Algorithms on graphs, excluding heuristics.

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Approximating the clique size of the graph

Let $G=(V,E)$ be a graph. For a given $\rho \leq |V|$ and $\epsilon$ with $(0<\epsilon<1)$, is there any sublinear query algorithm known/possible to decide if the graph has a clique of size ...
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77 views

Simple path on dag with backward edges

What is the complexity of the following problem ($\in$ P? NP-hard?): Input: a directed acyclic graph $D=(V,E)$, a set of backward edges $E'\subset V\times V$, and two distinct nodes $s$ and $t$. ...
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105 views

Special properties of bipartite expanders

It is well known that expanders, and often the special case of bipartite expanders, have found many uses in derandomization, coding, etc. However, I am curious if there are any special properties of ...
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115 views

What is the intuition behind “hardness of approximation”? [closed]

I am reading a paper about graph matching problem. Which is, to some extent, an optimization version of the graph isomorphism problem. To my surprise, some closely related NP-hard problems are quite ...
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129 views

How does Camerini's algorithm for minimum-bottleneck-spanning-tree run in linear time?

I'm having a difficult time understanding Camerini's algorithm because there are very few clear explanations online. The goal is to find a minimum-bottleneck spanning tree in linear time. Camerini's ...
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Lower hull of a convex hull in 3D in LEDA

I want to compute the lower hull of a convex hul in 3D. I am working in LEDA. So, the function used in LEDA for computing the convex hull returns the planar graph of the convex hull. I can get the ...
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231 views

Can short-distance connectivity be harder than connectivity?

Has anybody seen the following (or similar) question being considered: Can it be easier to determine the presence/absence of $s$-$t$ paths than to determine the presence/absence of short $s$-$t$ ...
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Hardness of approximately counting independent sets with a PRAS, rather than FPRAS

It is known that approximately counting the independent sets of a graph is hard, even if randomness can be used, and even if we restrict ourselves to bounded degree graphs with degree bound at least ...
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Applications of network clustering coefficient

Consider the global clustering coefficient of a graph as defined here . The clustering coefficient describes how likely it is for a random connected triplet of vertices to be closed. My question ...
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Are there sparsifiers that approximate vertices rather than edges?

Originally introduced by Benczur and Karger, cut sparsifiers let one take a dense graph $G=(V,E)$ and produce a weighted sparse graph on the same vertex set, where - only knowing the sparse graph ...
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74 views

CSP-problem, based on context-free grammar

I'm trying to solve a CSP (Constraint-Satisfaction-Problem), which is based on arbitrary context-free grammars. A quick example: Let's say we have a context-free grammar with the following production ...
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About the sparsest-cut question

Can someone kindly help clarify as to exactly what is the generally accepted definition of the "sparsest cut" problem for a graph? (Isn't the set which achieves the Cheeger constant for a graph, ...
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Efficient generation of Tournament Graphs

How to generate all non-isomorphic tournament graphs of order $n$ in an "efficient" way ? nauty (http://cs.anu.edu.au/~bdm/nauty/) can generate non-isomorphic tournaments, what is the complexity of ...
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577 views

Time complexity of counting triangles in planar graphs

Counting triangles in general graphs can be done trivially in $O(n^3)$ time and I think that doing much faster is hard (references welcome). What about planar graphs? The following straightforward ...
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297 views

Generalized Geography on graphs of bounded treewidth

Generalized Geography (GG) is played on a directed graph where a token is moved along arcs alternatively by two players. The vertices from which the token leaves are deleted. When a player cannot play ...
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81 views

Maximum number of geometrically disjoint paths - is the complexity known?

Let $G$ be an undirected graph, given with a planar drawing. We do not assume $G$ is a planar graph, we just fix a planar representation of it, such that each vertex is represented by a distinct ...
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107 views

Does it help for clique if the vertices are partitioned into 3 cliques?

