Trivial Algorithm Research Articles

Approximation Algorithms for Preference Aggregation Using CP-Nets

This paper studies the design and analysis of approximation algorithms for aggregating preferences over combinatorial domains, represented using Conditional Preference Networks (CP-nets). Its focus is on aggregating preferences over so-called swaps, for which optimal solutions in general are already known to be of exponential size. We first analyze a trivial 2-approximation algorithm that simply outputs the best of the given input preferences, and establish a structural condition under which the approximation ratio of this algorithm is improved to 4/3. We then propose a polynomial-time approximation algorithm whose outputs are provably no worse than those of the trivial algorithm, but often substantially better. A family of problem instances is presented for which our improved algorithm produces optimal solutions, while, for any ε, the trivial algorithm cannot attain a (2- ε)-approximation. These results may lead to the first polynomial-time approximation algorithm that solves the CP-net aggregation problem for swaps with an approximation ratio substantially better than 2.

Filling crosswords is very hard

We revisit a classical crossword filling puzzle which already appeared in Garey& Jonhson's book. We are given a grid with n vertical and horizontal slots and a dictionary with m words. We are asked to place words from the dictionary in the slots so that shared cells are consistent. We attempt to pinpoint the source of intractability of this problem by carefully taking into account the structure of the grid graph, which contains a vertex for each slot and an edge if two slots intersect. Our main approach is to consider the case where this graph has a tree-like structure. Unfortunately, if we impose the common rule that words cannot be reused, we discover that the problem remains NP-hard under very severe structural restrictions, namely, if the grid graph is a union of stars and the alphabet has size 2, or the grid graph is a matching (so the crossword is a collection of disjoint crosses) and the alphabet has size 3. The problem does become slightly more tractable if word reuse is allowed, as we obtain an mtw algorithm in this case, where tw is the treewidth of the grid graph. However, even in this case, we show that our algorithm cannot be improved to obtain fixed-parameter tractability. More strongly, we show that under the ETH the problem cannot be solved in time mo(k), where k is the number of horizontal slots of the instance (which trivially bounds tw).Motivated by these mostly negative results, we also consider the much more restricted case where the problem is parameterized by the number of slots n. Here, we show that the problem does become FPT (if the alphabet has constant size), but the parameter dependence is exponential in n2. We show that this dependence is also justified: the existence of an algorithm with running time 2o(n2), even for binary alphabet, would contradict the randomized ETH. After that, we consider an optimization version of the problem, where we seek to place as many words on the grid as possible. Here it is easy to obtain a 12-approximation, even on weighted instances, simply by considering only horizontal or only vertical slots. We show that this trivial algorithm is also likely to be optimal, as obtaining a better approximation ratio in polynomial time would contradict the Unique Games Conjecture. The latter two results apply whether word reuse is allowed or not.Finally, we present some special cases where the problem is decidable in polynomial time. In particular, we present three reductions, one to 2-SAT, the second to Maximum Matching and the third to Exact Matching.

An Augmented Lagrangian Model for Signal Segmentation

In this paper, we provide a new insight to the two-phase signal segmentation problem. We propose an augmented Lagrangian variational model based on Chan–Vese’s original one. Using both energy methods and PDE methods, we show, in the one-dimensional case, that the set of minimizers to the proposed functional contains only binary functions and it coincides with the set of minimizers to Chan–Vese’s one. This fact allows us to obtain two important features of the minimizers as a by-product of our analysis. First of all, for a piecewise constant initial signal, the jump set of any minimizer is a subset of the jump set of the given signal. Second, all of the jump points of the minimizer belong to the same level set of the signal, in a multivalued sense. This last property permits to design a trivial algorithm for computing the minimizers.

Counting Non-Convex 5-Holes in a Planar Point Set

Let S be a set of n points in the general position, that is, no three points in S are collinear. A simple k-gon with all corners in S such that its interior avoids any point of S is called a k-hole. In this paper, we present the first algorithm that counts the number of non-convex 5-holes in S. To our best knowledge, prior to this work there was no known algorithm in the literature except a trivial brute force algorithm. Our algorithm runs in time O(T+Q), where T denotes the number of 3-holes, or empty triangles, in S and Q that denotes the number of non-convex 4-holes in S. Note that T+Q ranges from Ω(n2) to O(n3), while its expected number is Θ(n2logn) when the points in S are chosen uniformly and independently at random from a convex and bounded body in the plane.

