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Approximation and Hardness of Token Swapping

Tillmann Miltzow, Lothar Narins, Yoshio Okamoto, Günter Rote, Antonis Thomas, Takeaki Uno
unpublished
We consider the algorithmic question of finding a shortest sequence of swaps such that token T i is on vertex i.  ...  . , n}, we place on every vertex a token T 1 ,. .. , T n. A swap is an exchange of tokens on adjacent vertices.  ...  Both of these constructions 66:8 Approximation and Hardness of Token Swapping involve some auxiliary input tokens that have set destinations within the gadget.  ... 
fatcat:xnf4ri3xyrdh5fr3ruszp46cra

Approximation and Hardness for Token Swapping

Tillmann Miltzow, Lothar Narins, Yoshio Okamoto, Günter Rote, Antonis Thomas, Takeaki Uno
2016 arXiv   pre-print
,n}, we place on every vertex a token T_1,...,T_n. A swap is an exchange of tokens on adjacent vertices.  ...  We consider the algorithmic question of finding a shortest sequence of swaps such that token T_i is on vertex i.  ...  Tillmann Miltzow is partially supported by the ERC grant PARAMTIGHT: "Parameterized complexity and the search for tight complexity results", no. 280152.  ... 
arXiv:1602.05150v2 fatcat:zxocac5lovgethdeqy4247gici

Hardness of Token Swapping on Trees [article]

Oswin Aichholzer, Erik D. Demaine, Matias Korman, Jayson Lynch, Anna Lubiw, Zuzana Masárová, Mikhail Rudoy, Virginia Vassilevska Williams, Nicole Wein
2022 arXiv   pre-print
In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph.  ...  In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching.  ...  We thank the other participants of that workshop for providing a stimulating research environment.  ... 
arXiv:2103.06707v2 fatcat:bgbbivofd5fa3exti6ho6uikou

Improving Quantum Computation by Optimized Qubit Routing [article]

Friedrich Wagner, Andreas Bärmann, Frauke Liers, Markus Weissenbäck
2022 arXiv   pre-print
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.  ...  Furthermore, we enhance an existing approximation algorithm. We present numerical results for the integrated allocation and token swapping problem.  ...  Acknowledgements This research has been supported by the Bavarian Ministry of Economic Affairs, Regional Development and Energy with funds from the Hightech Agenda Bayern as well as the Center for Analytics  ... 
arXiv:2206.01294v2 fatcat:hm44saanurgivg6mwrno2mreha

Token Swapping on Trees [article]

Ahmad Biniaz, Kshitij Jain, Anna Lubiw, Zuzana Masárová, Tillmann Miltzow, Debajyoti Mondal, Anurag Murty Naredla, Josef Tkadlec, Alexi Turcotte
2022 arXiv   pre-print
The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.  ...  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.  ...  of stars), and may help to distinguish hard and easy cases of coloured weighted token swapping on trees.  ... 
arXiv:1903.06981v4 fatcat:ridowc24lre5bd2kbrgixoilby

Swapping Colored Tokens on Graphs [article]

Katsuhisa Yamanaka and Takashi Horiyama and J. Mark Keil and David Kirkpatrick and Yota Otachi and Toshiki Saitoh and Ryuhei Uehara and Yushi Uno
2018 arXiv   pre-print
Each pair of adjacent vertices can swap their current colors. Our goal is to perform the minimum number of swaps so that the current and target colors agree at each vertex.  ...  We then show that 2-Colored Token Swapping can be solved in polynomial time for general graphs and in linear time for trees.  ...  Third, they proposed polynomial-time 4-approximation algorithms for Token Swapping on general graphs and showed the APX-hardness of Token Swapping. Independently, Kawahara et al.  ... 
arXiv:1803.06816v1 fatcat:7us46q2ynjgmzavbqdhtcijkhq

Swapping Colored Tokens on Graphs [chapter]

Katsuhisa Yamanaka, Takashi Horiyama, David Kirkpatrick, Yota Otachi, Toshiki Saitoh, Ryuhei Uehara, Yushi Uno
2015 Lecture Notes in Computer Science  
Each pair of adjacent vertices can swap their current colors. Our goal is to perform the minimum number of swaps so that the current and target colors agree at each vertex.  ...  We then show that 2-Colored Token Swapping can be solved in polynomial time for general graphs and in linear time for trees.  ...  Third, they proposed polynomial-time 4-approximation algorithms for Token Swapping on general graphs and showed the APX-hardness of Token Swapping. Independently, Kawahara et al.  ... 
doi:10.1007/978-3-319-21840-3_51 fatcat:gkksiqiwynhh5budmxmu7tvqli

Swapping Labeled Tokens on Graphs [chapter]

Katsuhisa Yamanaka, Erik D. Demaine, Takehiro Ito, Jun Kawahara, Masashi Kiyomi, Yoshio Okamoto, Toshiki Saitoh, Akira Suzuki, Kei Uchizawa, Takeaki Uno
2014 Lecture Notes in Computer Science  
We prove that every such puzzle is solvable in O(n 2 ) token swaps, and thus focus on the problem of minimizing the number of token swaps to reach the target token placement.  ...  Consider a puzzle consisting of n tokens on an n-vertex graph, where each token has a distinct starting vertex and a distinct target vertex it wants to reach, and the only allowed transformation is to  ...  Acknowledgment We are grateful to Takashi Horiyama, Shin-ichi Nakano and Ryuhei Uehara for their comments on related work and fruitful discussions with them.  ... 
doi:10.1007/978-3-319-07890-8_31 fatcat:pvhw6vtiobamzgd6vkcfapz6ba

