6,205 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth
Dynamic programming on various graph decompositions is one of the most
fundamental techniques used in parameterized complexity. Unfortunately, even if
we consider concepts as simple as path or tree decompositions, such dynamic
programming uses space that is exponential in the decomposition's width, and
there are good reasons to believe that this is necessary. However, it has been
shown that in graphs of low treedepth it is possible to design algorithms which
achieve polynomial space complexity without requiring worse time complexity
than their counterparts working on tree decompositions of bounded width. Here,
treedepth is a graph parameter that, intuitively speaking, takes into account
both the depth and the width of a tree decomposition of the graph, rather than
the width alone.
Motivated by the above, we consider graphs that admit clique expressions with
bounded depth and label count, or equivalently, graphs of low shrubdepth (sd).
Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a
bounded-depth analogue of treewidth. We show that also in this setting,
bounding the depth of the decomposition is a deciding factor for improving the
space complexity. Precisely, we prove that on -vertex graphs equipped with a
tree-model (a decomposition notion underlying sd) of depth and using
labels, we can solve
- Independent Set in time using
space;
- Max Cut in time using space; and
- Dominating Set in time using space via
a randomized algorithm.
We also establish a lower bound, conditional on a certain assumption about
the complexity of Longest Common Subsequence, which shows that at least in the
case of IS the exponent of the parametric factor in the time complexity has to
grow with if one wishes to keep the space complexity polynomial.Comment: Conference version to appear at the European Symposium on Algorithms
(ESA 2023
Finding Small Complete Subgraphs Efficiently
(I) We revisit the algorithmic problem of finding all triangles in a graph
with vertices and edges. According to a result of Chiba and
Nishizeki (1985), this task can be achieved by a combinatorial algorithm
running in time, where is the
graph arboricity. We provide a new very simple combinatorial algorithm for
finding all triangles in a graph and show that is amenable to the same running
time analysis. We derive these worst-case bounds from first principles and with
very simple proofs that do not rely on classic results due to Nash-Williams
from the 1960s.
(II) We extend our arguments to the problem of finding all small complete
subgraphs of a given fixed size. We show that the dependency on and
in the running time of the algorithm of
Chiba and Nishizeki for listing all copies of , where , is
asymptotically tight.
(III) We give improved arboricity-sensitive running times for counting and/or
detection of copies of , for small . A key ingredient in
our algorithms is, once again, the algorithm of Chiba and Nishizeki. Our new
algorithms are faster than all previous algorithms in certain high-range
arboricity intervals for every .Comment: 14 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:2105.0126
Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth
Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels, - Independent Set can be solved in time 2^(dk) ⋅ n^(1) using (dk²log n) space; - Max Cut can be solved in time n^(dk) using (dk log n) space; and - Dominating Set can be solved in time 2^(dk) ⋅ n^(1) using n^(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial
Nonlocal games and their device-independent quantum applications
Device-independence is a property of certain protocols that allows one to ensure their proper execution given only classical interaction with devices and assuming the correctness of the laws of physics. This scenario describes the most general form of cryptographic security, in which no trust is placed in the hardware involved; indeed, one may even take it to have been prepared by an adversary.
Many quantum tasks have been shown to admit device-independent protocols by augmentation with "nonlocal games". These are games in which noncommunicating parties jointly attempt to fulfil some conditions imposed by a referee. We introduce examples of such games and examine the optimal strategies of players who are allowed access to different possible shared resources, such as entangled quantum states. We then study their role in self-testing, private random number generation, and secure delegated quantum computation. Hardware imperfections are naturally incorporated in the device-independent scenario as adversarial, and we thus also perform noise robustness analysis where feasible.
We first study a generalization of the Mermin–Peres magic square game to arbitrary rectangular dimensions. After exhibiting some general properties, these "magic rectangle" games are fully characterized in terms of their optimal win probabilities for quantum strategies. We find that for m×n magic rectangle games with dimensions m,n≥3, there are quantum strategies that win with certainty, while for dimensions 1×n quantum strategies do not outperform classical strategies. The final case of dimensions 2×n is richer, and we give upper and lower bounds that both outperform the classical strategies. As an initial usage scenario, we apply our findings to quantum certified randomness expansion to find noise tolerances and rates for all magic rectangle games. To do this, we use our previous results to obtain the winning probabilities of games with a distinguished input for which the devices give a deterministic outcome and follow the analysis of C. A. Miller and Y. Shi [SIAM J. Comput. 46, 1304 (2017)].