A graph is $(p,q)$-colorable if its vertices can be partitioned into $p$ cliques and $q$ independent sets. For $(2,0)$-colorable graphs clique is polynomial. I am interested how easier (if any) is ...
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488 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 ...
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Generalized geography on solid grid graphs

A post here For which families of graphs is Generalized Geography in $P$? mentioned that generalized geography on solid grid graphs is open. Is the question still open? A quick search on Google shows ...
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139 views

Efficient recalculation of the maximum flow when edge capacities are reduced

Assume that we have a (directed) graph $G(V \cup \{s, t\}, E)$ and an (integer) capacity function $c : E \mapsto \mathbb{N}$. Let $f : E \mapsto \mathbb{N}$ be a maximum $s-t$ flow on this graph. ...
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110 views

Many-one reduction from inequality problem to equality problem

Let the k-inequality-MIS problem be the decision problem whether an arbitrary graph $G=(V, E)$ contains a maximal independent set of at least size $k$, that is the corresponding language is: ...
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261 views

Has this formulation of pursuit evasion been researched? Similar to Helicopter Cops and Robbers Game

There are pursuers and evaders in the vertices of a directed graph G with one component. Each vertex must have atleast one outgoing edge (can be a loop). At each time t: The evaders must move ...
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Detect highly weighted but also densely inter-connected subnetworks

In a connected / undirected / node weighted (with both positive and negative weights) network, there are many papers studied about the 'Maximum weight connected subgraph' problem. But are there any ...
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Fastest polynomial time algorithm for solving minimum cost maximum flow problems in bipartite graphs

Orlin's algorithm is known to solve minimum cost maximum flow problems in $O(|E|^2 \log |V| + |E| \; |V| \log^2 |V|)$ time, where $|E|$ and $|V|$ respectively denote the cardinalities of the edge and ...
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Transitive reduction not provably minimal

Working on finding minimal equivalent graphs, which unlike transitive reductions only allows for edge removals from the original graph. I was under the impression that if you allow for new edges to be ...
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Is 4-colors precoloring extension for planar graphs fixed parameter tractable?

Given a planar graph $G=(V,E)$, there exists a quadratic algorithms for 4-coloring $G$ (and $G$ is surely 4-colorable). Assume you are given a set of $k$ constraints of the form "$v_i \text{ is ...
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132 views

Finding a minimum directed cut that splits a weakly connected bipartite graph into two such non-edgeless graphs

Consider a directed, weakly connected bipartite graph $G = (U, V, E)$ where $U$ and $V$ are sets comprising the nodes of $G$, and $E \subseteq U \times V$ is the set of edges. The task is to find a ...
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140 views

Algorithms for computing the minimal vertex separator of a graph

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 ...
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Maximum flow for all edges in an undirected graph

I was wondering if the following problem has been studied in the past, and what are some of the best ways people can come up with to solve it: Let $G=(V,E)$ be an undirected graph, which we use as a ...
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107 views

What is the Algorithm to find all the possible chordal graphs which can be formed by a given 'n' number of vertices

A chordal Graph is a connected graph which contains no chord-less cycle of size greater than three. They are also called as Triangulated graphs. All Paths are Chordal Graphs (No cycles). All Trees ...
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Is the complexity of this path problem known?

Instance: An undirected graph $G$ with two distinguished vertices $s\neq t$, and an integer $k\geq 0$. Question: Does there exist an $s-t$ path in $G$, such that the path intersects at most $k$ ...
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Which graph problems are $W[1]$-Hard on directed(/weighted) graphs but FPT on undirected(/unweighted) graphs?

Following the equivalent questions regarding NP-Completeness (see the weight question and the directed question), I was wondering how parameterized problems are affected by these attributes. ...
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DAG reduce edges by transitivity

I have a DAG like this $G_1 = \lbrace A \to B \to C \rbrace$ My algorithm modify $G_1$ so it will be like this $ G_2 = \lbrace A \to C, B \to C \rbrace $ I now that $G_2$ is not a transitive ...
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How much faster is solving Clique in properly colored graph?

