Questions tagged [hypergraphs]

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Are there classes for that FO-model checking is FPT on hypergraphs?

For graphs, there are many classes that admit FPT-algorithms for model checking of first order logic, e.g. the class of nowhere dense graphs by Grohe et. al. Are there similar results for ($k$-uniform)...
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2 votes
0 answers
135 views

Which hypergraphs can be simplified by alternatively removing a hyperedge and an isolated vertex?

Let $H = (V, E)$ be a hypergraph, with $V$ the set of vertices and $E \subseteq 2^V$ the set of hyperedges. An elimination sequence on $H$ consists of alternatively removing hyperedges. Specifically, ...
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  • 7,455
1 vote
0 answers
70 views

Hardness of vertex colouring on hypergraphs with $O(\log n)$ edges

I'm interested to know whether there has been any work done on the problem in the title. For the problem to be meaningful, we would naturally need that the hyperedges must have large ($\omega(1)$) ...
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  • 331
3 votes
1 answer
138 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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1 vote
0 answers
47 views

K-Uniform Hypergraph Strong Coloring [closed]

I want to ask if strong coloring of a k-uniform hypergraph using only k colors is NP-Hard or NP-Complete? If you can add a reference this will be helpful.
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3 votes
1 answer
229 views

Shortest path on a hypergraph with no leftovers

In quantum computing, determining the code distance of a stabilizer code is similar to the shortest path problem on a hypergraph. Each node in the graph would be some sort of parity check performed by ...
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  • 1,458
3 votes
1 answer
135 views

When is hypertree width more useful than generalized hypertree width?

In analysis of CSPs, there are three width notions that are analogous to treewidth: hypertree width (hw), generalized hypertree width (ghw) and fractional hypertree width (fhw). Moreover the ...
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  • 1,422
5 votes
1 answer
184 views

Expander Graph from Hypergraph

I came up with this problem while thinking about an optimizing compiler. Let $H$ be a hypergraph. From this we construct a graph $G_H$ as follows the vertices are the hyperedges of the hypergraph. ...
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  • 153
5 votes
2 answers
196 views

Minimising the root-set of a spanning hyperforest of a hypergraph

I am interested in the complexity of a problem involving spanning hyperforests (a union of hypertrees, which covers all of the vertices) of a $k$-hypergraph. I describe the relevant definitions for ...
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3 votes
2 answers
181 views

Reduction of graph chromatic number to hypergraph 2-colorability

I'm following this paper titled "Coverings and colorings of hypergraphs" by Lovasz 1973, which is referenced in Garey and Johnson's Computers and Intractability, for the Set Splitting Problem. In this ...
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  • 141
3 votes
1 answer
99 views

Almost regular subhypergraph of hypergraph with large minimal degree

I am interested in knowing whether the following conjecture is true or not: For every $d \geq 1$ there exist constants $\delta,M_0 > 0$ such that the following holds for all $M \geq M_0$. Let $\...
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2 votes
1 answer
97 views

Counting xyz-graphs in $\mathbb{Z}_n^3$

This is a followup question to: Lower bound on the largest restrained cubic subset How many distinct xyz-graphs exist in $\mathbb{Z}_n^3$? We denote this number as $C(n)$ This question may be seen ...
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3 votes
2 answers
130 views

Lower bound on the largest restrained cubic subset

Consider an $n \times n \times n$ cube. I would like to consider subsets of points in the cube with the two following constraints: Each row in the cube (in any of the three directions) has exactly 2 ...
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1 vote
0 answers
113 views

Detecting bridges in Hypergraph S-t Reachability

Is there a fast algorithm to detect possible bridge arcs in hyperpaths from set $S$ of nodes to node $t$ in a hypergraph $G$. That is, given a hypergraph $G$, source nodes $S$, and target node $t$, ...
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  • 338
1 vote
2 answers
239 views

Reduction from independent set in hypergraphs to independent set in graphs

Let me introduce my notations. IS-H : Input : an hypergraph $G=(V,H)$, an integer $k$ Question : is there a (weak) independent set of size $k$, i.e. a set $S \subseteq V$ such that $|S| \ge k$ and ...
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  • 137
3 votes
1 answer
136 views

What is the recognition complexity of k-uniform k-partite hypergraphs? [duplicate]

We can easily recognize bipartite graphs, but I surprisingly couldn't find anything on the recognition complexity of 3-uniform tripartite hypergraphs, though I'm sure this has been studied. It's also ...
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8 votes
2 answers
279 views

Complexity of testing if a hypergraph has generalized hypertreewidth $2$

A hypergraph $H = (V, E)$ consists of a set of vertices $V$ and a set $E$ of hyperedges, i.e., subsets of $V$. The generalized hypertreewidth (GHW) parameter is a measure of the cyclicity of a ...
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  • 7,455
6 votes
1 answer
244 views

Complexity of #SAT for monotone DNF formulae whose hypergraph is a hypertree

A monotone DNF on variables $x_1, \ldots, x_n$ is a disjunction of clauses, each clause being a conjunction of some of the $x_1, \ldots, x_n$. The #SAT problem asks, given a monotone DNF $\Phi$, how ...
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  • 7,455
5 votes
1 answer
981 views

Is there an approximation algorithm for MAX k DOUBLE SET COVER?

