Approximation Algorithms for Minimizing Average Distortion

The minimum latency problem, also known as the traveling repairman problem, is a variant of the traveling salesman problem in which the starting node of the tour is given and the goal is to minimize the sum of the arrival times at the other nodes. We present a quasi-polynomial time approximation scheme (QPTAS) for this problem when the instance is a weighted tree, when the nodes lie in $\mathbb{R}^d$ for some fixed d, and for planar graphs. We also present a polynomial time constant factor approximation algorithm for the general metric case. The currently best polynomial time approximation algorithm for general metrics, due to Goemans and Kleinberg, computes a 3.59-approximation.

Approximation schemes for Min-Sum [formula omitted]-Clustering

Recently, Adamaszek and Wiese [1, 2] presented a quasi-polynomial time approximation scheme (QPTAS) for the problem of computing a maximum weight independent set for certain families of planar objects. This major advance on the problem was based on their proof that a certain type of separator exists for any independent set. Subsequently, Har-Peled [22] simplified and generalized their result. Mustafa et al. [36] also described a simplification, and somewhat surprisingly, showed that QPTAS's can be obtained for certain, albeit special, type of covering problems.Building on these developments, we revisit two NP-hard geometric partitioning problems -- convex decomposition and surface approximation. Partitioning problems combine the features of packing and covering. In particular, since the optimal solution does form a packing, the separator theorems are potentially applicable. Nevertheless, the two partitioning problems we study bring up additional difficulties that are worth examining in the context of the wider applicability of the separator methodology. We show how these issues can be handled in presenting quasi-polynomial time algorithms for these two problems with improved approximation guarantees.

We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given a set of n items, each associated with a non-negative weight, and T time periods with non-decreasing capacities \(W_1 \le \dots \le W_T\). When item i is inserted at time t, we gain a profit of \(p_{it}\); however, this item remains in the knapsack for all subsequent periods. The goal is to decide if and when to insert each item, subject to the time-dependent capacity constraints, with the objective of maximizing our total profit. Interestingly, this setting subsumes as special cases a number of recently-studied incremental knapsack problems, all known to be strongly NP-hard. Our first contribution comes in the form of a polynomial-time \((\frac{1}{2}-\epsilon )\)-approximation for the generalized incremental knapsack problem. This result is based on a reformulation as a single-machine sequencing problem, which is addressed by blending dynamic programming techniques and the classical Shmoys–Tardos algorithm for the generalized assignment problem. Combined with further enumeration-based self-reinforcing ideas and new structural properties of nearly-optimal solutions, we turn our algorithm into a quasi-polynomial time approximation scheme (QPTAS). Hence, under widely believed complexity assumptions, this finding rules out the possibility that generalized incremental knapsack is APX-hard.

A note on a QPTAS for maximum weight triangulation of planar point sets

In this paper, we study the following vehicle routing problem: given n vertices in a metric space, a specified root vertex r (the depot), and a length bound D, find a minimum cardinality set of r-paths that covers all vertices, such that each path has length at most D. This problem is $\mathcal{NP}$-complete, even when the underlying metric is induced by a weighted star. We present a 4-approximation for this problem on tree metrics. On general metrics, we obtain an O(logD) approximation algorithm, and also an $(O(\log \frac{1}{\epsilon}),1+\epsilon)$ bicriteria approximation. All these algorithms have running times that are almost linear in the input size. On instances that have an optimal solution with one r-path, we show how to obtain in polynomial time, a solution using at most 14 r-paths. We also consider a linear relaxation for this problem that can be solved approximately using techniques of Carr & Vempala [7]. We obtain upper bounds on the integrality gap of this relaxation both in tree metrics and in general.

In this paper, we consider the problem of rate control for video transmission over variable bit rate (VBR) channels. We focus on finding off-line optimal rate control for VBR transmission with a token bucket policing function. To ensure a maximum minimum quality is obtained over all data units, we introduce a minimum maximum distortion (MMAX) criterion in this channel-constrained problem. We show that, due to the channel constraints, a MMAX solution leads to a relatively low average distortion, because the total rate budget is not completely utilized. Therefore, after finding a MMAX solution, an additional minimization of average distortion (MMAX+) criterion is proposed to increase overall quality of the data sequence by using remaining resources. The proposed algorithms lead to an increase in average quality with respect to the MMAX solution, while providing a much more constant quality than MMSE solutions. Moreover we show how the MMAX+ approach can be implemented with low complexity.

While there have been many successful applications of machine learning methods to tasks in NLP, learning algorithms are not typically designed to optimize NLP performance metrics. This paper evaluates an ensemble selection framework designed to optimize arbitrary metrics and automate the process of algorithm selection and parameter tuning. We report the results of experiments that instantiate the framework for three NLP tasks, using six learning algorithms, a wide variety of parameterizations, and 15 performance metrics. Based on our results, we make recommendations for subsequent machine-learning-based research for natural language learning.