Self-testing is a method to verify that one has a particular quantum state from purely classical statistics. For practical applications, such as device-independent delegated verifiable quantum computation, it is crucial that one self-tests multiple Bell states in parallel while keeping the quantum capabilities required of one side to a minimum. We use our 3×n magic rectangle games to obtain a self-test for n Bell states where one side needs only to measure single-qubit Pauli observables. The protocol requires small input sizes [constant for Alice and O(log n) bits for Bob] and is robust with robustness O(n⁵/²√ε), where ε is the closeness of the ideal (perfect) correlations to those observed. To achieve the desired self-test, we introduce a one-side-local quantum strategy for the magic square game that wins with certainty, we generalize this strategy to the family of 3×n magic rectangle games, and we supplement these nonlocal games with extra check rounds (of single and pairs of observables).
Finally, we introduce a device-independent two-prover scheme in which a classical verifier can use a simple untrusted quantum measurement device (the client device) to securely delegate a quantum computation to an untrusted quantum server. To do this, we construct a parallel self-testing protocol to perform device-independent remote state preparation of n qubits and compose this with the unconditionally secure universal verifiable blind quantum computation (VBQC) scheme of J. F. Fitzsimons and E. Kashefi [Phys. Rev. A 96, 012303 (2017)]. Our self-test achieves a multitude of desirable properties for the application we consider, giving rise to practical and fully device-independent VBQC. It certifies parallel measurements of all cardinal and intercardinal directions in the XY-plane as well as the computational basis, uses few input questions (of size logarithmic in n for the client and a constant number communicated to the server), and requires only single-qubit measurements to be performed by the client device
Jake pozicione igre
In this thesis, we study 2-player combinatorial games on graphs. We devote a lot of attention to strong positional games, where both players have the same goal. First, we consider the so-called fixed graph strong Avoider-Avoider game in which two players called Red and Blue alternately claim edges of the complete graph Kn, and the player who first completes a copy of a fixed graph F loses the game. If neither of the players claimed a copy of F in his graph and all the elements of the board are claimed, the game is declared a draw. Even though these games have been studied for decades, there are very few known results. We make a step forward by proving that Blue has a winning strategy it two different games of this kind. Furthermore, we introduce strong CAvoiderCAvoider F games where the claimed edges of each player must form a connected graph throughout the game. This is a natural extension of the strong Avoider-Avoider games, with a connectedness constraint. We prove that Blue can win in three standard CAvoider-CAvoider F games. Next, we study strong Maker-Maker F games, where now, the player who first occupies a copy of F is the winner. It is well-known that the outcome of these games when both players play optimally can be either the first player's win or a draw. We are interested in finding the achievement number a(F) of a strong Maker-Maker F game, that is, the smallest n for which Red has a winning strategy. We can find the exact value a(F) for several graphs F, including paths, cycles, perfect matchings, and a subclass of trees on n vertices. We also give the upper and lower bounds for the achievement number of stars and trees. Finally, we introduce generalized saturation games as a natural extension of two different types of combinatorial games, saturation games and Constructor-Blocker games. In the generalized saturation game, two graphs H and F are given in advance. Two players called Max and Mini alternately claim unclaimed edges of the complete graph Kn and together gradually building the game graph G, the graph that consists of all edges claimed by both players. The graph G must never contain a copy of F, and the game ends when there are no more moves, i.e. when G is a saturated F-free graph. We are interested in the score of this game, that is, the number of copies of the graph H in G at the end of the game. Max wants to maximize this score, whereas Mini tries to minimize it. The game is played under the assumption that both players play optimally. We study several generalized saturation games for natural choices of F and H, in an effort to locate the score of the game as precisely as possible.