The maximum independent set problem is a classic and fundamental combinatorial challenge, where the objective is to find the largest subset of vertices in a graph such that no two vertices are adjacent. In this paper, we introduce a novel... more
The maximum independent set problem is a classic and fundamental combinatorial challenge, where the objective is to find the largest subset of vertices in a graph such that no two vertices are adjacent. In this paper, we introduce a novel linear prioritized local algorithm tailored to address this problem on random d-regular graphs with a small and fixed degree d. Through exhaustive numerical simulations, we empirically investigated the independence ratio, i.e., the ratio between the cardinality of the independent set found and the order of the graph, which was achieved by our algorithm across random d-regular graphs with degree d ranging from 5 to 100. Remarkably, for every d within this range, our results surpassed the existing lower bounds determined by theoretical methods. Consequently, our findings suggest new conjectured lower bounds for the MIS problem on such graph structures. This finding has been obtained using a prioritized local algorithm. This algorithm is termed ‘prior...
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Finding a Maximum Clique is a classic property test from graph theory; find any one of the largest complete subgraphs in an Erdös-Rényi G(N, p) random graph. It is the simplest of many such problems in which algorithms requiring only a... more
Finding a Maximum Clique is a classic property test from graph theory; find any one of the largest complete subgraphs in an Erdös-Rényi G(N, p) random graph. It is the simplest of many such problems in which algorithms requiring only a small power of N steps cannot reach solutions which probabilistic arguments show must exist, exposing an inherently "hard" phase within the solution space of the problem. Such "hard" phases are seen in many NP-Complete problems, in the limit when N → ∞. But optimization problems arise and must be solved at finite N. We use this simplest case, MaxClique, to explore the structure of the problem as a function of N and K, the clique size. It displays a complex phase boundary, a staircase of steps at each of which 2log2 N and Kmax, the maximum size of clique that can be found, increase by 1. Each of its boundaries have finite width, and these widths allow local algorithms to find cliques beyond the limits defined by the study of infin...
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The MaxClique problem, finding the largest complete subgraph in an Erdös-RényiG(N, p) random graph in the large N limit, is a well-known example of a simple problem for which finding any approximate solution within a factor of 2 of the... more
The MaxClique problem, finding the largest complete subgraph in an Erdös-RényiG(N, p) random graph in the large N limit, is a well-known example of a simple problem for which finding any approximate solution within a factor of 2 of the known, probabilistically determined limit, appears to require P=NP. This type of search has practical importance in very large graphs. Algorithmic approaches run into phase boundaries long before they reach the size of the largest likely solutions. And, most intriguing, there is an extensive literature of challenges posed for concrete methods of finding maximum naturally occurring as well as artificially hidden cliques, with computational costs that are at most polynomial in the size of the problem. We use the probabilistic approach in a novel way to provide a more insightful test of constructive algorithms for this problem. We show that extensions of existing methods of greedy local search will be able to meet the challenges for practical problems of...
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Research Interests: Computer Science, Machine Learning, Computational Complexity, General Relativity, Computational Biology, and 15 moreStatistical Physics, Multidisciplinary, Nature, Phase Transitions, Hardware Design, Computer Model, Combinatorial Problems, Efficient Algorithm for ECG Coding, Search Algorithm, Phase transition, Analytical Solution, Second Order, Granular Material, Theory of Computing, and Polynomial Time
We give analytical and numerical results concerning a new Satisfiability problem for random Boolean expressions. Our model smoothly interpolates between the polynomial 2--SAT problem and the NP--complete 3--SAT problem. Possible... more
We give analytical and numerical results concerning a new Satisfiability problem for random Boolean expressions. Our model smoothly interpolates between the polynomial 2--SAT problem and the NP--complete 3--SAT problem. Possible consequences on the relationship between the statistical mechanics characterization of phase transitions --- particularly smooth second order RSB and first order RSB transitions --- and the onset of exponential behaviour in search algorithms are identified. CNRS-Laboratoire de Physique Th'eorique de l'ENS, Paris y Physics Department, Politecnico di Torino, Torino z IBM, Thomas J. Watson Research Center, Yorktown Heights, NY x AT&T Laboratories, Florham Park, NJ -- Institute of Computer Science and Center for Neural Computation, Hebrew University, Jerusalem 1 Introduction Complexity theory, as arising from Cook's theorem of 1971 [1], deals with the classification of problems according to the running time or memory requirement necessary for their ...
