Hyperbolic intersection graphs and (quasi)-polynomial time

Kisfaludi-Bak, Sándor

doi:10.1137/1.9781611975994.100

Cited by 10 publications

(15 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This in turn could allow for a higher code dimension than codes embedded in R D . Our results follow from some recent work by Kisfaludi-Bak [KB20] who proved that certain classes of hyperbolic graphs have bounded separators.…”

Section: Bounds On Transversal Gatessupporting

confidence: 71%

“…The motivation for considering codes embeddable in a hyperbolic manifold of even dimension is that such codes would have constant rate [BT16,LL17,GL14,Has13]. We use recent results by Kisfaludi-Bak [KB20] who showed that graphs locally embedded in D-dimensional hyperbolic space H D have bounded separators. For D ≥ 3, we will find that the distance of a D-dimensional hyperbolic code is upper bounded by O(n 1− 1 D−1 ).…”

Section: Summary Of Resultsmentioning

confidence: 99%

“…This leads to definition 27. A recent result by Kisfaludi-Bak [KB20] demonstrates that (ρ, w)-local graphs embedded in H D have small separators. We begin by repeating Theorem 2 from [KB20] as it applies to this class of graphs.…”

Section: Bounds On Transversal Gatesmentioning

confidence: 98%

See 2 more Smart Citations

Connectivity constrains quantum codes

Baspin,

Krishna

2021

Preprint

View full text Add to dashboard Cite

Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to process encoded information in quantum LDPC codes are poorly understood. Making progress on this problem is essential for using quantum LDPC codes in scalable quantum computers. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on geometric locality. As an application, we present limitations of quantum LDPC codes associated with local graphs in D-dimensional hyperbolic space.

show abstract

Section: Bounds On Transversal Gatessupporting

confidence: 71%

Section: Summary Of Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Connectivity constrains quantum codes

Baspin,

Krishna

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The first hopeful sign is that H 2 exhibits special properties when it comes to intersection graphs. Recently, the present author has given quasi-polynomial algorithms for several classic graph problems in certain hyperbolic intersection graphs of ball-like objects [19]. The studied problems include Independent Set, Dominating Set, Steiner Tree, Hamiltonian Cycle and several other problems that are NP-complete in general graphs.…”

Section: Introductionmentioning

confidence: 99%

“…It has already been observed in [19] that for any set P ⊂ H 2 of n points there exists a point q ∈ H 2 such that for any line through q the two open half-planes with boundary both contain at most 2 3 n points from P , that is, the line is a 2/3-balanced separator of P . Such a point q is called a centerpoint of P .…”

mentioning

confidence: 99%

A quasi-polynomial algorithm for well-spaced hyperbolic TSP

Kisfaludi-Bak

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We study the traveling salesman problem in the hyperbolic plane of Gaussian curvature −1. Let α denote the minimum distance between any two input points. Using a new separator theorem and a new rerouting argument, we give an n O(log 2 n) max(1,1/α) algorithm for Hyperbolic TSP. This is quasi-polynomial time if α is at least some absolute constant, and it grows to n O( √ n) as α decreases to log 2 n/ √ n. (For even smaller values of α, we can use a planarity-based algorithm of Hwang et al. (1993), which gives a running time of n O( √ n) .

show abstract

Clique-Based Separators for Geometric Intersection Graphs

et al. 2022

View full text Add to dashboard Cite

Let F be a set of n objects in the plane and let $$\mathcal {G}^{\times }(F)$$ G × ( F ) be its intersection graph. A balanced clique-based separator of $$\mathcal {G}^{\times }(F)$$ G × ( F ) is a set $$\mathcal {\mathcal {S}}$$ S consisting of cliques whose removal partitions $$\mathcal {G}^{\times }(F)$$ G × ( F ) into components of size at most $$\delta n$$ δ n , for some fixed constant $$\delta <1$$ δ < 1 . The weight of a clique-based separator is defined as $$\sum _{C\in \mathcal {\mathcal {S}}}\log (|C|+1)$$ ∑ C ∈ S log ( | C | + 1 ) . Recently De Berg et al. (SIAM J. Comput. 49: 1291-1331. 2020) proved that if S consists of convex fat objects, then $$\mathcal {G}^{\times }(F)$$ G × ( F ) admits a balanced clique-based separator of weight $$O(\sqrt{n})$$ O ( n ) . We extend this result in several directions, obtaining the following results. (i) Map graphs admit a balanced clique-based separator of weight $$O(\sqrt{n})$$ O ( n ) , which is tight in the worst case. (ii) Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight $$O(n^{2/3}\log n)$$ O ( n 2 / 3 log n ) . If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to $$O(\sqrt{n}\log n)$$ O ( n log n ) . (iii) Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight $$O(n^{2/3}\log n)$$ O ( n 2 / 3 log n ) . (iv) Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight $$O(\sqrt{n}+r\log (n/r))$$ O ( n + r log ( n / r ) ) , which is tight in the worst case. These results immediately imply sub-exponential algorithms for Maximum Independent Set (and, hence, Vertex Cover), for Feedback Vertex Set, and for q-Coloring for constant q in these graph classes.

show abstract

Hyperbolic intersection graphs and (quasi)-polynomial time

Cited by 10 publications

References 39 publications

Connectivity constrains quantum codes

Connectivity constrains quantum codes

A quasi-polynomial algorithm for well-spaced hyperbolic TSP

Clique-Based Separators for Geometric Intersection Graphs

Contact Info

Product

Resources

About