Let @math be a partially ordered set. We prove that if @math is sufficiently large, then there exists a packing @math of copies of @math in the Boolean lattice @math that covers almost every element of @math : @math might not cover the minimum and maximum of @math , and at most @math additional points due to divisibility. In particular, if @math divides @math , then the truncated Boolean lattice @math can be partitioned into copies of @math . This confirms a conjecture of Lonc from 1991.
Recently, problems in which the goal is to partition certain product structures, such as @math @cite_14 , the graph of the hypercube @cite_2 @cite_13 , the Boolean lattice @cite_0 @cite_3 , graph powers of cycles @cite_5 , into copies the definition of copy varies according to the structure we are interested in of some set gained increased interest due to a remarkable result of Gruslys, Leader and Tan @cite_14 . Let us briefly outline their main lemma. While this lemma proved to be powerful in partitioning problems, we were unable to adapt it for "almost partitioning" problems. That is, for problems in which there exists no desired partition for some obvious reason (for example divisibility is not satisfied), but we are still hoping to find a packing that covers almost every element. See @cite_13 @cite_3 for such results. Therefore, in this paper we develop our own packing method to deal with Theorem and Theorem , which exploits the flexibility of copies of posets, and we do not utilize the previously described lemma in any way.
@cite_14 @cite_3 @cite_0 @cite_2 @cite_5 @cite_13
2041869086 904137795 1995006434 2409645877
Graph partitioning requires the division of a graph's vertex set into k equally sized subsets s.t. some objective function is optimized. High-quality partitions are important for many applications, whose objective functions are often NP-hard to optimize. Most state-of-the-art graph partitioning libraries use a variant of the Kernighan-Lin (KL) heuristic within a multilevel framework. While these libraries are very fast, their solutions do not always meet all user requirements. Moreover, due to its sequential nature, KL is not easy to parallelize. Its use as a load balancer in parallel numerical applications therefore requires complicated adaptations. That is why we developed previously an inherently parallel algorithm, called Bubble-FOS C [H. Meyerhenke, B. Monien, S. Schamberger, Accelerating shape optimizing load balancing for parallel FEM simulations by algebraic multigrid, in: Proceedings of the 20th IEEE International Parallel and Distributed Processing Symposium, IPDPS'06, IEEE Computer Society, 2006, p. 57 (CD)], which optimizes partition shapes by a diffusive mechanism. However, it is too slow for practical use, despite its high solution quality. In this paper, besides proving that Bubble-FOS C converges towards a local optimum of a potential function, we develop a much faster method for the improvement of partitionings. This faster method called TruncCons is based on a different diffusive process, which is restricted to local areas of the graph and also contains a high degree of parallelism. By coupling TruncCons with Bubble-FOS C in a multilevel framework based on two different hierarchy construction methods, we obtain our new graph partitioning heuristic DibaP. Compared to Bubble-FOS C, DibaP shows a considerable acceleration, while retaining the positive properties of the slower algorithm. Experiments with popular benchmark graphs show that DibaP computes consistently better results than the state-of-the-art libraries METIS and JOSTLE. Moreover, with our new algorithm, we have improved the best known edge-cut values for a significant number of partitionings of six widely used benchmark graphs. A wide class of problems in combinatorics, computer science and physics can be described along the following lines. There are a large number of variables ranging over a finite domain that interact through constraints that each bind a few variables and either encourage or discourage certain value combinations. Examples include the k-SAT problem or the Ising model. Such models naturally induce a Gibbs measure on the set of assignments, which is characterised by its partition function. The present paper deals with the partition function of problems where the interactions between variables and constraints are induced by a sparse random (hyper)graph. According to physics predictions, a generic recipe called the “replica symmetric cavity method” yields the correct value of the partition function if the underlying model enjoys certain properties [, PNAS (2007) 10318–10323]. Guided by this conjecture, we prove general sufficient conditions for the success of the cavity method. The proofs are based on a “regularity lemma” for probability measures on sets of the form Ωn for a finite Ω and a large n that may be of independent interest. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016 In this paper we study the prevalent problem of graph partitioning by analyzing the diffusion-based partitioning heuristic Bubble-FOS C, a key component of a practical successful graph partitioner ( in J. Parallel Distrib. Comput. 69(9):750---761, 2009). We begin by studying the disturbed diffusion scheme FOS C, which computes the similarity measure used in Bubble-FOS C and is therefore the most crucial component. By relating FOS C to random walks, we obtain precise characterizations of the behavior of FOS C on tori and hypercubes. Besides leading to new knowledge on FOS C (and therefore also on Bubble-FOS C), these characterizations have been recently used for the analysis of load balancing algorithms ( in Proceedings of the 22nd Annual Symposium on Discrete Algorithms, pp. 429---439, 2011). We then regard Bubble-FOS C, which has been shown in previous experiments to produce solutions with good partition shapes and other favorable properties. In this paper we prove that it computes a relaxed solution to an edge cut minimizing binary quadratic program (BQP). This result provides the first substantial theoretical insight why Bubble-FOS C yields good experimental results in terms of graph partitioning metrics. Moreover, we show that in bisections computed by Bubble-FOS C, at least one of the two parts is connected. Using the aforementioned relation between FOS C and random walks, we prove that in vertex-transitive graphs both parts must be connected components. We show that the Graph Isomorphism (GI) problem and the more general problems of String Isomorphism (SI) andCoset Intersection (CI) can be solved in quasipolynomial(exp((logn)O(1))) time. The best previous bound for GI was exp(O( √n log n)), where n is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, exp(O (√ n)), where n is the size of the permutation domain (Babai, 1983). Following the approach of Luks’s seminal 1980 82 paper, the problem we actually address is SI. This problem takes two strings of length n and a permutation group G of degree n (the “ambient group”) as input (G is given by a list of generators) and asks whether or not one of the strings can be transformed into the other by some element of G. Luks’s divide-and-conquer algorithm for SI proceeds by recursion on the ambient group. We build on Luks’s framework and attack the obstructions to efficient Luks recurrence via an interplay between local and global symmetry. We construct group theoretic “local certificates” to certify the presence or absence of local symmetry, aggregate the negative certificates to canonical k-ary relations where k = O(log n), and employ combinatorial canonical partitioning techniques to split the k-ary relational structure for efficient divide-and- conquer. We show that in a well–defined sense, Johnson graphs are the only obstructions to effective canonical partitioning. The central element of the algorithm is the “local certificates” routine which is based on a new group theoretic result, the “Unaffected stabilizers lemma,” that allows us to construct global automorphisms out of local information.