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Understanding the detailed behavior of an operating system is crucial for making informed design decisions. But such an understanding is very hard to achieve, due to the increasing complexity of such systems and the fact that they are... more
Understanding the detailed behavior of an operating system is crucial for making informed design decisions. But such an understanding is very hard to achieve, due to the increasing complexity of such systems and the fact that they are implemented and maintained by large and diverse groups of developers. Tools like KLogger --- presented in this paper --- can help by enabling fine-grained logging of system events and the sharing of a logging infrastructure between multiple developers and researchers, facilitating a methodology where design evaluation can be an integral part of kernel development. We demonstrate the need for such methodology by a host of case studies, using KLogger to better understand various subsystems in the Linux kernel, and pinpointing overheads and problems therein.
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this paper, we give a preliminary discussion of a newSAT model, hereafter referred to as 2+p--SAT model. Themodel interpolates smoothly between 2--SAT and 3--SAT,and allows us to address the above issue.
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We have pioneered the deployment of EverLab, a production level private PlanetLab system using high-end clusters spread over Europe. EverLab supports both experimentation and computational work, incorporating many of the features found on... more
We have pioneered the deployment of EverLab, a production level private PlanetLab system using high-end clusters spread over Europe. EverLab supports both experimentation and computational work, incorporating many of the features found on Grid systems. This paper describes the ...
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Research Interests: Cognitive Science, Mathematics, Computer Science, Computational Complexity Theory, Artificial Intelligence, and 11 moreStatistical Mechanics, Kinetics, Complexity, Statistical Physics, Critical phenomena, Phase transition, Scaling, Spin Glass, Boolean Satisfiability, Finite Size Scaling, and critical behavior
Survey propagation is a powerful technique from statistical physics that has been applied to solve the 3-SAT problem both in principle and in practice. We give, using only probability arguments, a common deriva- tion of survey... more
Survey propagation is a powerful technique from statistical physics that has been applied to solve the 3-SAT problem both in principle and in practice. We give, using only probability arguments, a common deriva- tion of survey propagation, belief propagation and several interesting hy- brid methods. We then present numerical experiments which use WSAT (a widely used random-walk based SAT solver)
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In this paper, we describe the Smart Profile Management application that was designed to help minimize mobile phone disruptions. The system does this by making phone profile changes depending on the type and content of calendar entries it... more
In this paper, we describe the Smart Profile Management application that was designed to help minimize mobile phone disruptions. The system does this by making phone profile changes depending on the type and content of calendar entries it sees as well as an analysis of past usage history. A prototype was developed in Python for S60 and an informal usability
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Solvable Fractal Family, and Its Possible Relation to the Backbone at Percolation. Yuval Gefen and Amnon Aharony Department of Physics ...
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Physical and geometrical properties are studied on self similar fractal lattices. Properties of spin systems are shown to depend on various topological factors, in addition to the fractal dimensionality. A (non random) fractal model is... more
Physical and geometrical properties are studied on self similar fractal lattices. Properties of spin systems are shown to depend on various topological factors, in addition to the fractal dimensionality. A (non random) fractal model is proposed for the backbone of the infinite cluster near percolation in d dimensions, and its properties agree with those of the backbone for d 4.
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Analysis of an unusually detailed telephone call data set a month of nearly all mobile and landline phone calls placed during August 2005 the United Kingdom allows us to identify several different types of social networks that are... more
Analysis of an unusually detailed telephone call data set a month of nearly all mobile and landline phone calls placed during August 2005 the United Kingdom allows us to identify several different types of social networks that are formed, and relate them to different ...