THE DECISION PROBLEM FOR RESTRICTED UNIVERSAL QUANTIFICATION IN SET THEORY AND THE AXIOM OF FOUNDATION

The still unsettled decision problem for the restricted purely universal formulae ((?)₀-formulae) of the first order set-theoretic language based over =, ∈ is discussed in relation with the adoption or rejection of the axiom of foundation. Assuming the axiom of foundation, the related finite set-satisfiability problem for the very significant subclass of the (?)₀-formulae consisting of the formulae involving only nested variables of level 1 is proved to be semidecidable on the ground of a reflection property over the hereditarily finite sets, and various extensions of this result are obtained. When variables are restricted to range only over sets, in universes with infinitely many urelements the set-satisfiability problem is shown to be solvable provided the axiom of foundation is assumed; if it is not, then the decidability of a related derivability problem still holds. That, in turn, suggests the alternative adoption of an antifoundation axiom under which the set-satisfiability problem is also solvable (of course with different answers). Turning to set theory without urelements, assuming a form of Boffa's antifoundation axiom, the complement of the set-satisfiability problem for the full class of Δ₀-formulae is shown to be semidecidable; a result that is known not to hold, for the set-satisfiability problem itself, even for a very restricted subclass of the Δ₀-formulae.     

一类特殊二次分配问题的线性化求解新方法

许多抽象于实际的二次分配问题，其流矩阵与距离矩阵中有很多零元素，求解该类二次分配问题时，可通过先行利用零元素的信息减小问题规模，缩短计算时间．以二次分配问题的线性化模型为基础，提出了一种求解流矩阵与距离矩阵中同时存在大量零元素的二次分配问题新方法，不仅从理论上证明了方法的可行性，而且从实验的角度说明了该方法比以往方法更加优越．     

混合图上最小-最大圈覆盖问题的近似算法

考虑一个混合图上的最小-最大圈覆盖问题.给定一个正整数k和一个混合加权图G=(V E,A),这里V表示顶点集,E表示边集,A表示弧集.E中的每条边和A中的每条弧关联一个权重.问题的要求是确定k个环游,使得这k个环游能够经过A中的所有弧.目标是极小化最大环游的权重.该问题是运筹学和计算机科学中一个重要的组合优化问题,它和...     

Simultaneously lifting sets of binary variables into cover inequalities for knapsack polytopes

Cover inequalities are commonly used cutting planes for the 0–1 knapsack problem. This paper describes a linear-time algorithm (assuming the knapsack is sorted) to simultaneously lift a set of variables into a cover inequality. Conditions for this process to result in valid and facet-defining inequalities are presented. In many instances, the resulting simultaneously lifted cover inequality cannot be obtained by sequentially lifting over any cover inequality. Some computational results demonstrate that simultaneously lifted cover inequalities are plentiful, easy to find and can be computationally beneficial.     

The basis decision problem in λ-calculus

We show that the problem of deciding if a finite set of closed terms in normal form is a basis is recursively unsolvable. The restricted problem concerning one element sets is still recursively unsolvable. MSC: 03B40, 03D35.     

The property of being polynomial for Mal’tsev constraint satisfaction problems

A combinatorial constraint satisfaction problem aims at expressing in unified terms a wide spectrum of problems in various branches of mathematics, computer science, and AI. The generalized satisfiability problem is NP-complete, but many of its restricted versions can be solved in a polynomial time. It is known that the computational complexity of a restricted constraint satisfaction problem depends only on a set of polymorphisms of relations which are admitted to be used in the problem. For the case where a set of such relations is invariant under some Mal’tsev operation, we show that the corresponding constraint satisfaction problem can be solved in a polynomial time. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 655–686, November–December, 2006.     

A new family of proximity graphs: Class cover catch digraphs

Motivated by issues in machine learning and statistical pattern classification, we investigate a class cover problem (CCP) with an associated family of directed graphs—class cover catch digraphs (CCCDs). CCCDs are a special case of catch digraphs. Solving the underlying CCP is equivalent to finding a smallest cardinality dominating set for the associated CCCD, which in turn provides regularization for statistical pattern classification. Some relevant properties of CCCDs are studied and a characterization of a family of CCCDs is given.     

