A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bound for occupancy and the satisfiability threshold conjecture. In Proc. of the 35th IEEE Annual Symposium on Foundations of Computer Science, 1994, pp. 592--603.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....zerotemperature entropy of the satisfying assignments. This approximation ignores the fluctuations or higher order moments of the ensemble. Indeed, both numerical simulations and analytical results suggest that the transition occurs way before this annealed bound, at least for small values of K [10]. To gain better understanding of the SAT UNSAT transition we investigate the next three moments of the distribution of satisfying assignments and sketch the method for calculating other moments. The higher moments are then used to bound the number of formulae with very large number of satisfying ....

Kamath, Amil, Rajeev Motwani, Krishna Palem and Paul Spirakis, "Tail bounds for occupancy and the satisfiability thresholds conjecture", Proc. 35rd IEEE symposium on Foundation of Computer Science (1994), 592-- 603.

....that a sequence of events En occurs almost certainly if Pr[E n ] tends to 1 as n 1. For k = 2, a sharp threshold was proved in [9, 16] a random instance of 2 SAT is almost certainly satisfiable if r 1 and almost certainly unsatisfiable if r 1. For k = 3, there has been a series of results [6, 7, 5, 13, 19, 15, 21, 11] narrowing the area for which we do not have almost certain (un)satisfiability. The best bounds currently known come from [15] where it was proven that a random instance of 3 SAT is almost certainly satisfiable if r 3:003, and [21] where it was proven that a random instance of 3 SAT is almost ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis, "Tail bounds for occupancy and the satisfiability threshold conjecture," Random Structures and Algorithms, Vol. 7 (1995) 59--80.

....of this new class converge to zero is weakened, and consequently, the upper bound for is lowered. As we show in the next section, the bound for obtained by this sharpened first moment technique is equal to 4.667. This improves the previous best bound due to Kamath, Motwani, Palem, and Spirakis [9] of 4.758, which was obtained by non elementary means. Moreover our method is not computational, i.e. it does not use any mechanical computations that do not have provable accuracy and correctness (the fact that in our method we use a computer program to find a solution of an equation with one ....

.... provable accuracy and correctness (the fact that in our method we use a computer program to find a solution of an equation with one unknown does not render our proof computational, because the algorithms that find solutions to such equations have provable accuracy) The bound that Kamath et al. [9] attain with a noncomputational proof is equal to 4.87. In Section 3 we show how to further improve the bound to 4.598 by defining an even smaller subset of Sn . This is achieved by increasing the range of locality when selecting the local maxima that represent Sn . Actually, we define a ....

[Article contains additional citation context not shown here]

A. Kamath, R. Motwani, K. Palem, and P. Spirakis, "Tail bounds for occupancy and the satisfiability threshold conjecture," Random Structures and Algorithms 7, pp 59--80, 1995. Also in: Proc. 35th FOCS, IEEE, pp 592--603, 1994

....Naturally, the case k = 3 has received most attention. In [10] it was observed that for r 5:191 the expected number of satisfying truth assignments tends to 0 and hence, so does the probability of satisfiability . El Maftouhi and Fernandez de la Vega [8] lowered this to 5.08. Kamath et al. [14] lowered 5.08 to 4.758 by proving that if a random 3 SAT formula is satisfiable then, with extremely high probability, it has an exponential number of satisfying truth assignments. Hence, providing some account for the wastefulness of the first moment method. Finally, Kirousis et al. [16] ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis, Tail bounds for occupancy and the satisfiability threshold conjecture, Random Structures and Algorithms, 7 (1995), 59--80.

....last ten years, the satisfiability threshold conjecture has received attention in theoretical computer science, mathematics and, more recently, statistical physics. A large fraction of this attention has been devoted to the first computationally non trivial case, k = 3 and a long series of results [4, 5, 16, 1, 3, 21, 10, 22, 19, 23, 20, 11, 13] has narrowed the potential range of r 3 . Currently this is pinned between 3:42 by Kaporis, Kirousis and Lalas [21] and 4:506 by Dubois and Boufkhad [10] All upper bounds for r 3 come from probabilistic counting arguments, refining the idea of counting the expected number of satisfying truth ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. Random Structures Algorithms, 7(1):59--80, 1995.