Given a graph $G$ and a proper vertex coloring $C$ with the minimum numbers of colors, how much faster can a maximum clique be found than when just $G$ is given? Additional information doesn't make ...
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48 views

polytime transformation from a graph to a set of binary strings

$d_H$ denotes the Hamming distance between two binary strings of the same size. The problem is stated as follows. Given any undirected graph $(V, A)$, does there always exist a one-to-one ...
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Minimum vertex cover on k-regular graphs, for fixed k>2 NP-hard proof?

Good Evening! I am aware that the minimum (cardinality) vertex cover problem on cubic graphs ($3$-regular) graphs is $NP$-hard. Say positive integer $k>2$ is fixed. Has there been any ...
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199 views

Graph with minimum number of edges having given sets of nodes as its paths

Consider the following problem: Input: a list of subsets $P_1, P_2, \ldots, P_k \subseteq V = \{1, \ldots, n\}$ Output: a graph $G = (V,E)$ with minimum number of edges such that for every $P_i$ ...
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Leader election algorithm in a grid

I have to write a leader election algorithm in an unoriented mesh (a grid a*b), with many initiators. Someone give me an indication to wake up each node and then make an election in the exterior ring ...
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Equivalence of deterministic finite transducers over finite/infinite words

Equivalence of deterministic finite transducers - a special case of single-valued finite transducers - is decidable because it is decidable whether a transducer is single-valued. Note that two ...
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What mathematical models can analyze and optimize such message passing system?

I look for a mathematical model that can accommodate, analyze and suggest optimizations for a system that can be humanly described as message passing black box programs to which where optimal message ...
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Graph generation model effect on performance of Spectral graph algorithms

I use spectral graph algorithms for finding community structures, specifically the Leading Eigenvector Method (http://arxiv.org/abs/physics/0605087). I try analyzing the performance of these ...
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Is this dense version of Kruskal's algorithm well-known?

About a year ago, a friend and I thought of a way to implement Kruskal's algorithm for dense graphs in better than the usual $O(m \log m)$ bound (without assuming pre-sorted edges). Specifically, we ...
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Existing technique to detect the clique and biclique cover the edges in a graph?

I wonder if there is any existing method to find the clique and biclique structures in a graph that can cover all the edges in it, and every edge belongs to exactly one of the cliques or bicliques ...
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116 views

Why does the transformation in the proof for SL=L preserve connectedness of s and t?

I'm currently working on this paper showing that $\mathcal{SL} = \mathcal{L}$. But I keep wondering how the transformation $\tau$ in definition 3.1 preserves the connectivity of $s$ and $t$. $\tau$ ...
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Realization of a bipolar orientation by a mixed graph

Given an undirected graph $G(V,E)$ and a bipolar orientation $s$ over $G$, consider the problem of identifying $s$ by finding the minimum number of edges such that when orienting them in a particular ...
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Total number of spanning trees of a set of graphs with constraint

This is an extension of the question "Total number of spanning trees of a set of graphs". The original problem has been shown to be #P-complete. Now a new constraint is added to the problem. I have ...
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Maximum weight “fair” matching

I'm interested in a variant of the maximum weight matching in a graph, which I call "Maximum Fair Matching". Assume that the graph is full (i.e. $E=V\times V$), has even number of vertices, and that ...
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Extending clique-percolation to bipartite graphs

Clique percolation is well-defined for general graphs. In bigraphs, however, there are only 2-cliques, so the algorithm does not work. There a several extensions of the definition of a clique for ...
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130 views

checking isomorphism between K regular graph

Problem Input is a k regular graph of n vertices and I have to check whether this is isomorphic to another given k regular graph G. This is a restricted version of graph isomorphism in the sense that ...
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State of the art algorithms for community detection in graphs

Is anyone aware of the must read papers to get knowledge of the most recent algorithms and method for community detection in graphs, especially those that represent social networks?