Swapping labeled tokens on graphs

Katsuhisa Yamanaka, Erik D. Demaine, Takehiro Ito, Jun Kawahara, Masashi Kiyomi, Yoshio Okamoto, Toshiki Saitoh, Akira Suzuki, Kei Uchizawa, Takeaki Uno
2015 Theoretical Computer Science  
We prove that every such puzzle is solvable in O(n 2 ) token swaps, and thus focus on the problem of minimizing the number of token swaps to reach the target token placement.  ...  Consider a puzzle consisting of n tokens on an n-vertex graph, where each token has a distinct starting vertex and a distinct target vertex it wants to reach, and the only allowed transformation is to  ...  Acknowledgment We are grateful to Takashi Horiyama, Shin-ichi Nakano and Ryuhei Uehara for their comments on related work and fruitful discussions with them.  ... 
doi:10.1016/j.tcs.2015.01.052 fatcat:gtob2vbtqfbjpiqouvanoipjra

The Time Complexity of Permutation Routing via Matching, Token Swapping and a Variant [article]

Jun Kawahara, Toshiki Saitoh, Ryo Yoshinaka
2017 arXiv   pre-print
The problems of Permutation Routing via Matching and Token Swapping are reconfiguration problems on graphs. This paper is concerned with the complexity of those problems and a colored variant.  ...  While all pairs of tokens on a matching can be exchanged at once in Permutation Routing via Matching, Token Swapping allows only one pair of tokens can be swapped.  ...  The work was supported in part by JSPS KAKENHI Grant Number 16K16006, CREST Grant Numbers JPMJCR1401 and JPMJCR1402, Japan.  ... 
arXiv:1612.02948v2 fatcat:k6ffhl5sn5hcdf6dcf4wrfxfne

On Liquidity Mining for Uniswap v3 [article]

Jimmy Yin, Mac Ren
2021 arXiv   pre-print
We analyze the Nash Equilibrium and the corresponding strategy, approximately, deploying the liquidity proportional to the reward distribution, in some special cases and use it to guide the general situations  ...  The recently proposed Uniswap v3 replaces the fungible liquidity provider token (LP token) into non-fungible ones, making the design for liquidity mining more difficult.  ...  When the price changes from 0 to 1 and 1 < 0 < , a mount of tokens are swapped into passively for the liquidity providers. We have: |Δ | = 0 − 1 = √ · ( √ 0 − √ 1 ).  ... 
arXiv:2108.05800v1 fatcat:6sh5wc4tj5dslhfjyjeha7usaa

Oolong: Investigating What Makes Crosslingual Transfer Hard with Controlled Studies [article]

Zhengxuan Wu and Isabel Papadimitriou and Alex Tamkin
2022 arXiv   pre-print
Little is known about what makes cross-lingual transfer hard, since factors like tokenization, morphology, and syntax all change at once between languages.  ...  effect from switching tokenizers (<2% drop) or word morphologies (<2% drop).  ...  Acknowledgements We would like to thank Nelson Liu, Mirac Suzgun and Tolúlo . pé . Ògúnrè . mí for useful discussions and comments on drafts.  ... 
arXiv:2202.12312v1 fatcat:5ftoq5ad6vhm7p7mztllnyfcwa

Circuit Transformations for Quantum Architectures [article]

Andrew M. Childs and Eddie Schoute and Cem M. Unsal
2019 arXiv   pre-print
Secondly, for serial routing, we consider the token swapping framework and extend a 4-approximation algorithm for general graphs to support partial permutations.  ...  We give new routing procedures for common architecture graphs and for the generalized hierarchical product of graphs, which produces subgraphs of the Cartesian product.  ...  Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Algorithms Teams and Quantum Testbed Pathfinder (award number DE-SC0019040) programs.  ... 
arXiv:1902.09102v1 fatcat:7tb5l55zp5gdpaxh3zh7566mqq

Mobile Device Security Using Transient Authentication

A.J. Nicholson, M.D. Corner, B.D. Noble
2006 IEEE Transactions on Mobile Computing  
When the user departs, the token and device lose contact and the device secures itself.  ...  Transient Authentication lifts the burden of authentication from the user by use of a wearable token that constantly attests to the user's presence.  ...  Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Security Agency or the US National Science  ... 
doi:10.1109/tmc.2006.169 fatcat:nhkpbwy52fgvvequ3vdlly4pqy

Finding Optimal Solutions to Token Swapping by Conflict-based Search and Reduction to SAT [article]

Pavel Surynek
2018 arXiv   pre-print
A colored token is placed in each vertex. A pair of tokens can be swapped between adjacent vertices. The goal is to perform a sequence of swaps so that token and vertex colors agree across the graph.  ...  The minimum number of swaps is required in the optimization variant of the problem.  ...  It is also known that finding swapping sequence that has as few swaps as possible is an NP-hard problem.  ... 
arXiv:1806.09487v1 fatcat:t2o5pkdto5fh5p6ebdvdh57wsq
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