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Questions tagged [graph-algorithms]

Algorithms on graphs, excluding heuristics.

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Restricted Reachability Problem

Let $G$ be a directed acyclic graph with $V$ vertices and $E$ edges. Choose some subset of $n\leq V$ "special" vertices $\{v_i\}_{i=1}^n$. How efficiently can we preprocess $(G, \{v_i\})$ so that we ...
Shaun Harker's user avatar
5 votes
0 answers
103 views

Fine-Grained Hardness for Undirected Hamiltonicity

The fastest known algorithm for detecting Hamiltonian cycles in directed graphs on $n$ nodes runs in essentially $2^n\text{poly}(n)$ time. However, for undirected graphs on $n$ nodes, there is an ...
Naysh's user avatar
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5 votes
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187 views

Minimum spanning tree, but with an unusual objective function

This is a problem that came up in my study of rumour networks. I was wondering if anyone had thoughts or references on this problem. If we have a rooted tree $T = (V,E)$ with root $r$, I first label ...
Karagounis Z's user avatar
5 votes
0 answers
193 views

Is this problem in P? Given a bipartite graph, find a minimum cardinality set of edges which intersect every vertex cover

This problem came up in my study of digraphs: Given a connected bipartite graph $G = (A \cup B, E)$, a vertex cover is a set $S$ of vertices such that every edge has some endpoint in $S$. Note that $A$...
Karagounis Z's user avatar
5 votes
0 answers
320 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 ...
Qian's user avatar
  • 51
5 votes
0 answers
88 views

Complexity of bounded degree full contraction

This paper defines the problem $\mathrm{B{\scriptsize OUNDED} \ D{\scriptsize EGREE}\ C{\scriptsize ONTRACTION}}$ as follows: Instance: A graph $G$ and two integers $d$ and $k$. Question: Is there a ...
delete000's user avatar
  • 828
5 votes
0 answers
94 views

Series-parallel extension of a partial order respecting a given total order

Consider a partial order $P$, a series-parallel order $Q$ and a total order $R$, such that $P \subseteq Q \subseteq R$. Given $P$ and $R$, we are asked to find $Q$ of minimum length. An $O(n^3)$ ...
Alexander Tiskin's user avatar
5 votes
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111 views

Optimal polynomial time algorithm to determine if a random graph is $k$-colorable

Let $G(n, d/n)$ be an Erdos-Renyi graph with edge probablity $p = d/n$. For any fixed $k$ sufficiently large, it is known that $d_{k-col} \sim 2 k\log k$ is the sharp threshold for $G(n, d/n)$ to be $...
Steve's user avatar
  • 451
5 votes
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84 views

Finding almost minimum cycle

Given an undirected unweighted graph, the almost minimum cycle is defined as the cycle whose length is greater than that of a minimum cycle by at most one. Itai and Rodeh in a seminal paper in 1978 ...
Sankardeep Chakraborty's user avatar
5 votes
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151 views

Online triangle counting

Please consider the following problem. It can (but probably shouldn't) be called offline version of online triangle detection on subgraphs. Given a graph $G$ and a collection $C$ of subsets of ...
ivmihajlin's user avatar
5 votes
0 answers
121 views

The distribution on the solution space induced by randomized rounding

Consider the Goemans-Williamson algorithm for the MAX-CUT problem. It is known, that if $maxcut(G) \geq 1-\epsilon$, then the algorithm returns a cut $S$ of fractional size at least $1-\sqrt{\epsilon}$...
Lior Eldar's user avatar
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165 views

Indexing structure for all-pairs min-cuts in a huge DAG

I have a huge DAG - e.g., the dependency graph of all packages in a linux distribution. Suppose I'd like to make a user-friendly tool that makes it very easy to understand how to break the transitive ...
jkff's user avatar
  • 8,951
5 votes
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132 views

Spectrum of absorbing random walk for regular graphs

I have a symmetric Markov chain given by the matrix $P$. Let $M$ be a set of special states of the chain (called marked states, they correspond to solutions to some problem). We can write $P$ as \...
costelus's user avatar
5 votes
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131 views

Why is it necessary to maintain a collection of forests in the dynamic graph data structure?