.... Basic Research in Computer Science BRICS Report Series RS 96 27 ISSN 0909 0878 July 1996 Copyright c fl 1996, BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a ....

.... Basic Research in Computer Science BRICS Report Series RS 96 27 ISSN 0909 0878 July 1996 Copyright c fl 1996, BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a ....

[Article contains additional citation context not shown here]

Kamath, A., Motwani, R., Palem, K. and Spirakis, P. (1995) Tail bounds for occupancy and the satisfiability threshold conjecture. Random Structures Algorithms 7 59--80

....CH bound extends to sums of such variables. This extension is useful but somewhat ad hoc. Here we can see clearly that it is no co incidence that CH bounds can be applied in their case. The same indicator variables for empty bins also underlie results related to the Satisfiability Threshold in [23]. The analysis in both these papers can be streamlined and simplified. The key idea is that the variables E i satisfy in fact the much stronger properties of negative dependence, negative association and negative regression: Theorem 46 The empty bins indicator variables E 1 ; En satisfy ....

....numbers transfers to E 1 ; En via Corollary 25. One can now apply the CH bound to get tail estimates for Pr[E 1 Delta Delta Delta En s] Note that in this proof, we avoid any expansion and manipulations of Taylor series, as in [32, 33] The Occupancy bounds Theorems 2 and 3 in [23] follow directly as well. 4.3 Load Balancing Consider a scenario in which one has to allocate various jobs to available servers, for example, programs requesting data from disc drives, or user queries to a database system. It is desirable to perform the allocation dynamically in such a way that ....

A. Kamath, R. Motwani, K. Palem and P. Spirakis, "Tail Bounds for Occupancy and the Satisfiability Threshold Conjecture", FOCS 1994.

....First moment methods can be used to show that, at the satisfiability phase transition, the expected number of solutions for a problem is exponentially large. Kamath et al. proved that, whilst most of these problems have few or no solutions, a few have a very large number of clustered solutions [Kamath et al. 1995] . This was verified empirically by Parkes who showed that many variables are frozen although some are almost free [Parkes, 1997] He argued that such problems are hard for local search algorithms to solve as solutions are clustered and not distributed uniformly over the search space. Monasson et ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. Randomized Structure and Algorithms, 7:59--80, 1995.

....an instance of SAT. When k = 2 a sharp threshold has been proved in [9, 15] A random instance of 2 SAT is almost surely satisfiable if r 1 and almost surely unsatisfiable if r 1. For k = 3, such a sharp threshold behaviour is widely conjectured to exist and there has been a series of papers [6, 7, 5, 12, 18, 14, 19] narrowing the area for which we do not have almost surely (un)satisfiability. The best bounds currently known come from [14] where it was proven that a random instance of 3 SAT is almost surely satisfiable if r 3:003, and [19] where it was proven that a random instance of 3 SAT is almost surely ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis, "Tail bounds for occupancy and the satisfiability threshold conjecture," Random Structures and Algorithms 7, (1995) 59--80.

....ajoute une boule; la probabilit e de chaque couleur varie au cours du temps. Il s agit d etudier la proportion de boules blanches [27] 8.5 Urnes et satisfiabilit e Les mod eles d urnes peuvent servir a etudier le probl eme de satisfiabilit e d une formule bool eenne al eatoire. Kamath et al. [26] ont montr e comment la probabilit e qu une formule al eatoire soit satisfiable etait li ee au nombre d urnes vides, et ont etabli au passage des bornes sur la queue de sa distribution. Ils ont obtenu la borne C 4:758, am eliorant une etude ant erieure de El Maftouhi et De La Vega [31] qui ....

A. KAMATH, R. MOTWANI, K. PALEM, and P. SPIRAKIS. Tail bounds for occupancy and the satisfiability threshold conjecture. Random Structures and Algorithms, 7(1):59--80, 1995.

.... deviations provides a solid introduction into the entire area of large deviations, including Kurtz s work [SW95] There are by now many examples of works that use large deviation bounds and differential equations for a variety of problems, including but in no way limited to [AM97, AH90, AFP98, KMPS95, LMS 97] 38 Given this framework, it is easy to find the limiting fraction of empty bins after m = cn balls have been thrown, by solving the differential equation dy dt = 1 Gamma y d with the initial condition y(0) 0 at time c. This can be done by using the nonrigorous high school ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. Random Structures and Algorithms, 7(1):59--80, August 1995.

