(Almost) tight bounds for randomized and quantum Local Search on hypercubes and grids

The Local Search problem, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to many areas in computer science and natural sciences. In this paper, we show that for the Boolean hypercube \B^n, the randomized query complexity of Local Search is $\Theta(2^{n/2}n^{1/2})$ and the quantum query complexity is $\Theta(2^{n/3}n^{1/6})$. We also show that for the constant dimensional grid $[N^{1/d}]^d$, the randomized query complexity is $\Theta(N^{1/2})$ for $d \geq 4$ and the quantum query complexity is $\Theta(N^{1/3})$ for $d \geq 6$. New lower bounds for lower dimensional grids are also given. These improve the previous results by Aaronson [STOC'04], and Santha and Szegedy [STOC'04]. Finally we show for $[N^{1/2}]^2$ a new upper bound of $O(N^{1/4}(\log\log N)^{3/2})$ on the quantum query complexity, which implies that Local Search on grids exhibits different properties at low dimensions.Comment: 18 pages, 1 figure. v2: introduction rewritten, references added. v3: a line for grant added. v4: upper bound section rewritte

Lower Bounds on Quantum Query Complexity

Shor's and Grover's famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers_cannot_ do, and specifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.Comment: survey, 23 page

Query Complexity of Approximate Equilibria in Anonymous Games

We study the computation of equilibria of anonymous games, via algorithms that may proceed via a sequence of adaptive queries to the game's payoff function, assumed to be unknown initially. The general topic we consider is \emph{query complexity}, that is, how many queries are necessary or sufficient to compute an exact or approximate Nash equilibrium. We show that exact equilibria cannot be found via query-efficient algorithms. We also give an example of a 2-strategy, 3-player anonymous game that does not have any exact Nash equilibrium in rational numbers. However, more positive query-complexity bounds are attainable if either further symmetries of the utility functions are assumed or we focus on approximate equilibria. We investigate four sub-classes of anonymous games previously considered by \cite{bfh09, dp14}. Our main result is a new randomized query-efficient algorithm that finds a $O(n^{-1/4})$-approximate Nash equilibrium querying $\tilde{O}(n^{3/2})$ payoffs and runs in time $\tilde{O}(n^{3/2})$. This improves on the running time of pre-existing algorithms for approximate equilibria of anonymous games, and is the first one to obtain an inverse polynomial approximation in poly-time. We also show how this can be utilized as an efficient polynomial-time approximation scheme (PTAS). Furthermore, we prove that $\Omega(n \log{n})$ payoffs must be queried in order to find any $\epsilon$-well-supported Nash equilibrium, even by randomized algorithms

Separations in proof complexity and TFNP

It is well-known that Resolution proofs can be efficiently simulated by Sherali–Adams (SA) proofs. We show, however, that any such simulation needs to exploit huge coefficients: Resolution cannot be efficiently simulated by SA when the coefficients are written in unary. We also show that Reversible Resolution (a variant of MaxSAT Resolution) cannot be efficiently simulated by Nullstellensatz (NS). These results have consequences for total search problems. First, we characterise the classes , , by unary-SA, unary-NS, and Reversible Resolution, respectively. Second, we show that, relative to an oracle, ⊈, ⊈, and ⊈. In particular, together with prior work, this gives a complete picture of the black-box relationships between all classical classes introduced in the 1990s

Kvantu klejošanas un varbūtisko algoritmu ierobežojumi

Šajā darbā tiek pētīta algoritmu sarežģītība dažādos skaitļošanas modeļos. Konkrētāk, tiek pētītas kvantu klejošanas algoritmu īpašības un ierobežojumi, kā arī varbūtisko vaicājumalgoritmu darbības laika novērtēšanas metodes. Pirmajā daļā tiek aplūkotas Grovera kvantu klejošana un meklēšana grafos. Darbā tiek sniegts vispārīgs matemātisks apraksts klejošanas lokalizācijai un meklēšanas stacionārajiem stāvokļiem. Otrajā daļā tiek aplūkotas apakšējo novērtējumu metodes varbūtisko vaicājumalgoritmu modelī. Darbā tiek pierādīta klasisko pretinieka metožu asimptotiskā ekvivalence visur definētām funkcijām, un aprakstītas to atšķirības daļēji definētām funkcijām. Tiek arī aplūkota saistība starp bloku jutīgumu un daļskaitļu bloku jutīgumu.In this work, we study the complexity of algorithms in different models of computation. Specifically, we investigate properties and limitations of quantum walk algorithms, as well as methods for estimating the running time of randomized algorithms in the query complexity model. In the first part of the thesis, we study Grover’s quantum walk and search. We develop complete mathematical characterizations of the localization in quantum walk and of the stationary states in quantum walk search. In the second part, we study lower bound methods in the randomized query complexity model. We prove that classical adversary lower bounds are asymptotically equivalent for total functions and show that they differ for partial functions. We also investigate the relationship between block sensitivity and fractional block sensitivity