In this paper, we explore the correlation between the quality of initial assignments provided to local search heuristics and that of the corresponding final assignments. We restrict our attention to the Max r-Sat problem and to one of the leading local search heuristics - Configuration Checking Local Search (CCLS). We use a tailored version of the Method of Conditional Expectations (MOCE) to generate initial assignments of diverse quality. We show that the correlation in question is significant and long-lasting. Namely, even when we delve deeper into the local search, we are still in the shadow of the initial assignment. Thus, under practical time constraints, the quality of the initial assignment is crucial to the performance of local search heuristics. To demonstrate our point, we improve CCLS by combining it with MOCE. Instead of starting CCLS from random initial assignments, we start it from excellent initial assignments, provided by MOCE. Indeed, it turns out that this kind of initialization provides a significant improvement of this state-of-the-art solver. This improvement becomes more and more significant as the instance grows.

We introduce the AdaptiveClimb cache policy, a new policy for cache management. This policy improves LRU to gain higher performance in a fixed probability scenario, without maintaining statistics for each item. Rather, it stores a single value (the current jump), while preserving the fast adaptation of probability changes of LRU. AdaptiveClimb is a modification of CLIMB cache policy, that unlike CLIMB changes the number of positions shifts according to whether the last request has been a hit or a miss. Performance of CLIMB is close to that of the optimal off-line algorithm, but its stabilization time is long. LRU is much more sensitive to changes, but it is sensitive to noise. Thus, AdaptiveClimb combines these two advantages in a single algorithm with both good performance and short stabilization time.

For both historical and practical reasons, the Boolean satisfiability problem (SAT) has become one of central importance in computer science. One type of instances arises when the clauses are chosen uniformly randomly \textendash{} random SAT. Here, a major problem, recently solved for sufficiently large clause length, is the satisfiability threshold conjecture. The value of this threshold is known exactly only for clause length $2$, and there has been a lot of research concerning its value for arbitrary fixed clause length. In this paper, we endeavor to study the satisfiability threshold for random industrial SAT. There is as yet no generally accepted model of industrial SAT, and we confine ourselves to one of the more common features of industrial SAT: the set of variables consists of a number of disjoint communities, and clauses tend to consist of variables from the same community. Our main result is that the threshold of random community-structured SAT tends to be smaller than its counterpart for random SAT. Moreover, under some conditions, this threshold even vanishes.

We analyze the effect of network topology on the pattern stability of the Hopfield neural network in the case of general graphs. The patterns are randomly selected from a uniform distribution. We start the Hopfield procedure from some pattern v. An error in an entry e of v is the situation where, if the procedure is started at e, the value of e flips. Such an entry is an instability point. Note that we disregard the value at e by the end of the procedure, as well as what happens if we start the procedure from another pattern v' or another entry e' of v. We measure the instability of the system by the expected total number of instability points of all the patterns. Our main result is that the instability of the system does not depend on the exact topology of the underlying graph, but rather only on its degree sequence. Moreover, for a large number of nodes, the instability can be approximated by mΣni=1(1 − Φ(√δi/m−1)), where is the standard normal distribution function and δ1, . . . , δn are the degrees of the nodes.

Pinsker's widely used inequality upper-bounds the total variation distance ∥P - Q∥1in terms of the Kullback-Leibler divergence D(P∥Q). Although, in general, a bound in the reverse direction is impossible, in many applications the quantity of interest is actually D*(v, Q)-defined, for an arbitrary fixed Q, as the infimum of D(P∥Q) over all distributions P that are at least v-far away from Q in total variation. We show that D*(v, Q) ≤ Cv2+ O(v3), where C = C(Q) = 1/2 for balanced distributions, thereby providing a kind of reverse Pinsker inequality. Some of the structural results obtained in the course of the proof may be of independent interest. An application to large deviations is given.

Finding the convex hull of n points in the plane requires O ( n log n ) time in general. In Devroye and Toussaint [1993] and Golin et al. [2002] the problem of computing the convex hull of the intersection points of n lines was considered, where the lines are chosen randomly according to two various models. In both models, linear-time algorithms were developed. Here we improve the results of Devroye and Toussaint [1993] by giving a universal algorithm for a wider range of distributions.

The design and analysis of approximation algorithms for NP -hard problems is perhaps the most active research area in the theory of combinatorial algorithms. In this article, we study the notion of a combinatorial dominance guarantee as a way for assessing the performance of a given approximation algorithm. An f ( n ) dominance bound is a guarantee that the heuristic always returns a solution not worse than at least f ( n ) solutions. We give tight analysis of many heuristics, and establish novel and interesting dominance guarantees even for certain inapproximable problems and heuristic search algorithms. For example, we show that the maximal matching heuristic of VERTEX COVER offers a combinatorial dominance guarantee of 2 n − (1.839 + o (1)) n . We also give inapproximability results for most of the problems we discuss.

Variants of the classical Tower of Hanoi problem evolved in various directions. Allowing more than 3 pegs, and imposing limitations on the possible moves among the pegs, are two of these. Here, we deal with the case of h ≥3 pegs arranged on a circle, where moves are allowed only from a peg to the next peg (in the clockwise direction). Unlike the multi-peg problem without restrictions on moves between pegs, the complexity of this variant as a function of the number of disks is exponential. We find explicit lower and upper bounds for its complexity for any h , and show how this complexity can be estimated arbitrarily well for any specific h .