....[1] 2 Preliminaries In this section we present tail bounds for hypergeometrical (and hypergeometrical like) probability distributions. These bounds will be very useful for proving both the upper and the lower bound for the path coloring problem. Our approach is similar to the one used in [13] (see also [17] to calculate the tail bounds of a well known occupancy problem. We exploit the properties of special sequences of random variables called martingales, using Azuma s inequality [2] for their analysis. Similar results in a more general context are presented in [21] Consider the ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. In Proc. of the 35th Annual IEEE Symposium on Foundations of Computer Science (FOCS '94), pp. 592--603, 1994.

....laws that do not hold for dependent random variables. The well known Chernoff Hoeffding bounds from the theory of large deviations are an excellent example in this respect. Much effort has been made to salvage these sharp bounds in the more general situation under consideration; see, for example, [11, 13, 3, 8]. It turns out that one can apply the Chernoff Hoeffding bounds to sums of strongly negatively dependent random variables just as one would apply them to independent random variables; see Section 6. Hence, it is useful to establish negative dependence among random variables. However, this can ....

Kamath, A., Motwani, R., Palem, K. and Spirakis, P. (1995) Tail bounds for occupancy and the satisfiability threshold conjecture. Random Structures Algorithms 7 59--80

....approach focusing on how a random 3 SAT formula can be satisfied by binary assignments (possibly of a particular kind) to its variables. Thus, two lower bounds of were calculated : 1:63 and 3:003 [2] And starting from an easy upper bound, 5:19, three others were successively obtained : 4:76 [3], 4:643 [1] and 4:601 [4] However this semantic approach may be subject to diminishing returns in view of the complexity of the calculations for the latest bounds. In this paper, in order to estimate better, a new syntactic approach is proposed: This is suggested, e.g. simply by practical ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. Random Structures and Algorithms, 7(1):59--80, 1995.

....distributions In the following we give tail bounds for hypergeometrical (and hypergeometrical like) probability distributions. These bounds will be very useful for proving both the upper and the lower bound for the path coloring problem. Our approach is similar to the one used in [11] (see also [15] to calculate the tail bounds of a well known occupancy problem. In the following we assume familiarity of the reader with concepts of probability theory (see [6,15] The analysis of the tail bounds of hypergeometrical distribution uses properties of special sequences of random ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. In Proc. of the 35th Annual IEEE Symposium on Foundations of Computer Science (FOCS '94), pp. 592--603, 1994.

....has exactly 3 literals taken uniformly and independently from a set V of n Boolean variables and complemented independently with probability 1 2. Below, we refer to this model of generation of a random instance as M(m; n; 3) Suppose m=n is held constant as m and n tend to 1. A series of papers [7, 3, 5, 10] ended with the currently best result ( 10] that I is unsatisfiable, with probability tending to 1, if m=n 4:75. Another series of papers [1, 2, 3, 8] ended with the currently best result ( 8] that I is satisfiable, Computer Science Department, University of Cincinnati, Cincinnati, Ohio 45221 ....

....from a set V of n Boolean variables and complemented independently with probability 1 2. Below, we refer to this model of generation of a random instance as M(m; n; 3) Suppose m=n is held constant as m and n tend to 1. A series of papers [7, 3, 5, 10] ended with the currently best result ([10]) that I is unsatisfiable, with probability tending to 1, if m=n 4:75. Another series of papers [1, 2, 3, 8] ended with the currently best result ( 8] that I is satisfiable, Computer Science Department, University of Cincinnati, Cincinnati, Ohio 45221 (franco franco.csm.uc.edu) Supported in ....

Kamath, A., Motwani, R., Palem, K., and Spirakis, P., "Tail bounds for occupancy and the satisfiability threshold conjecture," Stanford University Tech Report (1994).