A goal-driven prototype column generation strategy for the multiple container loading cost minimization problem

In the multiple container loading cost minimization problem (MCLCMP), rectangular boxes of various dimensions are loaded into rectangular containers of various sizes so as to minimize the total shipping cost. The MCLCMP can be naturally modeled as a set cover problem. We generalize the set cover formulation by introducing a new parameter to model the gross volume utilization of containers in a solution. The state-of-the-art algorithm tackles the MCLCMP using the prototype column generation (PCG) technique. PCG is an effective technique for speeding up the column generation technique for extremely hard optimization problems where their corresponding pricing subproblems are NP-hard. We propose a new approach to the MCLCMP that combines the PCG technique with a goal-driven search. Our goal-driven prototype column generation (GD-PCG) algorithm improves the original PCG approach in three respects. Computational experiments suggest that all three enhancements are effective. Our GD-PCG algorithm produces significantly better solutions for the 350 existing benchmark instances than all other approaches in the literature using less computation time. We also generate two new set instances based on industrial data and the classical single container loading instances.     

From the restricted to the full three-body problem

The three-body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three-body problem. We use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three-body problem with one small mass is to the first approximation the product of the restricted three-body problem and a harmonic oscillator. This allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three-body problem.

For example, all the non-degenerate periodic solutions, generic bifurcations, Hamiltonian-Hopf bifurcations, bridges and natural centers known to exist in the restricted problem can be continued into the reduced three-body problem. The classic normalization calculations of Deprit and Deprit-Bartholomé show that there are two-dimensional KAM invariant tori near the Lagrange point in the restricted problem. With the above result this proves that there are three-dimensional KAM invariant tori near the Lagrange point in the reduced three-body problem.     

A branch-and-cut-and-price approach for the pickup and delivery problem with shuttle routes

The Pickup and Delivery Problem with Shuttle routes (PDPS) is a special case of the Pickup and Delivery Problem with Time Windows (PDPTW) where the trips between the pickup points and the delivery points can be decomposed into two legs. The first leg visits only pickup points and ends at some delivery point. The second leg is a direct trip – called a shuttle – between two delivery points. This optimization problem has practical applications in the transportation of people between a large set of pickup points and a restricted set of delivery points.     

A matheuristic based on Lagrangian relaxation for the multi-activity shift scheduling problem

The multi-activity shift scheduling problem involves assigning a sequence of activities to a set of employees. In this paper, we consider the variant where the employees have different qualifications and each activity must be performed in a specified time window; i.e., we specify the earliest start period and the latest finish period. We propose a matheuristic in which Lagrangian relaxation is used to identify a subset of promising shifts, and a restricted set covering problem is solved to find a feasible solution. Each shift is represented by a context-free grammar. Computational tests are carried out on two sets of instances from the literature. For the first set, the matheuristic finds a solution with an optimality gap less than 0.01% for 70% of the instances and improves the best-known solution for 16% of them; for the second set, the matheuristic reaches the best-known solutions for 55% of the instances and finds better solutions for 37.5% of them.     

Isolating blocks near the collinear relative equilibria of the three-body problem

The collinear relative equilibrium solutions are among the few explicitly known periodic solutions of the Newtonian three-body problem. When the energy and angular momentum constants are varied slightly, these unstable periodic orbits become normally hyperbolic invariant spheres whose stable and unstable manifolds form separatrices in the integral manifolds. The goal of this paper is to construct simple isolating blocks for these invariant spheres analogous to those introduced by Conley in the restricted three-body problem. This allows continuation of the invariant set and the separatrices to energies and angular momenta far from those of the relative equilibrium.