Guessing fractions of online sequences

An online algorithm is typically unaware of the length of the input request sequence that it is called upon. Consequently, it cannot determine whether it has already processed most of its input or whether the bulk of work is still ahead.In this paper, we are interested in whether some sort of orientation within the request sequence is nevertheless possible. For a real number 0<f<1, our objective is to preemptively guess the position fn within the request sequence of unknown length n: While processing the input, the online algorithm maintains a guess and is only allowed to update its guess to the position of the current element under investigation.We give a complete characterization of the optimal guessing strategies with respect to f: For f≤f0≈0.1867, the trivial algorithm that never updates its initial guess is (asymptotically) optimal. For f>f0, we give a randomized algorithm that in expectation places the guess at distance β(f)n from fn, where β is a non-elementary function, and we prove that this is optimal. Our findings show that the position fmaxn≈0.2847n is hardest to guess. We also give upper and lower bounds for a natural extension to weighted sequences.Our work can also be seen as the first randomized approach to the online checkpointing problem.

A tight $$\sqrt{2}$$-approximation for linear 3-cut

We investigate the approximability of the linear 3-cut problem in directed graphs. The input here is a directed graph $$D=(V,E)$$ with node weights and three specified terminal nodes $$s,r,t\in V$$ , and the goal is to find a minimum weight subset of non-terminal nodes whose removal ensures that s cannot reach r and t, and r cannot reach t. The precise approximability of linear 3-cut has been wide open until now: the best known lower bound under the unique games conjecture (UGC) was 4 / 3, while the best known upper bound was 2 using a trivial algorithm. In this work we completely close this gap: we present a $$\sqrt{2}$$ -approximation algorithm and show that this factor is tight under UGC. Our contributions are twofold: (1) we analyze a natural two-step deterministic rounding scheme through the lens of a single-step randomized rounding scheme with non-trivial distributions, and (2) we construct integrality gap instances that meet the upper bound of $$\sqrt{2}$$ . Our gap instances can be viewed as a weighted graph sequence converging to a “graph limit structure”. We complement our results by showing connections between the linear 3-cut problem and other fundamental cut problems in directed graphs.

Characterization of Advanced Encryption Standard (AES) for Textual and Image data

— Safekeeping of multimedia data is an imperious concern for the reason that fast progression of digital data exchanges (DDE) over unsafe network. Multimedia data safety is accomplished by methods of cryptography, which is concerned with encryption of data. Standard symmetric encryption algorithms are responsible to enhance security for the multimedia data. The problem of computational head is always on the way. To overwhelmed that delinquent, we examine the Advanced Encryption Standard (AES) and amend it, to diminish the calculation of algorithm and for enlightening the encryption performance. In this paper, an improved AES algorithm has been investigated. As an alternative of using Mix column we practice with modified step, takes a stable modified step key Mk, and perform expansion routine by using Randal’s key schedule to generate a modified step schedule. When the Mk length is 128 bit, then the Expansion generates a total 11 sub Mk arrays of 128 bits, and Xor with state array in every rounds except final round. Hypothetical analysis and experimental results provide evidence that this technique has the capability of high speed as well as fewer overheads on data. Modified-AES algorithm is a quick trivial encryption algorithm for security of multimedia data. All above advantages put together algorithm highly appropriate for the images and plain text transfer as well.

Blind, Greedy, and Random: Algorithms for Matching and Clustering Using Only Ordinal Information

We study the Maximum Weighted Matching problem in a partial information setting where the agents' utilities for being matched to other agents are hidden and the mechanism only has access to ordinal preference information. Our model is motivated by the fact that in many settings, agents cannot express the numerical values of their utility for different outcomes, but are still able to rank the outcomes in their order of preference. Specifically, we study problems where the ground truth exists in the form of a weighted graph, and look to design algorithms that approximate the true optimum matching using only the preference orderings for each agent (induced by the hidden weights) as input. If no restrictions are placed on the weights, then one cannot hope to do better than the simple greedy algorithm, which yields a half optimal matching. Perhaps surprisingly, we show that by imposing a little structure on the weights, we can improve upon the trivial algorithm significantly: we design a 1.6-approximation algorithm for instances where the hidden weights obey the metric inequality. Our algorithm is obtained using a simple but powerful framework that allows us to combine greedy and random techniques in unconventional ways. These results are the first non-trivial ordinal approximation algorithms for such problems, and indicate that we can design robust matchings even when we are agnostic to the precise agent utilities.

Why Do Simple Algorithms for Triangle Enumeration Work in the Real World?

