A generalized @math puzzle' consists of an @math numbered grid, with one missing number. A move in the game switches the position of the empty square with the position of one of its neighbors. We solve Diaconis' 15 puzzle problem' by proving that the asymptotic total variation mixing time of the board is at least order @math when the board is given periodic boundary conditions and when random moves are made. We demonstrate that for any @math with @math , the number of fixed points after @math moves converges to a Poisson distribution of parameter 1. The order of total variation mixing time for this convergence is @math without cut-off. We also prove an upper bound of order @math for the total variation mixing time.
Much of the prior work regarding @math puzzles has focused on sorting strategies for a puzzle in general position, and on the hardness of finding shortest sorting algorithms. For instance, it is known that any @math puzzle may be returned to sorted order in order @math steps (and fewer are not always possible) @cite_14 , and that finding the shortest solution is NP-hard @cite_6 , @cite_13 .
@cite_14 @cite_13 @cite_6
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In this paper we study noisy sorting without re-sampling. In this problem there is an unknown order @math for all pairs @math , where @math is a constant and @math for all @math and @math . It is assumed that the errors are independent. Given the status of the queries the goal is to find the maximum likelihood order. In other words, the goal is find a permutation @math that minimizes the number of pairs @math where @math . The problem so defined is the feedback arc set problem on distributions of inputs, each of which is a tournament obtained as a noisy perturbations of a linear order. Note that when @math and @math is large, it is impossible to recover the original order @math . It is known that the weighted feedback are set problem on tournaments is NP-hard in general. Here we present an algorithm of running time @math and sampling complexity @math that with high probability solves the noisy sorting without re-sampling problem. We also show that if @math is an optimal solution of the problem then it is close'' to the original order. More formally, with high probability it holds that @math and @math . Our results are of interest in applications to ranking, such as ranking in sports, or ranking of search items based on comparisons by experts. We prove depth optimality of sorting networks from "The Art of Computer Programming".Sorting networks posses symmetry that can be used to generate a few representatives.These representatives can be efficiently encoded using regular expressions.We construct SAT formulas whose unsatisfiability is sufficient to show optimality.Resulting algorithm is orders of magnitude faster than prior work on small instances. We solve a 40-year-old open problem on depth optimality of sorting networks. In 1973, Donald E. Knuth detailed sorting networks of the smallest depth known for n ź 16 inputs, quoting optimality for n ź 8 (Volume 3 of "The Art of Computer Programming"). In 1989, Parberry proved optimality of networks with 9 ź n ź 10 inputs. We present a general technique for obtaining such results, proving optimality of the remaining open cases of 11 ź n ź 16 inputs. Exploiting symmetry, we construct a small set R n of two-layer networks such that: if there is a depth-k sorting network on n inputs, then there is one whose first layers are in R n . For each network in R n , we construct a propositional formula whose satisfiability is necessary for the existence of a depth-k sorting network. Using an off-the-shelf SAT solver we prove optimality of the sorting networks listed by Knuth. For n ź 10 inputs, our algorithm is orders of magnitude faster than prior ones. An important question in theoretical computer science is to determine the best possible running time for solving a problem at hand. For geometric optimization problems, we often understand their complexity on a rough scale, but not very well on a finer scale. One such example is the two-dimensional knapsack problem for squares. There is a polynomial time (1 + ϵ)-approximation algorithm for it (i.e., a PTAS) but the running time of this algorithm is triple exponential in 1 ϵ, i.e., Ω(n221 ϵ). A double or triple exponential dependence on 1 ϵ is inherent in how this and several other algorithms for other geometric problems work. In this paper, we present an EPTAS for knapsack for squares, i.e., a (1+ϵ)-approximation algorithm with a running time of Oϵ(1)·nO(1). In particular, the exponent of n in the running time does not depend on ϵ at all! Since there can be no FPTAS for the problem (unless P = NP) this is the best kind of approximation scheme we can hope for. To achieve this improvement, we introduce two new key ideas: We present a fast method to guess the Ω(221 ϵ) relatively large squares of a suitable near-optimal packing instead of using brute-force enumeration. Secondly, we introduce an indirect guessing framework to define sizes of cells for the remaining squares. In the previous PTAS each of these steps needs a running time of Ω(n221 ϵ) and we improve both to Oϵ(1) · nO(1). We complete our result by giving an algorithm for two-dimensional knapsack for rectangles under (1 + ϵ)-resource augmentation. In this setting, we also improve the best known running time of Ω(n1 ϵ1 ϵ) to Oϵ(1) · nO(1) and compute even a solution with optimal profit, in contrast to the best previously known polynomial time algorithm for this setting that computes only an approximation. We believe that our new techniques have the potential to be useful for other settings as well. Given @math elements with non-negative integer weights @math and an integer capacity @math , we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most @math . We give the first deterministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number of solutions to any knapsack constraint (our estimate has relative error @math ). Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes (FPRAS) were known first by Morris and Sinclair via Markov chain Monte Carlo techniques, and subsequently by Dyer via dynamic programming and rejection sampling. In addition, we present a new method for deterministic approximate counting using read-once branching programs. Our approach yields an FPTAS for several other counting problems, including counting solutions for the multidimensional knapsack problem with a constant number of constraints, the general integer knapsack problem, and the contingency tables problem with a constant number of rows.