....that a sequence of events En occurs almost certainly if Pr[E n ] tends to 1 as n 1. For k = 2, a sharp threshold was proved in [10, 17] a random instance of 2 SAT is almost certainly satisfiable if r 1 and almost certainly unsatisfiable if r 1. For k = 3, there has been a series of results [7, 8, 6, 14, 20, 16, 22, 12] narrowing the area for which we do not have almost certain (un)satisfiability. The best bounds currently known come from [16] where it was proven that a random instance of 3 SAT is almost certainly satisfiable if r 3:003, and [22] where it was proven that a random instance of 3 SAT is almost ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis, "Tail bounds for occupancy and the satisfiability threshold conjecture," Random Structures and Algorithms, Vol. 7 (1995) 59--80.

....that a sequence of events En occurs almost certainly if Pr[E n ] tends to 1 as n 1. For k = 2, a sharp threshold was proved in [11, 19] a random instance of 2 SAT is almost certainly satisfiable if r 1 and almost certainly unsatisfiable if r 1. For k = 3, there has been a series of results [8, 9, 7, 15, 22, 17, 25, 13] narrowing the area for which we do not have almost certain (un)satisfiability. The best bounds currently known come from [17] where it was proven that a random instance of 3 SAT is almost certainly satisfiable if r 3:003, and [25] where it was proven that a random instance of 3 SAT is almost ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis, "Tail bounds for occupancy and the satisfiability threshold conjecture," Random Structures and Algorithms, Vol. 7 (1995) 59--80.

....of 8ln2 was proven by Franco and Paul [7] and independently by Chvatal and Reed [5] and Simon, Carlier, Dubois, and Moulines [21] using the first moment method. Improvements on the upper bound have been made by Maftouhi and Vega (5.08) 18] by Kamath, Motwani, Palem, and Spirakis (4. 758) [15], and most recently by Kirousis, Kranakis, Krizanc (4.598) 14] The last result represents the current best upper bound. Constructive approaches have been used to prove lower bounds on the conjectured value of c . The results obtained are based on the probabilistic analysis of iterative ....

A. Kamath, R. Motwani, K. Palem, P. Spirakis, Tail Bounds for Occupancy and the Satisfiability Threshold Conjecture, Random Structures and Algorithms 7, pg. 59-80, 1995

....needed to generate an instance that is not positive. Empirical estimates of the 0 1 threshold in Section 3.1 confirm the form of Equation 1. While the above calculation gives an upper bound on m as a function of t and n, this upper bound is likely larger than the actual number. In several papers (Kamath et al. 1995; Kirousis et al. 1998) it has been demonstrated that the expected number of satisfying assignments is larger than the number of satisfying assignments almost certainly, because for those formulae that are satisfiable, they actually have many satisfying assignments (thus skewing the average) In ....

....al. 1998) it has been demonstrated that the expected number of satisfying assignments is larger than the number of satisfying assignments almost certainly, because for those formulae that are satisfiable, they actually have many satisfying assignments (thus skewing the average) In particular, Kamath et al. 1995) show that if a random k CNF formula is satisfiable, then with exponentially high probability (probability at least 1 Gamma 2 Gammaffln ) it has an exponential number of satisfying truth assignments. Let Z be the number of variables that are not mentioned at all when m clauses are chosen ....

[Article contains additional citation context not shown here]

Kamath, A.; Motwani, R.; Palem, K.; and Spirakis, P. 1995. Tail bounds for occupancy and the satisfiability threshold conjecture. Random Structures and Algorithms 7(1):59--80.

....needed to generate an instance that is not positive. Empirical estimates of the 0 1 threshold in Section 3.1 confirm the form of Equation 1. While the above calculation gives an upper bound on m as a function of t and n, this upper bound is likely larger than the actual number. In several papers (Kamath et al. 1995; Kirousis et al. 1998) it has been demonstrated that most formulae will have a much smaller number of satisfying assignments than the expected number of satisfying assignments, because for those formulae that are satisfiable, they actually have many satisfying assignments (thus skewing the ....

.... it has been demonstrated that most formulae will have a much smaller number of satisfying assignments than the expected number of satisfying assignments, because for those formulae that are satisfiable, they actually have many satisfying assignments (thus skewing the average) In particular, Kamath et al. 1995) show that if a random k CNF formula is satisfiable, then with exponentially high probability (probability at least 1 Gamma 2 Gammaffln ) it has an exponential number of satisfying truth assignments. Let Z be the number of variables that are not mentioned at all when m clauses are chosen ....