_Listing all triangles is a fundamental graph operation. Triangles can have important interpretations in real-world graphs, especially social and other interaction networks. Despite the lack of provably efficient (linear, or slightly super linear) worst-case algorithms for this problem, practitioners run simple, efficient heuristics to find all triangles in graphs with millions of vertices. How are these heuristics exploiting the structure of these special graphs to provide major speedups in running time_? _We study one of the most prevalent algorithms used by practitioners. A trivial algorithm enumerates all paths of length 2, and checks if each such path is incident to a triangle. A good heuristic is to enumerate only those paths of length 2 in which the middle vertex has the lowest degree. It is easily implemented and is empirically known to give remarkable speedups over the trivial algorithm_. _We study the behavior of this algorithm over graphs with heavy-tailed degree distributions, a defining feature of real-world graphs. The erased configuration model (ECM) efficiently generates a graph with asymptotically (almost) any desired degree sequence. We show that the expected running time of this algorithm over the distribution of graphs created by the ECM is controlled by the ℓ_4/3-norm of the degree sequence. Norms of the degree sequence are a measure of the heaviness of the tail, and it is precisely this feature that allows non trivial speedups of simple triangle enumeration algorithms. As a corollary of our main theorem, we prove expected linear-time performance for degree sequences following a power law with exponent α ≥ 7/3, and non trivial speedup whenever_ α ∈ (2, 3).

On Extracting Common Random Bits From Correlated Sources on Large Alphabets

Suppose Alice and Bob receive strings X=(X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,...,X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) and Y=(Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,...,Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) each uniformly random in [s] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , but so that X and Y are correlated. For each symbol i, we have that Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> =X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> with probability 1-ε and otherwise Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> is chosen independently and uniformly from [s]. Alice and Bob wish to use their respective strings to extract a uniformly chosen common sequence from [s] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> , but without communicating. How well can they do? The trivial strategy of outputting the first k symbols yields an agreement probability of (1-ε+ε/s) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> . In a recent work by Bogdanov and Mossel, it was shown that in the binary case where s=2 and k=k(ε) is large enough then it is possible to extract k bits with a better agreement probability rate. In particular, it is possible to achieve agreement probability (kε) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/2</sup> ·2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-kε/(2(1-ε/2))</sup> using a random construction based on Hamming balls, and this is optimal up to lower order terms. In this paper, we consider the same problem over larger alphabet sizes s and we show that the agreement probability rate changes dramatically as the alphabet grows. In particular, we show no strategy can achieve agreement probability better than (1-ε) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> (1+δ(s)) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> where δ(s)→ 0 as s→∞. We also show that Hamming ball-based constructions have much lower agreement probability rate than the trivial algorithm as s→∞. Our proofs and results are intimately related to subtle properties of hypercontractive inequalities.

On the NP-Hardness of Max-Not-2

We prove that, for any $\epsilon>0$, given a satisfiable instance of Max-NTW (Not-2), it is NP-hard to find an assignment that satisfies a fraction $\frac 58 +\epsilon$ of the constraints. This, up to the existence of $\epsilon$, matches the approximation ratio obtained by the trivial algorithm that just picks an assignment at random, and thus the result is tight. Said equivalently, the result proves that Max-NTW is approximation resistant on satisfiable instances, and this makes complete our understanding of arity three maximum constraint satisfaction problems with regards to approximation resistance.

Minimization of Side Effects of Knowledge Hiding Based On Association Rule Mining

Privacy is very important in Data Mining. Association rule mining, is a important technique, has already been applied in a wide range of areas. Large databases cannot be secured by encryption or other techniques because of complexities. These security issues can be divided into two categories: data hiding and knowledge hiding. This paper deals with the problem of association rule mining which preserves the confidentiality of each database i.e. Knowledge hiding. Knowledge hiding is concerned with the sanitization of confidential knowledge from the data. There are many methods to solve this problem. The one, presented in the paper is called support-based and confidence-based blocking schemes. Apriori algorithm is used for mining rule from database. Knowledge hiding is achieved using ISL and DSR. The improvement over sensitive rule hiding is proposed to offer more accuracy and security with smaller negative impact. In ever changing business environment like enterprises e-businesses, the old fashioned disclosure and database inference protection techniques are not capable to ensure complete data privacy. Privacy is important for the on line disclosure of private information Organizations should be able to evaluate the risk of disclosing information and should adopt new more efficient approaches for information disclosure control, in order to maintain their competitive edge in the market. Not only the data but also the hidden knowledge in the data should be made secure. The sensitive information can be extracted in the form of association rules with association rule mining tools. But this can jeopardize the privacy of the customers. Some other rule could be very critical for the company itself such as buying patterns of very rich customers. For example, an Internet based company may give up selling hardware and may concentrate on selling books and videos. Therefore a rule relating the customer buying patterns of hardware may no longer be sensitive for that company. It is desirable to apply a trivial algorithm that hides data by deleting it randomly. Data Mining is the process of discovering new patterns from large data sets involving methods from statistics and artificial intelligence but also database management. Privacy has become an important issue in Data Mining. Many methods have been brought out to solve this problem. This paper focuses on the problem of association rule mining, which hides the knowledge from each database. This paper reviews the major method of knowledge hiding. Association rule mining, as a very important technique, has already been applied in a wide range of areas. Knowledge hiding is concerned with the sanitization of confidential knowledge from the data. As a result of association rule mining, many useful association rules will be discovered, but at the same time, many privacy rules will also be exposed which do not want others to know. To solve this, limit the mining process, in order to keep these sensitive rules being hidden. There are so many methods to solve this problem. The one that is covered in the paper is just one kind of them, called support-based and confidence-based blocking schemes. Algorithm used For Knowledge Hiding is ISL algorithm or DSR algorithm (2). The main disadvantage of a block in algorithm is the fact that the dataset, apart from the blocked values, is not distorted. Thus, an adversary can disclose the hidden rules by identifying those generating item sets that contain question marks and lead to rules with a maximum confidence that lies above the minimum confidence threshold. If the number of these rules is small then the probability of identifying the sensitive ones among them becomes high. To avoid this problem, a method for selectively removing individual values from a database is introduced to prevent the discovery of a set of rules, while preserving the data for other applications. The problem of building privacy preserving algorithms is considered for one category of data mining techniques, the association rule mining. New metrics is introduced in order to demonstrate how security issues can be taken into consideration in the general framework of association rule mining (1).