У овој тези проучавамо комбинаторне игре на графовима које играју 2 играча. Посебну пажњу посвећујемо јаким позиционим играма, у којима оба играча имају исти циљ. Прво, посматрамо такозвану јаку Авојдер-Авојдер игру са задатим фиксним графом у којој два играча, Црвени и Плави наизменично селектују гране комплетног графа Kn, а играч који први селектује копију фиксног графа F губи игру. Ако ниједан од играча не садржи копију од F у свом графу и сви елементи табле су селектовани, игра се проглашава нерешеном. Иако су ове игре проучаване деценијама, врло је мало познатих резултата. Ми смо направили корак напред доказавши да Плави има победничку стратегију у две различите игре ове врсте. Такође, уводимо јаке ЦАвојдер-ЦАвојдер F игре у којима граф сваког играча мора остати повезан током игре. Ово је природно проширење јаких Авојдер-Авојдер игара, са ограничењем повезаности. Доказујемо да Плави може да победи у три стандардне ЦАвојдер-ЦАвојдер F игре. Затим проучавамо јаке Мејкер-Мејкер F игре, у којима је играч који први селектује копију од F победник. Познато је да исход ових игара уколико оба играча играју оптимално може бити или победа првог играча или нерешено. Циљ нам је да пронађемо ачивмент број а(F) јаке Мејкер-Мејкер F игре, односно најмање n за које Црвени има победничку стратегију. Дајемо тачну вредност a(F) за неколико графова F, укључујући путеве, циклусе, савршене мечинге и поткласу стабала са n чворова. Такође, дајемо горње и доње ограничење ачивмент броја за звезде и стабла. Коначно, уводимо уопштене игре сатурације као природно проширење две различите врсте комбинаторних игара, игара сатурације и Конструктор-Блокер игара. У уопштеној игри сатурације унапред су дата два графа H и F. Два играча по имену Макс и Мини наизменично селектују слободне гране комплетног графа Kn и заједно постепено граде граф игре G, који се састоји од свих грана које су селектовала оба играча. Граф G не сме да садржи копију од F, а игра се завршава када више нема потеза, односно када је G сатуриран граф који не садржи F. Занима нас резултат ове игре, односно, број копија графа H у G на крају игре. Макс жели да максимизира овај резултат, док Мини покушава да га минимизира. Игра се под претпоставком да оба играча играју оптимално. Проучавамо неколико уопштених игара сатурације за природне изборе F и H, у настојању да што прецизније одредимо резултат игре.U ovoj tezi proučavamo kombinatorne igre na grafovima koje igraju 2 igrača. Posebnu pažnju posvećujemo jakim pozicionim igrama, u kojima oba igrača imaju isti cilj. Prvo, posmatramo takozvanu jaku Avojder-Avojder igru sa zadatim fiksnim grafom u kojoj dva igrača, Crveni i Plavi naizmenično selektuju grane kompletnog grafa Kn, a igrač koji prvi selektuje kopiju fiksnog grafa F gubi igru. Ako nijedan od igrača ne sadrži kopiju od F u svom grafu i svi elementi table su selektovani, igra se proglašava nerešenom. Iako su ove igre proučavane decenijama, vrlo je malo poznatih rezultata. Mi smo napravili korak napred dokazavši da Plavi ima pobedničku strategiju u dve različite igre ove vrste. Takođe, uvodimo jake CAvojder-CAvojder F igre u kojima graf svakog igrača mora ostati povezan tokom igre. Ovo je prirodno proširenje jakih Avojder-Avojder igara, sa ograničenjem povezanosti. Dokazujemo da Plavi može da pobedi u tri standardne CAvojder-CAvojder F igre. Zatim proučavamo jake Mejker-Mejker F igre, u kojima je igrač koji prvi selektuje kopiju od F pobednik. Poznato je da ishod ovih igara ukoliko oba igrača igraju optimalno može biti ili pobeda prvog igrača ili nerešeno. Cilj nam je da pronađemo ačivment broj a(F) jake Mejker-Mejker F igre, odnosno najmanje n za koje Crveni ima pobedničku strategiju. Dajemo tačnu vrednost a(F) za nekoliko grafova F, uključujući puteve, cikluse, savršene mečinge i potklasu stabala sa n čvorova. Takođe, dajemo gornje i donje ograničenje ačivment broja za zvezde i stabla. Konačno, uvodimo uopštene igre saturacije kao prirodno proširenje dve različite vrste kombinatornih igara, igara saturacije i Konstruktor-Bloker igara. U uopštenoj igri saturacije unapred su data dva grafa H i F. Dva igrača po imenu Maks i Mini naizmenično selektuju slobodne grane kompletnog grafa Kn i zajedno postepeno grade graf igre G, koji se sastoji od svih grana koje su selektovala oba igrača. Graf G ne sme da sadrži kopiju od F, a igra se završava kada više nema poteza, odnosno kada je G saturiran graf koji ne sadrži F. Zanima nas rezultat ove igre, odnosno, broj kopija grafa H u G na kraju igre. Maks želi da maksimizira ovaj rezultat, dok Mini pokušava da ga minimizira. Igra se pod pretpostavkom da oba igrača igraju optimalno. Proučavamo nekoliko uopštenih igara saturacije za prirodne izbore F i H, u nastojanju da što preciznije odredimo rezultat igre
Physical sketching tools and techniques for customized sensate surfaces
