On the orthogonal rank and impossibility of quantum round elimination

Briët, Jop; Zuiddam, Jeroen

doi:10.26421/qic17.1-2-6

Cited by 8 publications

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“…The orthogonality dimension of a graph G over a field F, denoted by ξ(G, F), is the smallest integer t for which there exists a t-dimensional orthogonal representation of G over F. 1 The research on orthogonal representations and on the orthogonality dimension was initiated by Lovász [18] in the study of the Shannon capacity of graphs in information theory. Over the years, they have been found useful for applications in several areas of theoretical computer science, e.g., algorithms [2], circuit complexity [32], and communication complexity [7]. As for the computational perspective, it follows from a work of Peeters [23] that for every k ≥ 3 and a field F, it is NP-hard to decide whether an input graph G satisfies ξ(G, F) ≤ k (see [16] and [10] for related hardness of approximation results).…”

Section: Orthogonality Dimensionmentioning

confidence: 99%

Local Orthogonality Dimension

Inon¹,

Haviv²

2021

Preprint

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An orthogonal representation of a graph G over a field F is an assignment of a vector u v ∈ F t to every vertex v of G, such that u v , u v = 0 for every vertex v and u v , u v ′ = 0 whenever v and v ′ are adjacent in G. The locality of the orthogonal representation is the largest dimension of a subspace spanned by the vectors associated with a closed neighborhood in the graph. We introduce a novel graph parameter, called the local orthogonality dimension, defined for a given graph G and a given field F, as the smallest possible locality of an orthogonal representation of G over F. This is a local variant of the well-studied orthogonality dimension of graphs, introduced by Lovász (Trans. Inf. Theory, 1978), analogously to the local variant of the chromatic number introduced by Erd ös et al. (Discret. Math., 1986).We investigate the usefulness of topological methods for proving lower bounds on the local orthogonality dimension. Such methods are known to imply tight lower bounds on the chromatic number of several graph families of interest, such as Kneser graphs and Schrijver graphs. We prove that graphs for which topological methods imply a lower bound of t on their chromatic number have local orthogonality dimension at least ⌈t/2⌉ + 1 over every field, strengthening a result of Simonyi and Tardos on the local chromatic number (Combinatorica, 2006). We show that for certain graphs this lower bound is tight, whereas for others, the local orthogonality dimension over the reals is equal to the chromatic number. More generally, we prove that for every complement of a line graph, the local orthogonality dimension over R coincides with the chromatic number. This strengthens a recent result by Daneshpajouh, Meunier, and Mizrahi, who proved that the local and standard chromatic numbers of these graphs are equal (J. Graph Theory, 2021). As another extension of their result, we prove that the local and standard chromatic numbers are equal for some additional graphs, from the family of Kneser graphs. We also study the computational aspects of the local orthogonality dimension and show that for every integer k ≥ 3 and a field F, it is NP-hard to decide whether the local orthogonality dimension of an input graph over F is at most k. We finally present an application of the local orthogonality dimension to the index coding problem from information theory, extending a result of Shanmugam, Dimakis, and Langberg (ISIT, 2013).

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Section: Orthogonality Dimensionmentioning

confidence: 99%

Local Orthogonality Dimension

Inon¹,

Haviv²

2021

Preprint

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“…(1. 19) The only inequality in (1.6) not implied by considering G and w = 1 in (1. 19) is χ f (G) ≤ χ(G).…”

Section: Antiblocking Dualitymentioning

confidence: 99%

“…19) The only inequality in (1.6) not implied by considering G and w = 1 in (1. 19) is χ f (G) ≤ χ(G). However, it is easy to directly prove this inequality using (1.17).…”

Section: Antiblocking Dualitymentioning

confidence: 99%

“…The graph parameter ξ has been studied as the minimum positive semidefinite rank of a graph [10,35]. Briët and Zuiddam [19] bound ξ on Cayley graphs and use those bounds to prove a conjecture on quantum interactive protocols.…”

Section: The Orthogonal and Projective Rankmentioning

confidence: 99%

“…has nonempty interior, this implies that unit(β) has nonempty interior. We are using that β is positive definite to be sure that β(1) > 0 in (4 19…”

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Combinatorial and geometric dualities in graph homomorphism optimization problems

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Local orthogonality dimension

Inon

Haviv

2023

Journal of Graph Theory

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An orthogonal representation of a graph over a field is an assignment of a vector to every vertex of , such that for every vertex and whenever and are adjacent in . The locality of the orthogonal representation is the largest dimension of a subspace spanned by the vectors associated with a closed neighborhood in the graph. We introduce a novel graph parameter, called the local orthogonality dimension, defined for a given graph and a given field , as the smallest possible locality of an orthogonal representation of over . This is a local variant of the well‐studied orthogonality dimension of graphs, introduced by Lovász, analogously to the local variant of the chromatic number introduced by Erdös et al. We investigate the usefulness of topological methods for proving lower bounds on the local orthogonality dimension. Such methods are known to imply tight lower bounds on the chromatic number of several graph families of interest, such as Kneser graphs and Schrijver graphs. We prove that graphs for which topological methods imply a lower bound of on their chromatic number have local orthogonality dimension at least over every field, strengthening a result of Simonyi and Tardos on the local chromatic number. We show that for certain graphs this lower bound is tight, whereas for others, the local orthogonality dimension over the reals is equal to the chromatic number. More generally, we prove that for every complement of a line graph, the local orthogonality dimension over coincides with the chromatic number. This strengthens a recent result by Daneshpajouh, Meunier, and Mizrahi, who proved that the local and standard chromatic numbers of these graphs are equal. As another extension of their result, we prove that the local and standard chromatic numbers are equal for some additional graphs, from the family of Kneser graphs. We also study the computational aspects of the local orthogonality dimension and show that for every integer and a field , it is NP‐hard to decide whether the local orthogonality dimension of an input graph over is at most . We finally present an application of the local orthogonality dimension to the index coding problem from information theory, extending a result of Shanmugam, Dimakis, and Langberg.

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On the orthogonal rank and impossibility of quantum round elimination

Cited by 8 publications

References 0 publications

Local Orthogonality Dimension

Local Orthogonality Dimension

Combinatorial and geometric dualities in graph homomorphism optimization problems

Local orthogonality dimension

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