Replacement paths and k simple shortest paths in unweighted directed graphs

Let G = ( V,E ) be a directed graph and let P be a shortest path from s to t in G . In the replacement paths problem, we are required to find, for every edge e on P , a shortest path from s to t in G that avoids e . The only known algorithm for solving the problem, even for unweighted directed graphs, is the trivial algorithm in which each edge on the path, in its turn, is excluded from the graph and a shortest paths tree is computed from s . The running time is O ( mn + n 2 log n ). The replacement paths problem is strongly motivated by two different applications: (1) The fastest algorithm to compute the k simple shortest paths between s and t in directed graphs [Yen 1971; Lawler 1972] computes the replacement paths between s and t . Its running time is Õ ( mnk ). (2) The replacement paths problem is used to compute the Vickrey pricing of edges in a distributed network. It was raised as an open problem by Nisan and Ronen [2001] whether it is possible to compute the Vickrey pricing faster than n computations of a shortest paths tree. In this article we present the first nontrivial algorithm for computing replacement paths in unweighted directed graphs (and in graphs with small integer weights). Our algorithm is Monte-Carlo and its running time is Õ ( m √ n ). This result immediately improves the running time of the two applications mentioned above in a factor of √ n . We also show how to reduce the problem of computing k simple shortest paths between s and t to O ( k ) computations of a second simple shortest path from s to t each time in a different subgraph of G . The importance of this result is that computing a second simple shortest path may turn out to be an easier problem than computing the replacement paths, thus, we can focus our efforts to improve the k simple shortest paths algorithm in obtaining a faster algorithm for the second shortest path problem.

Affine‐invariant character recognition by progressive removing

AbstractRecognizing characters in scene images suffering from perspective distortion is a challenge. Although there are some methods to overcome this difficulty, they are time‐consuming. In this paper, we propose a set of affine‐invariant features and a new recognition scheme called “progressive removing” that can help reduce the processing time. Progressive removing gradually removes less feasible categories and skew angles by using multiple classifiers. We observed that progressive removing and the use of the affine invariant features reduced the processing time by about 60% in comparison to a trivial algorithm without decreasing the recognition rate. © 2012 Wiley Periodicals, Inc. Electr Eng Jpn, 180(2): 55–63, 2012; Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/eej.22276

A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set

Graphs and Algorithms An independent dominating set D of a graph G = (V,E) is a subset of vertices such that every vertex in V \ D has at least one neighbor in D and D is an independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum independent dominating set in a graph is an NP-hard problem. Whereas it is hard to cope with this problem using parameterized and approximation algorithms, there is a simple exact O(1.4423^n)-time algorithm solving the problem by enumerating all maximal independent sets. In this paper we improve the latter result, providing the first non trivial algorithm computing a minimum independent dominating set of a graph in time O(1.3569^n). Furthermore, we give a lower bound of \Omega(1.3247^n) on the worst-case running time of this algorithm, showing that the running time analysis is almost tight.