Sensate surfaces are a promising avenue for enhancing human interaction with digital systems due to their inherent intuitiveness and natural user interface. Recent technological advancements have enabled sensate surfaces to surpass the constraints of conventional touchscreens by integrating them into everyday objects, creating interactive interfaces that can detect various inputs such as touch, pressure, and gestures. This allows for more natural and intuitive control of digital systems. However, prototyping interactive surfaces that are customized to users' requirements using conventional techniques remains technically challenging due to limitations in accommodating complex geometric shapes and varying sizes. Furthermore, it is crucial to consider the context in which customized surfaces are utilized, as relocating them to fabrication labs may lead to the loss of their original design context. Additionally, prototyping high-resolution sensate surfaces presents challenges due to the complex signal processing requirements involved. This thesis investigates the design and fabrication of customized sensate surfaces that meet the diverse requirements of different users and contexts. The research aims to develop novel tools and techniques that overcome the technical limitations of current methods and enable the creation of sensate surfaces that enhance human interaction with digital systems.Sensorische Oberflächen sind aufgrund ihrer inhärenten Intuitivität und natürlichen Benutzeroberfläche ein vielversprechender Ansatz, um die menschliche Interaktionmit digitalen Systemen zu verbessern. Die jüngsten technologischen Fortschritte haben es ermöglicht, dass sensorische Oberflächen die Beschränkungen herkömmlicher Touchscreens überwinden, indem sie in Alltagsgegenstände integriert werden und interaktive Schnittstellen schaffen, die diverse Eingaben wie Berührung, Druck, oder Gesten erkennen können. Dies ermöglicht eine natürlichere und intuitivere Steuerung von digitalen Systemen. Das Prototyping interaktiver Oberflächen, die mit herkömmlichen Techniken an die Bedürfnisse der Nutzer angepasst werden, bleibt jedoch eine technische Herausforderung, da komplexe geometrische Formen und variierende Größen nur begrenzt berücksichtigt werden können. Darüber hinaus ist es von entscheidender Bedeutung, den Kontext, in dem diese individuell angepassten Oberflächen verwendet werden, zu berücksichtigen, da eine Verlagerung in Fabrikations-Laboratorien zum Verlust ihres ursprünglichen Designkontextes führen kann. Zudem stellt das Prototyping hochauflösender sensorischer Oberflächen aufgrund der komplexen Anforderungen an die Signalverarbeitung eine Herausforderung dar. Diese Arbeit erforscht dasDesign und die Fabrikation individuell angepasster sensorischer Oberflächen, die den diversen Anforderungen unterschiedlicher Nutzer und Kontexte gerecht werden. Die Forschung zielt darauf ab, neuartigeWerkzeuge und Techniken zu entwickeln, die die technischen Beschränkungen derzeitigerMethoden überwinden und die Erstellung von sensorischen Oberflächen ermöglichen, die die menschliche Interaktion mit digitalen Systemen verbessern
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When to Hold and When to Fold: Studies on the topology of origami and linkages
Linkages and mechanisms are pervasive in physics and engineering as models for avariety of structures and systems, from jamming to biomechanics. With the increasein physical realizations of discrete shape-changing materials, such as metamaterials,programmable materials, and self-actuating structures, an increased understandingof mechanisms and how they can be designed is crucial. At a basic level, linkagesor mechanisms can be understood to be rigid bars connected at pivots around whichthey can rotate freely. We will have a particular focus on origami-like materials, anextension to linkages with the added constraint of faces. Self-actuated versions typ-ically start flat and when exposed to an external stimulus - such as a temperaturechange or magnetic field - spontaneously fold. Since these structures fold all at once,and the number of folding patterns accessible to a given origami are exponential, theyare prone to folding to a configuration other than the desired one. Other work hassuggested methods for avoiding this misfolding, but it assumes ideal, rigid origami. Here, we expand on these models to account for the elasticity of real structures andintroduce methods for accounting for Gaussian curvature in them. We also explorehow to find and set an upper bound on minimal forcing sets, or the minimum set offolds required to force an origami, and present a graph theory algorithm for findingthem in arbitrary origami. Taken altogether, these origami studies give insight intohow the physical properties of origami influence folding and a new set of tools foravoiding misfolding. Next, we turn back to a more fundamental study of linkagesand present a new method for finding the manifold of their critical points. We thendemonstrate a design protocol that utilizes this manifold to create linkages with tun-able motions, before turning to several example structures, including the four-barlinkage and the Kane-Lubensky chain
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