Questions tagged [graph-algorithms]

Algorithms on graphs, excluding heuristics.

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granularity of bidirectional breadth-first search

I tried posting this on stack overflow and it got no takers, decided to cross post here: One thing that I've never seen discussed about bidirectional breadth-first-search (which I'll abbreviate as ...
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1answer
42 views

Sample graph dataset for testing algorithms

I hope I'm addressing the right community. For a project for my students, I need to find some weighted graphs (oriented or not) to benchmark their algorithms (shortest paths, flows...). There are a ...
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The Edge Cover Equilibrium Problem

Let the Edge Cover Equilibrium Problem be the following: INPUT: a simple undirected graph $G$. OUTPUT: YES, if the number of edge covers of $G$ having odd cardinality is equal to the number of edge ...
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Algorithm to decide percentage of data from previous node, current node and forward nodes

I have a graph-like structure, let's say there is one node $C$, who has 2 predecessors $A,B$ and 1 successor $D$. I have a value of $C$. Let's say value at $C=30$%. From this, I could infer that I ...
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Find min-weight simple path assuming no negative simple cycles

A simple path is a path that doesn't revisit a node. A simple cycle is a cycle that doesn't revisit an edge. You are given an undirected graph G which may contain edges of negative weight but which ...
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Argument that Graph Isomorphism is polynomial via reduction to CNF

In short we found 3 invertible transformations which imply that Graph Isomorphism is polynomial. Meta reasoning: Isomorphism preserving transformation CNF to "sparse" CNF is possible and ...
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52 views

Comparing two graphs when starting from a single edge

Let's assume that we are given two graphs $G_1$ and $G_2$ defined by the two following nicely drawn pictures. Black numbers label the nodes, red numbers show the edge weight between the nodes. $G_1$ ...
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1answer
40 views

Find a boundary from set of 3d line segments

I have a set of n 3d line segments ...
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1answer
60 views

approximate maximum clique given vertex cover

I have a non optimal vertex cover of size k of a graph G, and I want to get a (1+epsilon)-approximation kernel of size linear in k for maximum clique of G. One thing I got is that every clique in G ...
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1answer
125 views

Neighborly properties in a bipartite graph

Input: Let $G$ be a connected, bipartite graph with parts $A$ and $B$, each of size $n$. For a set of vertices $S$, let $N(S)$ be its set of neighbors. Question: Decide whether there exists a subset $...
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Graph problems in P with unknown lower bounds

I am looking for references to interesting graph problems, which are known to be in P, but their precise big-O lower bounds are elusive. I would split this into 2 classes: problems, where we know of ...
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1answer
91 views

Intuition behind the Charikar's LP formulation for densest subgraph problem

I understand why the LP gives the optimal solution for the densest subgraph problem. But don't understand the intuition behind the LP in this paper. Just mentioning the LP for maximum density of a ...
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2answers
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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 ...
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701 views

Algorithm for finding a 3-cycle cover

Given: An undirected, unweighted graph Looking for: A disjoint vertex cycle cover where every cycle has at least 3 edges Is there any algorithm that solves this problem, possibly with some ...
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NP-Hard or PTIME?

I am working on my research problem that essentially boils down to the following question. Consider an $N \times N$ matrix. There is a man at given a starting point $(x,y)$. In each unit of time, the ...
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33 views

Efficient algorithm for finding segregators in a directed acyclic graph

Given a directed acyclic graph $G=(V,E)$, we define a $(\alpha,\beta)$-segregator of $G$ to be a subset $S$ of $V$ of size $\alpha$ such that no vertex in $G\setminus S$ has more than $\beta$ ...
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Are the intermediary sets in maximum cardinality search optimal in some way?

The maximum cardinality search (MCS) algorithm works as follows. Given a weighted graph $G = (V, E)$ where $w(u, v)$ denotes the weight of the edge $\{u, v\}$, we select a start node $a \in V$ and do ...
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1answer
137 views

How to efficiently find a loop between two nodes in a directed graph?

Given two nodes in a directed graph, how can I find a loop (if exists) that pass these two nodes? The loop cannot pass a node more than once. And if there isn't such a loop, how to efficiently ...
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Uniformly sampling or counting connected graph partitions with any number of blocks

Question: Is it possible to uniformly sample in polynomial time from the set of all connected partitions of a graph? Or is there a JVV type argument that proves this to be NP-hard? To clarify: By a ...
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Incremental PDA emptiness testing?

Is there anything known about the problem of incremental emptiness testing for a pushdown automata? Suppose you have a PDA with (up to) $n$ states and transitions, but instead of being given the ...
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1answer
67 views

3-dimensional matching variant

In the normal version of the matching problem, we are given a set of vertices $X$, $Y$, and $Z$, each of size $n$, and a set of edges $E\subseteq X\times Y\times Z$. We need to find a matching $M\...
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1answer
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What is the complexity of this weighted b-edge matching problem?

I'm wondering about the complexity of the following variant of the Generalized Weighted b-edge Matching problem: Input: An undirected multigraph $G = (V, E)$ without loops, an edge partition $(E_1,...
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1answer
94 views

Reference request: Depth- (or Breadth-) first search with hints?