In their paper "Poly-Logarithmic Deterministic Fully-Dynamic Algorithms for Connectivity, Minimum Spanning Tree, 2-Edge, and Biconnectivity", Holm, de Lichtenberg, and Thorup describe a data structure ...
templatetypedef's user avatar
5 votes
0 answers
207 views

Standard reference for efficient computation of non-intersecting Eulerian circuit

A plane graph $G$ defines a cyclic ordering $O(v) = \langle v_1, v_2, \dotsc, n_{\deg(v)}\rangle$ on the neighborhood $N(v)$ of each vertex $v \in V(G)$. A non-intersecting Eulerian circuit $C$ is an ...
Tyson Williams's user avatar
5 votes
0 answers
155 views

Computing diameter of a 3D polyhedron

A polyhedron is given as a set of its vertex coordinates. Is it possible to find its diameter faster than $O(n^2)$? Or, maybe, some another common polyhedron representation would help fasten this?
sshilovsky's user avatar
5 votes
0 answers
389 views

Approximate c-chromatic number, each color class is P4-free (cograph)

The classic chromatic number of graph, $\chi(G)$, describes the minimum number of colors needed so that each color class is an independent set. There are many other graph coloring definitions. One of ...
Peng Zhang's user avatar
  • 1,453
5 votes
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132 views

Is minimum weight simple cycles through specified vertics fixed parameter tractable?

The problem formulation is as follows: Input: Undirected graph $G=(V,E)$, a set of vertices $S\subseteq |V|$, a weight function $w:E\to \mathbb{R}$ and a threshold $T\in \mathbb{R}$. Parameter: $|S|=...
R B's user avatar
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5 votes
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156 views

Union of two matching to maximize the number of cycles

Given $G$, $C$ and $M$, where $G$ is a graph with maximum degree $3$, $C$ is a hamiltonian cycle of $G$, and $M$ is a matching of $G$. Let $\mathcal{N}$ be the set of all matching of $C$ with size $|...
Chao Xu's user avatar
  • 4,479
5 votes
0 answers
1k views

Fast algorithm for successively merging k-overlapping sets?

Consider the following algorithm for clustering sets: Begin with $n$ sets, $S_1, S_2, \ldots,S_n$, such that $$\sum_{i = 1}^n |S_i| = m \,,$$ and successively merge sets with at least $k$ elements in ...
jonderry's user avatar
  • 747
5 votes
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213 views

Minimum infeasible subgraph in assignment problem

Given a bipartite graph $G$ with node set $(X+Y)$. Each node $x \in X$ has to be assigned to 1 node $y \in Y$. Assignment is only possible if there is an edge between $x \in X$ and $y\in Y$. ...
Joris's user avatar
  • 51
5 votes
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106 views

Quantized Unbounded Flow

I am interested in the following flow problem, since it turns out to be equivalent to a more general problem. INPUT: A graph where each edge $e$ has an integer multiplier $q_e$, and a lower bound $...
daveagp's user avatar
  • 1,100
4 votes
0 answers
94 views

Linear-time maze exploration for finite automaton with pebbles?

Blum and Kozen have shown that a robot with the computational capabilities of a finite automaton can visit all $n$ cells in a quadratic maze when the robot is equipped with two pebbles which it may ...
jfriemel's user avatar
4 votes
0 answers
98 views

Time Complexity of Pairwise Graph Connectedness

The Setup Consider the following algorithmic problem which, for now, I will call $\mathsf{2GraphConnector}$. Input: A natural number $|V|$, and a finite collection $\mathscr{E} = \left\{E_1, E_2, \...
k-variant's user avatar
4 votes
0 answers
206 views