     

A primal-dual method for approximating tree cover with two weights

The tree cover (TC) problem is to compute a minimum weight connected edge set, given a connected and edge-weighted graph G, such that its vertex set forms a vertex cover for G. Unlike related problems of vertex cover or edge dominating set, weighted TC is not yet known to be approximable in polynomial time as well as the unweighted version is. Moreover, the best approximation algorithm known so far for weighted TC is far from practical in its efficiency. In this paper we consider a restricted version of weighted TC, as a first step towards better approximation of general TC, where only two edge weights differing by at least a factor of 2 are available. It will be shown that a factor 2 approximation can be attained efficiently (in the complexity of max flow) in this case by a primal-dual method. Even under the limited weights as such, the primal-dual arguments used will be seen to be quite involved, having a nontrivial style of dual assignments as an essential part, unlike the case of uniform weights.     

ON GENERALIZED BUFFON NEEDLE PROBLEM FOR LATTICES

Two new concepts, the generalized support function and restricted chord function, both referring to a convex set, were introduced in [1]. General formulae to yield the kinematic measure of a segment of fixed length in a convex set were established based on these concepts. In this article , using the partial intersection method, we consider the generalized Buffon problem for three kinds of lattices. We determine the probability of intersection of a body test needle of length l, l < a.     

求解非减上模集函数最小值问题的近似算法及其性能保证

上模集函数的优化问题在组合优化问题中有广泛应用,许多组合优化问题,如设备选址问题、p-中心问题等都可化为上模集函数的优化问题.本文给出了求解非减上模集函数最小值问题的一种近似算法,并讨论了所给算法的性能保证.     

On Nuclearly Nilpotent Loops of Finite Exponent

We show that the restricted Burnside problem has a positive answer for suitable classes of nuclearly nilpotent loops. Using this technique we give a positive answer to the restricted Burnside problem for Moufang A-loops.     

带有边集限制的最均匀支撑树问题

本在无向网络中，建立了带有边集限制的最均匀支撑树问题的网络模型．中首先解决最均匀支撑树问题，并给出求无向网络中最均匀支撑树的多项式时间算法；然后，给出了求无向网络中带有边集限制的最小树多项式时间算法；最后，在已解决的两个问题的基础上解决了带有边集限制的最均匀支撑树问题．     

A modified greedy algorithm for dispersively weighted 3-set cover

The set cover problem is that of computing a minimum weight subfamily F^′, given a family F of weighted subsets of a base set U, such that every element of U is covered by some subset in F^′. The k-set cover problem is a variant in which every subset is of size at most k. It has been long known that the problem can be approximated within a factor of by the greedy heuristic, but no better bound has been shown except for the case of unweighted subsets. In this paper we consider approximation of a restricted version of the weighted 3-set cover problem, as a first step towards better approximation of general k-set cover problem, where any two distinct subset costs differ by a multiplicative factor of at least 2. It will be shown, via LP duality, that an improved approximation bound of H(3)-1/6 can be attained, when the greedy heuristic is suitably modified for this case. A key to our algorithm design and analysis is the Gallai-Edmonds structure theorem for maximum matchings.     

SUPERFLUOUS AND S-ESSENTIAL HOMOMORPHISMS

Let R be a ring and S a class of R-modules. S-superfluous epimorphisms and S-essential monomorphisms are introduced and studied in this article. As applications, some new characterizations of von Neumann regular rings and perfect rings are given. Finally, these notions are also used to study minimal homomorphisms.     

A prototype column generation strategy for the multiple container loading problem

The multiple container loading cost minimization problem (MCLCMP) is a practical and useful problem in the transportation industry, where products of various dimensions are to be loaded into containers of various sizes so as to minimize the total shipping cost. The MCLCMP can be naturally formulated as a set cover problem and solved using column generation techniques, which is a popular method for handling huge numbers of variables. However, the direct application of column generation is not effective because feasible solutions to the pricing subproblem is required, which for the MCLCMP is NP-hard. We show that efficiency can be greatly improved by generating prototypes that approximate feasible solutions to the pricing problem rather than actual columns. For many hard combinatorial problems, the subproblem in column generation based algorithms is NP-hard; if suitable prototypes can be quickly generated that approximate feasible solutions, then our strategy can also be applied to speed up these algorithms.