[Article contains additional citation context not shown here]

Kamath, A.; Motwani, R.; Palem, K.; and Spirakis, P. 1995. Tail bounds for occupancy and the satisfiability threshold conjecture. Random Structures and Algorithms 7(1):59--80.

.... (1 Gamma 1 2 k ) 65] A recent result of Friedgut proves that the width of the phase transition in the random 3 Sat model narrows as problems increases in size [22] Kamath et al. show that at the random 3Sat transition, the distribution in the number of satisfying assignments is very skewed [58]. An exponentially small number of problems have an exponentially large number of satisfying assignments. This makes it hard to calculate the exact location of the transition. A technique borrowed from statistical mechanics called finite size scaling [1] models the shape of the satisfiability ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. Randomized Structure and Algorithms, 7:59--80, 1995.

....will see that is typically between about 0.5 and 1 at the phase transition. More refined estimates can be achieved either by taking account of the variance in the number of solutions at the phase boundary [57, 51] or by finding an equivalent problem with fewer solutions at its phase transition [32, 13]. There is a subtle difference between the prediction of a phase transition at 1 and at hSoli 1. While hSoli at the phase transition can grow exponentially with N , the value of tends to vary more slowly. For example, with random 3 Sat problems, the expected number of solutions at the ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. Randomized Structure and Algorithms, 7:59--80, 1995.

....The theoretical status of the threshold is in flux. For 2SAT, is it proven that there is a threshold at the ratio of 1 (Chvatal and Reed 1992; Goerdt 1992) For 3SAT, there is theoretical progress, but the problem is very challenging. Almost all instances are unsatisfiable when the ratio 4. 8 (Kamath et al. 1994; previously 5.2 by Franco and Paull 1983) Almost all instances are satisfiable when the ratio 3.003 (Freize and Suen 1993; previously 1.63 by Broder et al. 1993) But no one has yet been able to prove that a threshold even exists. We noted that ratio shifts slightly as the number of ....

Kamath, A., Motwani, R., Palem, K., and Spirakis, P., Tail bounds for occupancy and the satisfiability threshold conjecture. Proc. 35th Annual Symp. on Found. of Comp. Sci., (1994).

....sequence of events E n occurs almost surely if Pr(E n ) tends to one as n tends to infinity. For k = 2 a sharp threshold was proved in [9, 16] A random instance of 2 SAT is almost surely satisfiable if r 1 and almost surely unsatisfiable if r 1. For k = 3, there has been a series of results [6, 7, 5, 13, 19, 15, 21, 11] narrowing the area for which we do not have almost surely (un)satisfiability. The best bounds currently known come from [15] where it was proven that a random instance of 3 SAT is almost surely satisfiable if r 3:003, and [21] where it was proven that a random instance of 3 SAT is almost surely ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis, "Tail bounds for occupancy and the satisfiability threshold conjecture," Random Structures and Algorithms, Vol. 7 (1995) 59--80.

.... allocation problems are naturally encountered in areas like statistical inference [21, 7, 4] where they are implicitly used to to prove lower rates of convergence for regression or estimation problems) data structure analysis [9, 10, 11] or average case analysis of hard combinatorial problems [15, 20]. Analyzing random allocation problems consist in investigating the random occupancy patterns created by sending balls at random into N urns. An allocation is an infinite sequence of numbers between 1 and N . k denotes the index of the urn that receives the k th ball. This research was ....

A. Kamath, R. Motwani, R. Palem and P. Spirakis. Tail Bounds for Occupancy and the Satisfiability Threshold Conjecture. Proceedings of 35th FOCS, pp. 592-603, 1995.

....zerotemperature entropy of the satisfying assignments. This approximation ignores the fluctuations or higher order moments of the ensemble. Indeed, both numerical simulations and analytical results suggest that the transition occurs way before this annealed bound, at least for small values of K [10]. To gain better understanding of the SAT UNSAT transition we investigate the next three moments of the distribution of satisfying assignments and sketch the method for calculating other moments. The higher moments are then used to bound the number of formulae with very large number of satisfying ....