Fastest Known Algorithm for $k$-Dimensional Matching and $k$-Exact Cover

Given a $k$-uniform hypergraph $G$ (i.e., each edge of $G$ contains precisely $k$ vertices) on $n$ vertices, the $k$-Exact Cover problem is the task of deciding if there exists $n/k$ edges in $G$ ...
Naysh's user avatar
  • 686
4 votes
0 answers
86 views

Flipping one bit to maximize BMM output

Consider a boolean matrix $A$ of size $N \times N$ and let $A^\top$ be its transpose. Let $C = AA^\top$ be the boolean matrix multiplication (BMM) result and let $c$ be the number of non-negative ...
karmanaut's user avatar
  • 1,177
4 votes
0 answers
1k 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 ...
cglacet's user avatar
  • 141
4 votes
0 answers
215 views

Find a pair of nodes with maximum sum of distances in k given trees

For k edge-weighted trees $T_1,T_2...T_k$ which contain the same set of nodes $\{1,2,... n \}$, I want to find a pair of nodes $(x,y)$ which maxifies $$\sum_{i=1}^k d_i(x,y)$$ where $d_i(x,y)$ ...
newbie's user avatar
  • 235
4 votes
0 answers
1k views

Optimizing Maximum Weighted Matching (Edmonds Blossom)

Background: I've ported Edmonds Blossom Algorithm with Maximum Weighted Matching to Java: https://github.com/simlu/EdmondsBlossom/blob/master/src/Blossom.java The original Python implementation ...
vincent's user avatar
  • 231
4 votes
0 answers
308 views

Computational Complexity of cycle double cover

Let $\mathcal{G}$ be the set of all finite simple graphs. Let graph $G\in \mathcal{G}$ and $C_G=\left <C_1,...,C_m \right >$ be a sequence of cycles of $G$ for some $m$. For every edge $e$ of $G$...
Erfan Khaniki's user avatar
4 votes
0 answers
155 views

Graph Isomorphism Algorithm of Vertex Transistive Graphs and other

What are the best known Graph-Isomorphism algorithms for below graph classes- 1.vertex-transitive, 2. edge-transitive, 3.arc-transitive (or symmetric) 4.distance-transitive. Are they GI Complete? ...
Michael's user avatar
  • 543
4 votes
0 answers
203 views

Graph Isomorphism of Strongly Regular Graph with fixed parameter

$G, H$ are strongly regular graphs with parameter $(n, r, \lambda, \mu)$ where $\lambda$ is constant. Here, $n$ is the number of total vertices. Each graph is $r$ regular. Every two adjacent ...
Michael's user avatar
  • 543
4 votes
0 answers
94 views

Polynomial Time Delay Enumeration of Maximal Bipartite Subgraphs

Let $G=(V, E)$ be an undirected simple graph. Is it known how to list all the maximal bipartite subgraphs of $G$, without repetitions, and with a polynomial time delay and a polynomial space ...
XORwell's user avatar
  • 650
4 votes
0 answers
298 views

Definition of Clique width of graph

The clique width of graph $G$ is defined as minimum number of labels required to construct $G$ by using four operations. I would like to know why the name clique width is given to this definition. ...
Kumar's user avatar
  • 2,044
4 votes
0 answers
1k views

K-path cover problem for a DAG

I am doing a little literature review and I was trying to know if, for a directed acyclic graph, the minimum k-path cover problem is solvable in polynomial time. A k-path cover is a set of paths with ...
user25296's user avatar
4 votes
0 answers
64 views

Has there been any work done on incremental connectivity in path graphs?

This set of lecture notes describes a data structure for decremental connectivity in path graphs that supports queries and removals in amortized O(1) each. Has there been any work done on incremental ...
templatetypedef's user avatar
4 votes
0 answers
202 views

Size of Independent set of sparse graphs with few triangles

Notations $\alpha(G) = $ Max sized independent set of graph $G$. $n(G) = $ Number of vertex in graph $G$. Theorem (by Ajtai et al.): For a triangle-free graph $G$ and max degree being $\Delta$, $$\...
Vivek Bagaria's user avatar
4 votes
0 answers
1k views