We give a polynomial time, (1 + ∊)-approximation algorithm for the traveling repairman problem (TRP) in the Euclidean plane, on weighted planar graphs, and on weighted trees. This improves on the known quasi-polynomial time approximation schemes for these problems. The algorithm is based on a simple technique that reduces the TRP to what we call the segmented TSP. Here, we are given numbers l1, …, lK and n1, …, nK and we need to find a path that visits at least nh points within path distance lh from the starting point for all h ∊ {1, …, K}. A solution is α-approximate if at least nh points are visited within distance αlh. It is shown that any algorithm that is α-approximate for every constant K in some metric space, gives an α(1 + ∊)-approximation for the TRP in the same metric space. Subsequently, approximation schemes are given for this segmented TSP problem in different metric spaces. The segmented TSP with only one segment (K = 1) is equivalent to the k-TSP for which a (2 + ∊)-approximation is known for a general metric space. Hence, this approach through the segmented TSP gives new impulse for improving on the 3.59-approximation for TRP in a general metric space. A similar reduction applies to many other minimum latency problems. To illustrate the strength of this approach we apply it to the well-studied scheduling problem of minimizing total weighted completion time under precedence constraints, 1|prec|Σ wjCj, and present a polynomial time approximation scheme for the case of interval order precedence constraints. This improves on the known 3/2-approximation for this problem. Both approximation schemes apply as well if release dates are added to the problem.

Marginal (contingency) tables are the method of choice for government agencies releasing statistical summaries of categorical data. In this paper, we derive lower bounds on how much distortion (noise) is necessary in these tables to ensure the privacy of sensitive data. We extend a line of recent work on impossibility results for private data analysis [9, 12, 13, 15] to a natural and important class of functionalities. Consider a database consisting of n rows (one per individual), each row comprising d binary attributes. For any subset of T attributes of size |T|=k, the marginal table for T has 2k entries; each entry counts how many times in the database a particular setting of these attributes occurs. We provide lower bounds for releasing all d k k-attribute marginal tables under several different notions of privacy. (1) We give efficient polynomial time attacks which allow an adversary to reconstruct sensitive information given insufficiently perturbed marginal table releases. In particular, for a constant k, we obtain a tight bound of ~Ω(min √n, √dk-1) on the average distortion per entry for any mechanism that releases all k-attribute marginals while providing attribute privacy (a weak notion implied by most privacy definitions). (2) Our reconstruction attacks require a new lower bound on the least singular value of a random matrix with correlated rows. Let M(k) be a matrix with d k rows formed by taking all possible k-way entry-wise products of an underlying set of d random vectors from {0,1}n. For constant k, we show that the least singular value of M(k) is ~Ω(√dk) with high probability (the same asymptotic bound as for independent rows). (3) We obtain stronger lower bounds for marginal tables satisfying differential privacy. We give a lower bound of ~Ω(min {√n, √ dk), which is tight for n ~Ω (dk). We extend our analysis to obtain stronger results for mechanisms that add instance-independent noise and weaker results when k is super-constant.

Linear predictive coding (LPC) parameters are widely used in various speech coding applications for representing the short-time spectral envelope information of speech [1]. For low bit rate speech coding applications, it is important to quantize these parameters using as few bits as possible. Considerable workhas been done in the past to develop both scalar and vector quantization procedures to quantize the LPC parameters [2, 3,4]. Scalar quantizers quantize each of the LPC parameters independently, while vector quantizers consider the entire set of LPC parameters as an entity and allow for direct minimization of quantization distortion. Because of this, the vector quantizers result in smaller distortion than the scalar quantizers at any given bit rate. The vector quantizers, however, have one major problem; their computational complexity is high. In our earlier paper [3], we have reported on a vector quantizer where the LPC parameter vector is split in the line spectral frequency (LSF) domain to overcome this complexity problem. We have shown that this quantizer can quantize the LPC parameters at 24 bits/frame with an average spectral distortion of 1 dB, less than 2% frames having spectral distortionl in the range 2-4 dB and no frame having spectral distortion greater than 4 dB.

