Hamiltonian cycle is in polynomial time on graphs of bounded tree width, either ...
0
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30 views

Alternating Delivery Problem

What is known about the complexity of the following problem: Suppose we have a complete bipartite graph $G(V,E)$ with disjoint sets $C$ and $T$. The candidate vertices, and the target vertices ...
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0answers
81 views

Minimum cut with nonlinear objective function

Let $G$ be an undirected graph. The classic minimum (cardinality) cut problem asks for a cut $C\subseteq E(G)$, such that $|C|$ is minimum. Let us generalize it the following way: let $f$ be a ...
0
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1answer
83 views

Densest k subgraph problem for outerplanar graphs?

The densest k subgraph problem aims to find a subgraph $H$ of a graph $G$ with exactly $k$ vertices that maximizes the number of edges $|E(H)|$. Does anyone know if there exists a polynomial-time ...
6
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0answers
182 views

Largest “non-disturbing” subset in a graph

The definition: The subset of vertices in a graph is called "non-disturbing" if any two vertices from this subset could be connected by a path not passing through other vertices of this subset. ...
5
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1answer
154 views

Minimum cost cut with discount - what is the complexity?

Consider an undirected graph $G=(V,E)$ with non-negative edge costs. Given an integer $k$ with $0\leq k\leq |E|$, let us call an edge set $C\subseteq E$ a $k$-discounted cut, if the following hold: $...
2
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0answers
127 views

Shortest s-t path when is allowed to ignore k weights

Given an undirected graph $G$ with $n$ vertices and $m$ edges, with non-negative weights on the edges, what's the best algorithm that computes the shortest path from $s$ to $t$, where you are allowed ...
1
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1answer
64 views

Complexity status of the Edge Deletion problem to bounded degree graphs

I'm interested in the complexity status of the following problem. Input: a graph $G=(V,E)$ and two natural numbers $k$ and $d$. Output: Yes, if there exists a subset $E' \subseteq E$ of cardinality ...
3
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1answer
146 views

How is SDP an extension of spectral algorithms?

In one of his lectures, Uri Feige described semidefinite programming (SDP) as ... an algorithmic technique that extends both linear programming and spectral algorithms. I know the basic ...
0
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1answer
154 views

Data Strcuture to represent dependencies amongst modules

Consider several software modules $m_1, m_2, ... m_n$. Each module has some inputs and outputs and the inputs to some of the modules are dependent on the outputs of some other modes. For example, in ...
3
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0answers
222 views

Generating a random connected bipartite graph

A (n, m, k)-bipartite graph is a bipartite graphs with: independent sets of size $\{n, m\}$ a total of $k \geq n+m-1$ edges We want an algorithm to generate a (n, m, k)-bipartite selected uniformly ...
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0answers
124 views

Star seperators to explain computational complexity of algorithms on a class of graphs?

A lot of NP-hard optimization problems on graphs which are perfect become solvable in polynomial time. Unfortunately, the class of graphs that arise in my problem are not perfect. The graphs can be ...
1
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1answer
83 views

Algorithm for K-best NON perfect bipartite matchings

I was reading this great article: https://core.ac.uk/download/pdf/82129717.pdf It solves a generalization of the maximum sum assignment problem by finding the k best assignments and not only the best....
6
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0answers
128 views

Grid-Minor Theorem of Robertson and Seymour and its Algorithmic Applications

Graph-Minor Theorem of Robertson and Seymour [1] states that if graph G has large treewidth, then it contains a large grid as minor. Most approximation results on general classes of graphs with ...
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0answers
40 views

Optimally fair stable matching

There's a nice post by Gil Kalai which outlines the inherent bias in stable matching algorithms quantitatively. In the traditional loyd shapeley algorithm for $n$ men and $n$ women, given randomly ...