Path finding algorithm to maximise points of interest along the route

I am trying to write an algorithm to find a path (not the shortest one) between a given start and end point. An user will enter the start location, the end location and the available time to travel. ...
Radu-Stefan Zugravu's user avatar
4 votes
0 answers
142 views

Min Weight Complete bipartite subgraph

Suppose we are given a large bipartite graph with weighted edges, and a small parameter $d$ (e.g. $d$ is 3 or 4). What is known about the run-time to find the minimum weight complete bipartite ...
Joe's user avatar
  • 1,327
4 votes
0 answers
666 views

What about apply maxplus algebra for all-pairs shortest paths?

I didn't find deep informations on Wikipedia about all-pairs shortest path, in particular I do not know what is the best algorithm to solve this problem beyond Floyd-Warshall's one, then I do not know ...
Immanuel Weihnachten's user avatar
4 votes
0 answers
2k views

Can the Hungarian method be used with real edge weights?

I had a problem where I need to apply bipartite weighted matching on a graph where the edge weights are real (positive and negative). I have looked at several implementations of the Hungarian method ...
stressed_geek's user avatar
4 votes
0 answers
98 views

Explicit combinatorial construction minimizing intersection of sets

I'd like to know if anything is known about the following problem: Suppose we choose positive integer $t$ to be constant. Let $S = \{1,2,\dots,n\}$, where $n$ is sufficiently large. Consider a ...
Jim's user avatar
  • 41
4 votes
0 answers
276 views

Integral k-multicommodity flow with demands on acyclic digraphs wirh maximum outdegree two

It is well-known that different variants of Multicommodity flow problem are NP-complete. What is the complexity of the following variant, that is, the integral k-multicommodity flow problem with ...
h.a's user avatar
  • 523
4 votes
1 answer
367 views

How do you achieve linear time complexity of greedy graph coloring?

In most resources I could find, greedy algorithm is described as follows: for every vertex $v$, assign the minimal color not used by its neighbors. The above could be implemented as: ...
Sebastian Szczepański's user avatar
3 votes
0 answers
47 views

Cuthill - Mckee Guarantees?

I'm interested in the following problem: given $M$, a $p \times p $ symmetric sparse matrix (the number of non-zero elements in each row is at most $s \ll p$), find a matrix $B = PMP^T$ where $P$ is a ...
WeakLearner's user avatar
3 votes
0 answers
85 views

FPRAS to estimate the probability to get a cyclic subgraph of a directed graph

Consider a directed graph $G = (V, E)$ whose edges are annotated with independent probabilities of existence. This gives a probability distribution on the subgraphs of $G$; for instance, if each edge ...
a3nm's user avatar
  • 9,517
3 votes
0 answers
69 views

Property testing algorithm for isomorphism to a balanced 3-sided complete graph

I am looking for testing algorithm in the dense graph model, that checks for a graph with $3n$ vertices whether it's isomorphic to a balanced 3-sided complete graph with $n$ vertices in each set. The ...
Z.L's user avatar
  • 31
3 votes
0 answers
118 views

Cover all triangles of a graph with n subgraphs as small as possible

What is the smallest number $s(n,\Delta)$ such that for any undirected simple graph $G=(V,E)$ with $n$ vertices and $\Delta$ triangles, there exist $n$ subgraphs of $G$ covering all triangles where ...
walydna's user avatar
  • 63
3 votes
0 answers
81 views

Is there an algorithm for reducing the average row width of a sparse matrix?

Suppose I have a sparse $M \times N$ matrix $A$ and I define the "width" of each row $i$ to be: $$w_i \equiv r(A_i) - l(A_i),$$ where $r(A_i)$ is the index of the rightmost nonzero element ...
Germ's user avatar
  • 191
3 votes
1 answer
370 views

Dynamic transitive closure with immediate new reachability facts

The typical definition of dynamic transitive closure (or reachability) uses two types of queries: the first one is an update (edge deletion/insertion) and the second one is a reachability query. Thus, ...
gsv's user avatar
  • 421