Given a set system $(X,\mathcal S)$, let us say that a subset $\mathcal C\subseteq \mathcal S$ doubly covers a vertex $x$ in $X$ if $x$ is contained in at least two sets of $\mathcal C$. Let us define ...
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6 votes
1 answer
301 views

NP-hardness of coloring uniform hypergraphs

Since a $2$-uniform hypergraphs are just graphs. The problem of deciding if $2$-uniform is $k$-colorable for $k=1,2$ is easy, and NP-hard for $k \geq 3$ colors. This is well know and I have seen ...
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4 votes
1 answer
1k views

Hardness of 3-dimensional matching variant

I am thinking about a variant of 3-dimensional matching. In normal 3d matching, we have three sets of vertices $X$, $Y$ and $Z$, and a set of edges $E \subseteq X \times Y \times Z$. We want to choose ...
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6 votes
1 answer
219 views

Intersection models for weakly chordal graphs?

Due to the apparent unpopularity of my previous posts, I'd like to post a question about graph theory which seems to be a popular topic here :) Among the many known classes of perfect graphs, there ...
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14 votes
4 answers
1k views

Which properties of planar graphs generalize to higher dimension / hypergraphs?

A planar graph is a graph which can be embedded in the plane, without having crossing edges. Let $G=(X,E)$ be a $k$-uniform-hypergraph, i.e. an hypergraph such that all its hyperedges have size k. ...
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  • 9,378
5 votes
1 answer
311 views

Best approximation for a HYPERGRAPH-MAXDICUT problem

Consider a $(c^a,(c+d)^a,1)$-regular directed hypergraph $\mathcal{H}(a)$ on $n^a$ vertices with fixed $n\geq c+d+1$, fixed $c\geq 2$, fixed $d\geq 0$ and variable parameter $a\geq 1$ (meaning every ...
13 votes
4 answers
2k views

hardness of approximating the chromatic number in graphs with bounded degree

I am looking for hardness results on vertex coloring of graphs with bounded degree. Given a graph $G(V,E)$, we know that for any $\epsilon>0$, it's hard to approximate $\chi(G)$ within a factor ...
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  • 271
21 votes
2 answers
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 $r$-...
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  • 2,467
6 votes
0 answers
285 views

k-uniform k-partite hypergraph matching in polynomial time

I have what seems like an elementary question, but google didn't throw up any answers for it. I would appreciate any pointers users here may provide. Please note that I have also asked this question ...
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  • 779
10 votes
4 answers
733 views

What are the root difficulties in going from graphs to hypergraphs?

There are many examples in combinatorics and computer science where we can analyze a graph-theoretic problem but for the problem's hypergraph analog, our tools are lacking. Why do you think problems ...
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  • 6,940
23 votes
2 answers
831 views

Are shift-chains two-colorable?

For $A\subset [n]$ denote by $a_i$ the $i^{th}$ smallest element of $A$. For two $k$-element sets, $A,B\subset [n]$, we say that $A\le B$ if $a_i\le b_i$ for every $i$. A $k$-uniform hypergraph ${\...
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  • 13.5k
10 votes
1 answer
313 views

CSPs with unbounded fractional hypertree width

At SODA 2006, Martin Grohe and D$\acute{\rm a}$niel Marx's paper "Constraint solving via fractional edge covers" (ACM citation) showed that for the class of hypergraphs $H$ with bounded fractional ...
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  • 5,963
8 votes
1 answer
743 views

Max-clique in line graph of hypergraph

Suppose we have a multigraph (later, a multihypergraph). An edge-clique is a set of edges which all pairwise intersect (have at least one common vertex). Then any edge-clique $C$ in a multigraph ...
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  • 1,100
4 votes
0 answers
229 views

Follow-up on Nair/Tetali's Correlation Decay Tree for hypergraphs?

I'm wondering if anybody has followed up on Nair/Tetali's correlation decay tree construction for hypergraphs. I didn't find anything relevant in back-citation on google scholar. Interesting question ...
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10 votes
2 answers
480 views

Consequences of lower bounds for $\epsilon$-nets on approximation

Many here are probably aware of Alon's recent super-linear lower bounds for $\epsilon$-nets in a natural geometric setting [PDF]. I would like to know what, if anything, such a lower bound implies ...
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12 votes
1 answer
411 views

Efficient algorithm for near-optimal edge-colourings of hypergraphs

Graph colouring problems are, already, hard enough for most people. Even so, I'm going to have to be difficult and ask a problem about hypergraph colouring. Question. What efficient algorithms are ...
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19 votes
4 answers
1k views

"All-different hypergraph coloring" - known problem?

I am interested in the following problem: Given a set X and subsets X_1, ..., X_n of X, find a coloring of the elements of X with k colors such that the elements in each X_i are all differently ...
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