Max Ones Research Articles

The exponential-time hypothesis and the relative complexity of optimization and logical reasoning problems

Obtaining lower bounds for NP-hard problems has for a long time been an active area of research. Algebraic techniques introduced by Jonsson et al. (2017) [4] show that the fine-grained time complexity of the parameterized ▪ problem correlates to the lattice of strong partial clones. With this ordering they isolated a relation R such that ▪ can be solved at least as fast as any other NP-hard ▪ problem. In this paper we extend this method and show that such languages also exist for the surjective SAT problem, the max ones problem, the propositional abduction problem, and the Boolean valued constraint satisfaction problem over finite-valued constraint languages. These languages may be interesting when investigating the borderline between polynomial time, subexponential time and exponential-time algorithms since they in a precise sense can be regarded as NP-hard problems with minimum time complexity. Indeed, with the help of these languages we relate all of the above problems to the exponential time hypothesis (ETH) in several different ways.

Modified Block Based Technique for Efficient Test Data Compression

Test data compression is one of the major challenging tasks in VLSI. This paper targets on Test Data compression by using block based method without any variation in the system performance. Test data sets are separated into five kinds of blocks like whole zeros, whole ones, min zero/min one, max ones and complement blocks. The blocks will be read based on the presence of 0s and 1s. The analysis made in the ISCAS89 benchmark circuit shows that the proposed method is more efficient and more flexible and produces a good compression ratio when compared to the existing methods.

Parameterized Complexity and Kernelizability of Max Ones and Exact Ones Problems

For a finite set Γ of Boolean relations, M ax O nes SAT(Γ) and E xact O nes SAT(Γ) are generalized satisfiability problems where every constraint relation is from Γ, and the task is to find a satisfying assignment with at least/exactly k variables set to 1, respectively. We study the parameterized complexity of these problems, including the question whether they admit polynomial kernels. For M ax O nes SAT(Γ), we give a classification into five different complexity levels: polynomial-time solvable, admits a polynomial kernel, fixed-parameter tractable, solvable in polynomial time for fixed k , and NP-hard already for k = 1. For E xact O nes SAT(Γ), we refine the classification obtained earlier by taking a closer look at the fixed-parameter tractable cases and classifying the sets Γ for which E xact O nes SAT(Γ) admits a polynomial kernel.

A practical method for driver sleepiness detection by processing the EEG signals stimulated with external flickering light

A novel steady state visual evoked potential (SSVEP)-based BCI system for driver’s sleepiness monitoring is proposed. Detecting the driver’s concentration is one of the most challenging assignments for researchers in these decades. A continuous attention and awareness is necessary to drive safely. Otherwise, one tragedy of drowsy driving would probably happen. Therefore, real-time sleepiness detection can restrain accidents effectively. In this study, SSVEPs are used for running the proposed system and two experimental setups consisting four single and paired light-emitting diodes (LEDs) using two different fast Fourier transform-based feature extraction methods, and three different classifiers of the linear discriminant analysis, the support vector machine (SVM) and the Max ones on the accuracy of the system are studied. For real-time application, related features are extracted from four different sweep lengths (temporal durations) of 0.5, 1, 2, and 3 s. The experimental results show that higher sweeps have higher accuracies and the SVM classifier, experimental setup of 4-paired LEDs in sweep length of 3 s has the highest accuracy of 98.2 %, while with the comparable information transfer rate (ITR) value of 24 bits/min within the sweep length of 1 s, this time is considered as the best response time. Therefore, this study demonstrates the feasibility of the proposed system in a practical driving application.

Approximability of Clausal Constraints

We study a family of problems, called Maximum Solution (Max Sol), where the objective is to maximise a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints. When the domain is Boolean (i.e. restricted to {0,1}), the maximum solution problem is identical to the well-studied Max Ones problem, and the complexity and approximability is completely understood for all restrictions on the underlying constraints. We continue this line of research by considering the Max Sol problem for relations defined by regular signed logic over finite subsets of the natural numbers; the complexity of the corresponding decision problem has recently been classified by Creignou et al. (Theory Comput. Syst. 42(2):239–255, 2008). We give sufficient conditions for when such problems are polynomial-time solvable and we prove that they are APX-hard otherwise. Similar dichotomies are also obtained for variants of the Max Sol problem.

MAX ONES Generalized to Larger Domains

We study a family of problems, called Maximum Solution, where the objective is to maximize a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints. When the domain is Boolean (i.e., restricted to $\{0,1\}$), the maximum solution problem is identical to the well-studied Max Ones problem, and the approximability is completely understood for all restrictions on the underlying constraints [S. Khanna, M. Sudan, L. Trevisan, and D. P. Williamson, SIAM J. Comput., 30 (2001), pp. 1863-1920]. We continue this line of research by considering domains containing more than two elements. We present two main results: a complete classification for the approximability of all maximal constraint languages over domains of cardinality at most $4$, and a complete classification of the approximability of the problem when the set of allowed constraints contains all permutation constraints. Under the assumption that a conjecture due to Szczepara [Minimal Clones Generated by Groupoids, Ph.D. thesis, Universite de Montreal, Montreal, QC, 1996] holds, we give a complete classification for all maximal constraint languages. These classes of languages are well studied in universal algebra and computer science; they have, for instance, been considered in connection with machine learning and constraint satisfaction. Our results are proved by using algebraic results from clone theory, and the results indicate that this approach is very powerful for classifying the approximability of certain optimization problems.

The Approximability of Constraint Satisfaction Problems

We study optimization problems that may be expressed as constraint satisfaction problems. An instance of a Boolean constraint satisfaction problem is given by m constraints applied to n Boolean variables. Different computational problems arise from constraint satisfaction problems depending on the nature of the underlying constraints as well as on the goal of the optimization task. Here we consider four possible goals: Max CSP (Min CSP) is the class of problems where the goal is to find an assignment maximizing the number of satisfied constraints (minimizing the number of unsatisfied constraints). Max Ones (Min Ones) is the class of optimization problems where the goal is to find an assignment satisfying all constraints with maximum (minimum) number of variables set to 1. Each class consists of infinitely many problems and a problem within a class is specified by a finite collection of finite Boolean functions that describe the possible constraints that may be used. Tight bounds on the of every problem in Max CSP were obtained by Creignou [ J. Comput. System Sci., 51 (1995), pp. 511--522]. In this work we determine tight bounds on the approximability (i.e., the ratio to within which each problem may be approximated in polynomial time) of every problem in Max Ones, Min CSP, and Min Ones. Combined with the result of Creignou, this completely classifies all optimization problems derived from Boolean constraint satisfaction. Our results capture a diverse collection of optimization problems such as MAX 3-SAT, Max Cut, Max Clique, Min Cut, Nearest Codeword, etc. Our results unify recent results on the (in-)approximability of these optimization problems and yield a compact presentation of most known results. Moreover, these results provide a formal basis to many statements on the behavior of natural optimization problems that have so far been observed only empirically.

Near-optimal nonapproximability results for some Npo PB-complete problems

We show that a number of Npo PB-complete problems, including Min Ones and Max Ones, are hard to approximate within n 1 − ε for arbitrary ε > 0.