Sorting on graphs by adjacent swaps using permutation groups

Kim, Dohan

doi:10.1016/j.cosrev.2016.09.003

Cited by 15 publications

(8 citation statements)

References 54 publications

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“…We begin with the setting Y = K n , which has been studied in the context of circular permutations. In particular, see Procedure 3.6 of [6] for an algorithm that achieves the minimal number of (Cycle n , K n )-friendly swaps between any two permutations in S n . Recall from Section 4 that we refer to a source-to-sink flip as an inflip, and a sink-to-source flip as an outflip.…”

Section: Cycle Graphsmentioning

confidence: 99%

Diameters of Components of Friends-and-Strangers Graphs are Not Polynomially Bounded

Ryan¹

2022

Preprint

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Given two graphs X and Y on n vertices, the friends-and-strangers graph FS(X, Y ) has as its vertices all n! bijections from V (X) to V (Y ), with bijections σ, τ adjacent if and only if they differ on two elements of V (X), whose mappings are adjacent in Y . In this work, we study the diameters of friends-and-strangers graphs, which correspond to the largest number of swaps necessary to achieve one configuration from another. We provide families of constructions XL and YL for all integers L ≥ 1 to show that diameters of connected components of friends-and-strangers graphs fail to be polynomially bounded in the size of X and Y , resolving a question raised by Alon, Defant, and Kravitz in the negative. Specifically, our construction yields that there exist infinitely many values of n for which there are n-vertex graphs X and Y with the diameter of a component of FS(X, Y ) at least n (log n)/(log log n) . We also study the diameters of components of friends-and-strangers graphs when X is taken to be a path graph or a cycle graph, showing that any component of FS(Pathn, Y ) has diameter at most |E(Y )|, and diam(FS(Cycle n , Y )) is O(n 3 ) whenever FS(Cycle n , Y ) is connected. We conclude the work with several conjectures that aim to generalize this latter result.

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Section: Cycle Graphsmentioning

confidence: 99%

Diameters of Components of Friends-and-Strangers Graphs are Not Polynomially Bounded

Ryan¹

2022

Preprint

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“…These include (in historical order): cliques [10], paths [25], cycles [21], stars [1,33,31], brooms [41,23], complete bipartite graphs [46], and complete split graphs [50]. See the survey by Kim [24].…”

Section: Token Swapping On Graphsmentioning

confidence: 99%

Token Swapping on Trees

Biniaz¹,

Jain²,

Lubiw³

et al. 2019

Preprint

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The input to the token swapping problem is a graph with vertices v 1 , v 2 , . . . , v n , and n tokens with labels 1, 2, . . . , n, one on each vertex. The goal is to get token i to vertex v i for all i = 1, . . . , n using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge.Token swapping on a tree, also known as "sorting with a transposition tree," is not known to be in P nor NP-complete. We present some partial results: * This work was partially supported by NSERC.

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“…Kim (see [13]) compares some of the aforementioned routing heuristics on benchmarks with known optimal solution. His results reveal a significant margin for improvement.…”

Section: Introductionmentioning

confidence: 99%

Improving Quantum Computation by Optimized Qubit Routing

Wagner¹,

Bärmann²,

Liers³

et al. 2022

Preprint

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In this work we propose a high-quality decomposition approach for qubit routing by swap insertion. This optimization problem arises in the context of compiling quantum algorithms formulated in the circuit model of computation onto specific quantum hardware. Our approach decomposes the routing problem into an allocation subproblem and a set of token swapping problems. This allows us to tackle the allocation part and the token swapping part separately. Extracting the allocation part from the qubit routing model of Nannicini et al. ([19]), we formulate the allocation subproblem as a binary linear program. Herein, we employ a cost function that is a lower bound on the overall routing problem objective. We strengthen the linear relaxation by novel valid inequalities. For the token swapping part we develop an exact branch-and-bound algorithm. In this context, we improve upon known lower bounds on the token swapping problem. Furthermore, we enhance an existing approximation algorithm which runs much faster than the exact approach and typically is able to determine solutions close to the optimum. We present numerical results for the fully integrated allocation and token swapping problem. Obtained solutions may not be globally optimal due to the decomposition and the usage of an approximation algorithm. However, the solutions are obtained fast and are typically close to optimal. In addition, there is a significant reduction in the number of artificial gates and output circuit depth when compared to various state-of-the-art heuristics. Reducing these figures is crucial for minimizing noise when running quantum algorithms on near-term hardware. As a consequence, using the novel decomposition approach leads to compiled algorithms with improved quality. Indeed, when compiled with the novel routing procedure and executed on real hardware, our experimental results for quantum approximate optimization algorithms show an significant increase in solution quality in comparison to standard routing methods.

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Sorting on graphs by adjacent swaps using permutation groups

Cited by 15 publications

References 54 publications

Diameters of Components of Friends-and-Strangers Graphs are Not Polynomially Bounded

Diameters of Components of Friends-and-Strangers Graphs are Not Polynomially Bounded

Token Swapping on Trees

Improving Quantum Computation by Optimized Qubit Routing

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