Complexity and parameterized algorithms for Cograph Editing

Cograph Editing is to find for a given graph G=(V,E) a set of at most k edge additions and deletions that transform G into a cograph. The computational complexity of this problem was open in the past. In this paper, we first show that this problem is NP-hard by a reduction from Exact 3-Cover. Subsequently, we present a parameterized algorithm based on a refined search tree technique with a running time of O(4.612k+|V|4.5), which improves the trivial algorithm of running time O(6k+|V|4.5).

A near-linear-time algorithm for computing replacement paths in planar directed graphs

Let ( G = ( V(G) , E(G) )) be a directed graph with nonnegative edge lengths and let P be a shortest path from s to t in G . In the replacement paths problem we are required to compute for every edge e in P , the length of a shortest path from s to t that avoids e . The fastest known algorithm for solving the problem in weighted directed graphs is the trivial one: each edge in P is removed from the graph in its turn and the distance from s to t in the modified graph is computed. The running time of this algorithm is O ( m n + n 2 log n ), where n = | V(G) | and m = | E(G) |. The replacement paths problem is strongly motivated by two different applications. First, the fastest algorithm to compute the k simple shortest paths from s to t in directed graphs [Yen 1971; Lawler 1972] repeatedly computes the replacement paths from s to t . Its running time is O ( kn ( m + n log n )). Second, the computation of Vickrey pricing of edges in distributed networks can be reduced to the replacement paths problem. An open question raised by Nisan and Ronen [2001] asks whether it is possible to compute the Vickrey pricing faster than the trivial algorithm described in the previous paragraph. In this article we present a near-linear time algorithm for computing replacement paths in weighted planar directed graphs. In particular, the algorithm computes the lengths of the replacement paths in O ( n log 3 n ) time (recall that in planar graphs m = O ( n )). This result immediately improves the running time of the two applications mentioned before by almost a linear factor. Our algorithm is obtained by combining several new ideas with a data structure of Klein [2005] that supports multisource shortest paths queries in planar directed graphs in logarithmic time. Our algorithm can be adapted to address the variant of the problem in which one is interested in the replacement path itself (rather than the length of the path). In that case the algorithm is executed in a preprocessing stage constructing a data structure that supports replacement path queries in time Õ ( h ), where h is the number of hops in the replacement path. In addition, we can handle the variant in which vertices should be avoided instead of edges.

A Spatiotemporal Framework for MEG/EEG Evoked Response Amplitude and Latency Variability Estimation

This paper presents a spatiotemporal framework for estimating single-trial response latencies and amplitudes from evoked response magnetoencephalographic/electroencephalographic data. Spatial and temporal bases are employed to capture the aspects of the evoked response that are consistent across trials. Trial amplitudes are assumed independent but have the same underlying normal distribution with unknown mean and variance. The trial latency is assumed to be deterministic but unknown. We assume that the noise is spatially correlated with unknown covariance matrix. We introduce a generalized expectation-maximization algorithm called Trial Variability in Amplitude and Latency ( TriViAL) that computes the maximum likelihood (ML) estimates of the amplitudes, latencies, basis coefficients, and noise covariance matrix. The proposed approach also performs ML source localization by scanning the TriViAL algorithm over spatial bases corresponding to different locations on the cortical surface. Source locations are identified as the locations corresponding to large likelihood values. The effectiveness of the TriViAL algorithm is demonstrated using simulated data and human evoked response experiments. The localization performance is validated using tactile stimulation of the finger. The efficacy of the algorithm in estimating latency variability is shown using the known dependence of the M100 auditory response latency to stimulus tone frequency. We also demonstrate that estimation of response amplitude is improved when latency is included in the signal model.

Bincoloring

We introduce a new problem that was motivated by a (more complicated) problem arising in a robotized assembly environment. The bin coloring problem is to pack unit size colored items into bins, such that the maximum number of different colors per bin is minimized. Each bin has size B ∈ N . The packing process is subject to the constraint that at any moment in time at most q ∈ N bins are partially filled. Moreover, bins may only be closed if they are filled completely. We settle the computational complexity of the problem and design an approximation algorithm for a natural version which gives a solution whose value is at most one greater than the optimal one. We also investigate the existence of competitive online algorithms, which must pack each item without knowledge of any future items. We prove an upper bound of 3 q − 1 and a lower bound of 2 q for the competitive ratio of a natural greedy-type algorithm, and show that surprisingly a trivial algorithm which uses only one open bin has a strictly better competitive ratio of 2 q − 1 . Moreover, we show that any deterministic algorithm has a competitive ratio Ω ( q ) and that randomization does not improve this lower bound even when the adversary is oblivious.

A new characterization of graphic matroids

We give a necessary and sufficient condition for a binary matroid to be graphic. The condition is very natural, but, unlike other similar results, it gives a trivial algorithm for testing graphicness.