Consider the standard s-t reachability problem of finding a path between nodes $s$ and $t$ in a directed graph $G$. A DFS or BFS could solve it. Would it be possible to pre-process the graph and ...
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Subgraph isomorphism on graph sequences

I'm looking for a subgraph isomorphism algorithm that takes advantage of properties of graph sequences. Say $\{G_i\}_{i=1}^k$ is a sequence of graphs on vertex set $\{1 ... n\}$, and two consecutive ...
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1answer
50 views

Distinguish Graph from Tree using Adjacency Matrix

Given an adjacency matrix, is there a way to determine if the graph will be a tree or a graph (whether or not there is a cycle). For example, given the adjacency matrix: ...
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76 views

Variation of edge-disjoint spanning trees

In a directed graph, I want to find 2 edge-disjoint spanning trees (arborescence), with the extra restrictions that edges in the 1st tree are not forward arcs in the 2nd tree. Are there existing ...
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165 views

Sublinear time path existence

Consider a graph $G = (V,E)$ with $N$ edges. Consider two vertices $u_1, v_1 \in V$. We wish to find whether there exists a path of length $4$ between these two vertices or not. This is easy to do in $...
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Alternative proofs of Savitch's theorem?

Question: Are there any known proofs of Savitch's theorem that $NL \subseteq L^2$ besides the usual one? By the usual one I mean the proof based on recursively querying whether there is a midpoint. ...
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1answer
56 views

Reachability Query for Tree

What is the best complexity for reachability queries on trees so far please? There is no constraint on the directions of the edges in the tree. According to Mikkel Thorup, there is an oracle of size $...
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38 views

Minimum rank graph cut

Consider the following problem: Input: A graph $G=(V,E)$ and a matroid $M$ on $E$, given by an independence oracle. Task: Find a cut $C\subseteq E$ in the graph, such that the rank of $C$ in the ...
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1answer
752 views

Sequential vs Distributed algo question

If a certain graph problem in the $\textbf{sequential}$ setting is proven to have "no" better constant-factor approximation algorithm than say a 2-approx. algorithm in polynomial time, then does this ...
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1answer
102 views

Is Hamiltonian cycle fixed parameter tractable with parameter clique cover?

We conjecture that Hamiltonian cycle is fixed parameter tractable with parameter clique cover, given $k$-clique cover. Let $G$ be connected simple graph. $k$-clique cover of graph $G$ is partition ...
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1answer
64 views

Does distance-2 coloring fit in Telle and Proskurowski 's algorithm for partial-k trees?

(This question is inteneded for people who have heard of "Vertex Partitioning Problems" framework of Telle and Proskurowski. Others may dig in only if they are interested in practical ...
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1answer
170 views

Minimum vertex cover and odd cycles

Suppose we have a graph G without odd cycles. Consider the minimum vertex cover problem of G formulated as a linear programming problem, that is for each vertex $v_{i}$ we have the variable $x_{i}$, ...
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1answer
102 views

How to choose good diagonals when partitioning an orthogonal polygon into rectangles?

Following this answer on MathOverflow and section 3 of the linked paper by David Eppstein on how to split an orthogonal polygon into rectangles I came to a point where I just fail to understand how to ...
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How to approach the “traveling salesman problem” with cost changing every time salesman reaches a new city

Let's say instead of finding the shortest path we have to maximize the profit in a year of the salesman under the following constraints. Salesman can go to a different city only on weekends, all ...
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1answer
64 views

Reduction graph to planar bounded treewidth and bounded diameter graph

We got reduction graph to planar bounded treewidth graph, but this is unlikely to be true. Let $H$, the planarizing gadget, be planar graph with four distinguished vertices $u,u',v,v'$ on the outer ...
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1answer
126 views

Hardness of finding if a vertex lies on a simple directed path between two vertices

Given a directed graph $G = (V, E)$ and three vertices $u, v, w \in V$. Is it NP-Hard to find whether there is a simple path from $u$ to $v$ passing through $w$? I found a couple of hardness ...
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1answer
52 views

Maximum subgraph problem with unknown complexity

Let $Q$ be a polynomial time decidable graph property. In a graph let us call a subgraph $S$ a $Q$-subgraph, if $S$ has the property $Q$. Consider the following optimization problem: Maximum $Q$-...
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Efficient game traversal of a DAG of 3-colorings

Let $X$ be a set of size $n$. Consider a game played on board $X$ by two players black and white. Starting with the empty board, each player chooses an empty spot to place a stone. Black moves ...
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1answer
65 views

Existence of graphs of every order related to Barnette’s conjecture

Consider the class C3CBP of $3$-connected cubic bipartite planar graphs. They form the class on which the (in)famous Barnette’s conjecture is based. My interest in C3CBP graphs is somewhat orthogonal ...
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2answers
57 views

Fast algorithm to find pair of triangles with a common edge in a complete graph

Suppose we have a complete graph with 4 nodes. To each triangle in this graph we assign a value $energy$ that is the multiplication of its edge weights. The question is to first find pair of triangles ...
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121 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'\...
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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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1answer
222 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 ...
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1answer
718 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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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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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 ...
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1answer
95 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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166 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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