Panoramic radiography is widely used as a tool supporting the diagnosis in dental practice. Currently the technology has evolved with the presence of radiographic imaging such as Cone Beam Computed Tomography (CBCT) that displays a three-dimensional picture. The purpose of this study was to determine large distortions that occur several regio in the mandible using digital panoramic radiographs and CBCT. Method experimental The research on one of the skull laboratories fitted staples vertically and horizontally in alveolar regions 31, 33, 35, 37, 41, 43, 45, 47, and then measured the length of the staples using the term digital. Furthermore, digital panoramic radiography performed with position 00, +50, +100, -50, -100 and CBCT radiography with 00 positions. Measurement description of radiopaque vertical and horizontal lines on digital panoramic radiographs and CBCT are computerized using the EZ software measurement menu. Calculating percentage distortion radiopaque overview of vertical and horizontal lines on digital panoramic radiographs and CBCT of the mandible by doing a comparison of the actual size of the skull. The results obtained on the percentage of minimal distortion of digital panoramic radiographs vertically was in the 31 region positioned +50at 0,23%, in the 33 region positioned 00at 5,99%, in the 35 region positioned -100at -6,33%, in the 37 region positioned +100at -1,46%, in the 41 region positioned -50at 0,46%, in the 43 region positioned 00at 0,52%, in the 45 region positioned +100at -0,45%, in the 47 region positioned +100at -4,76%. The percentage of minimal distortion of digital panoramic radiographs horizontally all of region was positioned 00. The percentage of minimal distortion of CBCT all of region was positioned 00 and different on each mandible alveolar region. The conclusion of this study is average distortion that occurs in the mandible using CBCT is more less than digital panoramic radiographs. That is mean CBCT more accurate than digital panoramic radiographs.

The fully polynomial-time approximation scheme (FPTAS) is a class of approximation algorithms that is able to deliver an approximate solution within any chosen ratio in polynomial time. By generalising Bird and de Moor's Thinning Theorem to a property between three orderings, we come up with a datatype-generic strategy for constructing fold-based FPTASs. Greedy, thinning, and approximation algorithms can thus be seen as a series of generalisations. Components needed in constructing an FPTAS are often natural extensions of those in the thinning algorithm. Design of complex FPTASs is thus made easier, and some of the resulting algorithms turn out to be simpler than those in previous works.

In this thesis, we study the problem of computing approximate equilibria in several classes of games. In particular, we study approximate Nash equilibria and approximate well-supported Nash equilibria in polymatrix and bimatrix games and approximate equilibria in Lipschitz games, penalty games and biased games. We construct algorithms for computing approximate equilibria that beat the cur- rent best algorithms for these problems. In Chapter 3, we present a distributed method to compute approximate Nash equilibria in bimatrix games. In contrast to previous approaches that analyze the two payoff matrices at the same time (for example, by solving a single LP that combines the two players’ payoffs), our algorithm first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then computes an approximate Nash equilibrium using only limited communication between the players. In Chapter 4, we present an algorithm that, for every δ in the range 0 < δ ≤ 0.5, finds a (0.5+δ)-Nash equilibrium of a polymatrix game in time polynomial in the input size and 1 . Note that our approximation guarantee does not depend on δ the number of players, a property that was not previously known to be achievable for polymatrix games, and still cannot be achieved for general strategic-form games. In Chapter 5, we present an approximation-preserving reduction from the problem of computing an approximate Bayesian Nash equilibrium (e-BNE) for a two-player Bayesian game to the problem of computing an e-NE of a polymatrix game and thus show that the algorithm of Chapter 4 can be applied to two-player Bayesian games. Furthermore, we provide a simple polynomial-time algorithm for computing a 0.5-BNE. In Chapter 5, we study games with non-linear utility functions for the players. Our key insight is that Lipschitz continuity of the utility function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly poly- nomial time algorithms for finding best responses in L1, L2, and L∞ biased games, and then use these algorithms to provide strongly polynomial algorithms that find 2/3, 5/7, and 2/3 approximations for these norms, respectively.

We present the first polynomial time algorithm to learn nontrivial classes of languages of infinite trees. Specifically, our algorithm uses membership and equivalence queries to learn classes of omega-tree languages derived from weak regular omega-word languages in polynomial time. The method is a general polynomial time reduction of learning a class of derived omega-tree languages to learning the underlying class of omega-word languages, for any class of omega-word languages recognized by a deterministic Büchi acceptor. Our reduction, combined with the polynomial time learning algorithm of Maler and Pnueli [Maler and Pneuli, Inform. Comput., 1995] for the class of weak regular omega-word languages yields the main result. We also show that subset queries that return counterexamples can be implemented in polynomial time using subset queries that return no counterexamples for deterministic or non-deterministic finite word acceptors, and deterministic or non-deterministic Büchi omega-word acceptors. A previous claim of an algorithm to learn regular omega-trees due to Jayasrirani, Begam and Thomas [Jayasrirani et al., ICGI, 2008] is unfortunately incorrect, as shown in [Angluin, YALEU/DCS/TR-1528, 2016].