Approximating the Orthogonality Dimension of Graphs and Hypergraphs

@inproceedings{Haviv2019ApproximatingTO,
  title={Approximating the Orthogonality Dimension of Graphs and Hypergraphs},
  author={I. Haviv},
  booktitle={MFCS},
  year={2019}
}
  • I. Haviv
  • Published in MFCS 2019
  • Mathematics, Computer Science
A $t$-dimensional orthogonal representation of a hypergraph is an assignment of nonzero vectors in $\mathbb{R}^t$ to its vertices, such that every hyperedge contains two vertices whose vectors are orthogonal. The orthogonality dimension of a hypergraph $H$, denoted by $\overline{\xi}(H)$, is the smallest integer $t$ for which there exists a $t$-dimensional orthogonal representation of $H$. In this paper we study computational aspects of the orthogonality dimension of graphs and hypergraphs. We… Expand
The (Generalized) Orthogonality Dimension of (Generalized) Kneser Graphs: Bounds and Applications
TLDR
It is proved that there exists a constant $c$ such that for every sufficiently large integer $t$, it is $\mathsf{NP}$-hard to decide whether the orthogonality dimension of an input graph over $\mathbb{R}$ is at most $t$ or at least $3t/2-c$. Expand
Topological Bounds for Graph Representations over Any Field
TLDR
The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries and the topological bound obtained for the minrank parameter over $R$ is improved. Expand

References

SHOWING 1-10 OF 58 REFERENCES
The Minrank of Random Graphs
TLDR
The lower bound matches the well-known upper bound obtained by the “clique covering” solution and settles the linear index coding problem for random knowledge graphs. Expand
Conditional Hardness for Approximate Coloring
TLDR
The AprxColoring problem is studied and tight bounds on generalized noise-stability quantities are extended, which extend the recent work of Mossel, O'Donnell, and Oleszkiewicz and should have wider applicability. Expand
Coloring Bipartite Hypergraphs
It is NP-Hard to find a proper 2-coloring of a given 2-colorable (bipartite) hypergraph H. We consider algorithms that will color such a hypergraph using few colors in polynomial time. The results ofExpand
Topological Bounds on the Dimension of Orthogonal Representations of Graphs
  • I. Haviv
  • Mathematics, Computer Science
  • Eur. J. Comb.
  • 2019
TLDR
General lower bounds on the dimension of orthogonal representations of graphs are proved using the Borsuk-Ulam theorem from algebraic topology to strengthen the Kneser conjecture and determine the integrality gap on the Shannon capacity of graphs and the quantum one-round communication complexity of certain promise equality problems. Expand
On the orthogonal rank of Cayley graphs and impossibility of quantum round elimination
TLDR
An exp$(n)$ lower bound is shown on the orthogonal rank of the graph on $\{0,1\}^n$ in which two strings are adjacent if they have Hamming distance at least $n/2$. Expand
Orthogonal representations over finite fields and the chromatic number of graphs
TLDR
It turns out that for some classes of matrices defined by a graph the 3-colorability problem is equivalent to deciding whether the class defined by the graph contains a matrix of rank 3 or not, which implies the NP-hardness of determining the minimum rank of a matrix in such a class. Expand
On the Quantum Chromatic Number of a Graph
We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince a referee that they have aExpand
The Complexity of Near-Optimal Graph Coloring
TLDR
It is proved that even coming close to khgr;(G) with a fast algorithm is hard, and it is shown that if for some constant r < 2 and constant d there exists a polynomial-time algorithm A which guarantees A(G). Expand
Colouring the Sphere
Let $G$ be the graph with the points of the unit sphere in $\mathbb{R}^3$ as its vertices, by defining two unit vectors to be adjacent if they are orthogonal as vectors. We present a proof, based onExpand
On the Hardness of 4-Coloring a 3-Colorable Graph
TLDR
A new proof showing that it is NP-hard to color a 3-colorable graph using just 4 colors is given, and it is pointed out that such graphs can always be colored using O(1) colors by a simple greedy algorithm, while the best known algorithm for coloring (general) 3- colorable graphs requires $n^{\Omega(1)}$ colors. Expand
...
1
2
3
4
5
...