Kamath, Amil, Rajeev Motwani, Krishna Palem and Paul Spirakis, "Tail bounds for occupancy and the satisfiability thresholds conjecture", Proc. 35rd IEEE symposium on Foundation of Computer Science (1994), 592-- 603.

....substantial recent progress in narrowing the theoretical bounds on the transition. As N gets large the probability that an instance of Random 3 SAT is satisfiable approaches 0 whenever the ratio of clauses to variables is less than 3.003 [15] and approaches 1 when this ratio is greater than 4. 758 [23]. As can be seen, our empirical results are in agreement with, though much more fine grained than, the best theoretical bounds. To get some further insight into the search performed by DP, we now consider what happens when we let DP search the full space, i.e. we do not stop the procedure as soon ....

....strategy requires exponential time with probability approaching 1 on unsatisfiable random 3 SAT formulas, when the ratio of clauses to variables is held constant. Random 3 SAT formulas are unsatisfiable with probability approaching 1 whenever the ratio of clauses to variables is greater than 4. 758 [23]. So, given that DP corresponds to a particular resolution strategy as mentioned above, it follows that the average time complexity on such formulas is exponential. This result may appear inconsistent with our claim that over constrained formulas are easy, but it is not. Our results show that ....

[Article contains additional citation context not shown here]

Kamath, A.P., Motwani, A., Palem, R., and Spirakis, P., Tail Bounds for Occupancy and the Satisfiability Threshold Conjecture. Proc. FOCS-94, (1994).

....are a.s. un)satisfiable. In [10] it was observed that for r 5:191 the expected number of satisfying truth assignments tends to 0 as n tends to infinity and, hence, so does the probability of satisfiability . El Maftouhi and Fernandez de la Vega [8] lowered this to 5.08. Kamath et al. [14] lowered 5.08 to 4.758 by proving that if a random 3 SAT formula is satisfiable then, with extremely high probability, it has an exponential number of satisfying truth assignments, hence providing some account for the wastefulness of the first moment method. Finally, Kirousis et al. 16] ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis, Tail bounds for occupancy and the satisfiability threshold conjecture, Random Structures and Algorithms, 7 (1995), 59--80.

....to almost 0. The location of this transition region varies with k, but for each k it occurs at approximately the same ratio for all values of n [5,14,20] The asymptotic location of the transition region is analytically bounded above and below: For k = 3, the currently known bounds are c = 4:758 [18] and c = 3:003 [10] respectively. The lower graph of Figure 2 shows the median number of DP steps required to test the same formulas. Note the logarithmic y axis, necessitated by the dramatic increase of peak difficulty with increasing k. As expected, the peak at each k lines up with the ....

Anil Kamath, Rajeev Motwani, Krishna Palem, and Paul Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. In Proc. FOCS-94, 1994.

....in an instance of k SAT. For k = 2 a sharp threshold was proved in [9, 15] A random instance of 2 SAT is almost surely satisfiable if r 1 and almost surely unsatisfiable if r 1. For k = 3, such a sharp threshold behavior is widely conjectured to exist and there has been a series of papers [6, 7, 5, 12, 18, 14, 19] narrowing the area for which we do not have almost surely (un)satisfiability. The best bounds currently known come from [14] where it was proven that a random instance of 3 SAT is almost surely satisfiable if r 3:003, and [19] where it was proven that a random instance of 3 SAT is almost surely ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis, "Tail bounds for occupancy and the satisfiability threshold conjecture," Random Structures and Algorithms, Vol. 7 (1995) 59--80.

....is approximately Poisson with mean m n . We formalize this relationship by adapting an argument used by Gonnet [6] to determine the expected maximum number of balls in a bin. While useful tail bounds on the distributions of balls in bins can be found with other methods, most notably martingales [8, 9], our method appears to be more general, and in some cases easier to apply. Although tighter probability bounds for specific problems can often be obtained with more detailed analyses, as can be seen for example in [3] for our purposes this simple approach is quite effective. As mentioned in [4] ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. In Proceedings of the 24th IEEE Symposium on Foundations of Computer Science, pages 592--603, 1994.

