Parameterized Distributed Algorithms

In this work, we initiate a thorough study of graph optimization problems parameterized by the output size in the distributed setting. In such a problem, an algorithm decides whether a solution of size bounded by k exists and if so, it finds one. We study fundamental problems, including Minimum Vertex Cover (MVC), Maximum Independent Set (MaxIS), Maximum Matching (MaxM), and many others, in both the LOCAL and CONGEST distributed computation models. We present lower bounds for the round complexity of solving parameterized problems in both models, together with optimal and near-optimal upper bounds. Our results extend beyond the scope of parameterized problems. We show that any LOCAL (1+epsilon)-approximation algorithm for the above problems must take Omega(epsilon^{-1}) rounds. Joined with the (epsilon^{-1}log n)^{O(1)} rounds algorithm of [Ghaffari et al., 2017] and the Omega (sqrt{(log n)/(log log n)}) lower bound of [Fabian Kuhn et al., 2016], the lower bounds match the upper bound up to polynomial factors in both parameters. We also show that our parameterized approach reduces the runtime of exact and approximate CONGEST algorithms for MVC and MaxM if the optimal solution is small, without knowing its size beforehand. Finally, we propose the first o(n^2) rounds CONGEST algorithms that approximate MVC within a factor strictly smaller than 2

Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack

The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+epsilon)-approximations in f(k,epsilon)n^O(1) time where k is some parameter of the input. The goal is to overcome lower bounds from either of the areas. We obtain the following results on parameterized approximability: - In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA\u2705]. The best-known polynomial-time approximation factor is O(log log n) [Chalermsook and Chuzhoy, SODA\u2709] and it admits a QPTAS [Adamaszek and Wiese, FOCS\u2713; Chuzhoy and Ene, FOCS\u2716]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant epsilon>0 and integer k>0, in time f(k,epsilon)n^g(epsilon), either outputs a solution of size at least k/(1+epsilon), or declares that the optimum solution has size less than k. - In the (2-dimensional) geometric knapsack problem (2DK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of 2DK with rotations (2DKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factor is 2+epsilon [Jansen and Zhang, SODA\u2704]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA\u2715]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for 2DKR. For all considered problems, getting time f(k,epsilon)n^O(1), rather than f(k,epsilon)n^g(epsilon), would give FPT time f\u27(k)n^O(1) exact algorithms by setting epsilon=1/(k+1), contradicting W[1]-hardness. Instead, for each fixed epsilon>0, our PASs give (1+epsilon)-approximate solutions in FPT time. For both MISR and 2DKR our techniques also give rise to preprocessing algorithms that take n^g(epsilon) time and return a subset of at most k^g(epsilon) rectangles/items that contains a solution of size at least k/(1+epsilon) if a solution of size k exists. This is a special case of the recently introduced notion of a polynomial-size approximate kernelization scheme [Lokshtanov et al., STOC\u2717]

Exact Performance of Concatenated Quantum Codes

When a logical qubit is protected using a quantum error-correcting code, the net effect of coding, decoherence (a physical channel acting on qubits in the codeword) and recovery can be represented exactly by an effective channel acting directly on the logical qubit. In this paper we describe a procedure for deriving the map between physical and effective channels that results from a given coding and recovery procedure. We show that the map for a concatenation of codes is given by the composition of the maps for the constituent codes. This perspective leads to an efficient means for calculating the exact performance of quantum codes with arbitrary levels of concatenation. We present explicit results for single-bit Pauli channels. For certain codes under the symmetric depolarizing channel, we use the coding maps to compute exact threshold error probabilities for achievability of perfect fidelity in the infinite concatenation limit.Comment: An expanded presentation of the analytic methods and results from quant-ph/0111003; 13 pages, 6 figure

Interpretable Transformations with Encoder-Decoder Networks

Deep feature spaces have the capacity to encode complex transformations of their input data. However, understanding the relative feature-space relationship between two transformed encoded images is difficult. For instance, what is the relative feature space relationship between two rotated images? What is decoded when we interpolate in feature space? Ideally, we want to disentangle confounding factors, such as pose, appearance, and illumination, from object identity. Disentangling these is difficult because they interact in very nonlinear ways. We propose a simple method to construct a deep feature space, with explicitly disentangled representations of several known transformations. A person or algorithm can then manipulate the disentangled representation, for example, to re-render an image with explicit control over parameterized degrees of freedom. The feature space is constructed using a transforming encoder-decoder network with a custom feature transform layer, acting on the hidden representations. We demonstrate the advantages of explicit disentangling on a variety of datasets and transformations, and as an aid for traditional tasks, such as classification.Comment: Accepted at ICCV 201