....at random and the distribution of Poisson random variables. We note that the close relationship between these two models has been observed and made use of previously, and tighter bounds on specific problems can often be obtained with more detailed analyses; see, for example [3, Chapter 6] 6] or [13]. Apart from enabling us to prove our bounds, the tool may be of independent interest. In Section 4 we describe an asynchronous parallelization of greedy for two rounds that matches the lower bound to within a constant factor for any fixed d. We also describe a more complicated extension of greedy ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. In Proceedings of the 35th IEEE Symposium on Foundations of Computer Science, 1994.

....(0:0; 10:0] needs to be explained. It is empirically shown that 3SAT exhibits phase transition at the clause variable ratio around 4.3 [CA96] i.e. there is a sudden change from satisfiable to unsatisfiable when the ratio passes the transition point. Weaker theoretical bounds are shown in [FS96,KMPS94] For practical SAT algorithms, the hardest instances concentrate at the transition point [SML96] We believe that sampling the ratio from the interval (0:0; 10:0] will allow us to sample instances that are the hardest and practically the most interesting. We generated the formulas by first ....

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 592--604. IEEE Computer Society Press, 1994.

....will almost always find solutions to these instances in polynomial time (although polynomial average time does not necessarily follow) The upper bound is due to Kirousis, Kranakas and Krizanc [KKK96] which improves previous bounds of 4.64 by Dubois and Boufkhad [DB96] 4. 78 by Kamath et al. [KMPS94] and 5.19 reported by several authors. The easy 5.19 bound comes from observing that a fixed assignment t satisfies a random 3 literal clause with probability 7 8 and hence t satisfies a random instance of 3 SAT with probability ( 7 8 ) cn . The expected number of assignments satisfying ....

Anil Kamath, Rajeev Motwani, Krishna Palem, and Paul Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. In Proc. FOCS-94, 1994.

No context found.

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bound for occupancy and the satisfiability threshold conjecture. In Proc. of the 35th IEEE Annual Symposium on Foundations of Computer Science, 1994, pp. 592--603.

No context found.

A. Kamath, R. Motwani,K. Palem ,P. Spirakis. "Tail bound for occupancy and the satisfiability threshold conjecture ". In Proc. of the 35th IEEE Annual Symposium on Foundations of Computer Science, 1994, pp. 592-603.

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A. Kamath, R. Motwani,K. Palem ,P. Spirakis. "Tail bound for occupancy and the satisfiability threshold conjecture ". In Proc. of the 35th IEEE Annual Symposium on Foundations of Computer Science, 1994, pp. 592-603.

No context found.

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. Randomized Structure and Algorithms, 7:59--80, 1995.

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A. Kamath, R. Motwani, K.V. Palem, and P.G. Spirakis, "Tail bounds for occupancy and the satisfiability threshold conjecture," Random Structures and Algorithms, vol. 7, no. 1, pp. 59--80, 1995.

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A. P. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. Random Structures and Algorithms, 7:59--80, 1995.

No context found.

Anil Kamath, Rajeev Motwani, Krishna Palem, and Paul Spirakis, Tail bounds for occupancy and the satisfiability threshold conjecture, Random Structures Algorithms 7 (1995), no. 1, 59--80.

No context found.

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. Randomized Structure and Algorithms, 7:59--80, 1995.

No context found.

A. Kamath, R. Motwani, K.V. Palem, and P.G. Spirakis, "Tail bounds for occupancy and the satisfiability threshold conjecture," Random Structures and Algorithms, vol. 7, no. 1, pp. 59--80, 1995.

No context found.

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. In Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, pages 592--603, 1994.

No context found.

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. In Proceedings of the 35th IEEE Symposium on Foundations of Computer Science,1994.

No context found.

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. In Proceeding of IEEE Foundations of Computer Science, pages 592--603, 1994.

No context found.

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. In Proceeding of IEEE Foundations of Computer Science, pages 592--603, 1994.

No context found.

A. Kamath, R. Motwani, K. Palem, and P. Spirakis. Tail bounds for occupancy and the satisfiability threshold conjecture. In Proceeding of IEEE Foundations of Computer Science, pages 592--603, 1994.

No context found.

Stat. 11, 286--295. A. Kamath, R. Motwani, K. Palem and P. Spirakis (1994), Tail bounds for occupancy and the satisfiability threshold conjecture, Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science.

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