The Conditional Covering Problem on Unweighted Interval Graphs with Nonuniform Coverage Radius

Let G = (V, E) be an interval graph with n vertices and m edges. A positive integer R(x) is associated with every vertex x ? V{x\in V}. In the conditional covering problem, a vertex x ? V{x \in V} covers a vertex y ? V{y \in V} (x ≠ y) if d(x, y) ≤ R(x) where d(x, y) is the shortest distance between the vertices x and y. The conditional covering problem (CCP) finds a minimum cardinality vertex set C í V{C\subseteq V} so as to cover all the vertices of the graph and every vertex in C is also covered by another vertex of C. This problem is NP-complete for general graphs. In this paper, we propose an efficient algorithm to solve the CCP with nonuniform coverage radius in O(n ²) time, when G is an interval graph containing n vertices.     

A Covering Problem for Tori

The problem of covering an n-dimensional torus with n-dimensional grid graphs is studied. This is the dual problem of a packing problem concerning the capacity of a graph, which has been studied in information theory. It is related to several other problems as well, including weighted coverings, Kellers cube-tiling problem, and the recreational problem of how to obtain zero correct predictions in the football pools. Motivated by the last problem, bounds on the minimum size of such coverings are tabulated for q = 3, p = 2, and small n.AMS Subject Classification: 05C70, 94B25.     

带核箱覆盖问题的在线算法

经典的箱覆盖问题是组合优化中一个著名的问题,并且得到了广泛的研究．本文主要讨论带核元的箱覆盖问题的复杂性和在线条件下的算法．指出了带核的箱覆盖问题是强ＮＰ－ｈａｒｄ的．给出了在不同的在线条件下可行算法渐近比的上界,指出仅在条件三下才存在渐近比好于０的在线算法,并给出了在此条件下一个渐近比为１／２的最好的在线算法。     

Algorithms for the Set Covering Problem

The Set Covering Problem (SCP) is a main model for several important applications, including crew scheduling in railway and mass-transit companies. In this survey, we focus our attention on the most recent and effective algorithms for SCP, considering both heuristic and exact approaches, outlining their main characteristics and presenting an experimental comparison on the test-bed instances of Beasley's OR Library.     

On Covering Groups

We study finite-sheeted covering mappings from compact connected spaces onto compact connected groups. For such a covering mapping we consider a method of supplying its covering space by a group structure. The covering space endowed with that group structure is a topological group, and a given covering mapping turns into a homomorphism of compact connected groups.

     

箱覆盖问题的半定松驰算法

箱覆盖问题是NP困难问题中的经典问题，得到了广泛地研究，九十年代以来，半定松驰策略被用来求解组合优化问题，取得了很好的结果[13]，本文首次给箱覆盖问题的半定松驰算法，算法的理论分析结果表明它适合于求解大规模的箱覆盖问题。     

The Minimum Covering l pb -Hypersphere Problem

The minimu vering hypersphere problem is defined as to find a hypersphere of minimum radius enclosing a finite set of given points in ⁿ. A hypersphere is a set S(c,r)={x ⁿ : d(x,c) r}, where c is the center of S, r is the radius of S and d(x,c) is the Euclidean distance between x and c, i.e.,d(x,c)=l ₂ (x-c). We consider the extension of this problem when d(x, c) is given by any l _pb -norm, where 1<p and b=(b ₁,...,b _n ) with b _j >0, j=1,...,n, then S(c,r) is called an l _pb -hypersphere, in particular for p=2 and b _j =1, j=1,..., n, we obtain the l ₂-norm. We study some properties and propose some primal and dual algorithms for the extended problem , which are based on the feasible directions method and on the Wolfe duality theory. By computational experiments, we compare the proposed algorithms and show that they can be used to approximate the smallest l _pb -hypersphere enclosing a large set of points in ⁿ.     

基于改进遗传算法的集合覆盖问题

集合覆盖问题是组合优化中的典型问题,在日常生活中有着广泛的应用.提出了一种改进遗传算法来解决集合覆盖问题.算法对标准遗传算法的改进主要表现在:1)结合启发式算法和随机生成,设计了新的产生初始种群的方法;2)引入修补操作处理不可行解使其转换成可行解;3)对重复个体进行处理再利用;4)对多点交叉进行推广,提出了新的交叉算子;5)针对可行解和不可行解,采取两种自适应多位变异操作.数值实验结果表明该算法对于解决规模较大的集合覆盖问题是有效的.     

A Note on the Set Covering Problem

A previously published work on a ‘probabilistic’ formulation of the set covering problem is discussed. Attention is drawn to the dependence of the feasible locations of facilities on the way the continuous space of incidents is divided into subregions when using either the above formulation or the standard deterministic formulation of the set covering problem.     

Existence Problem of a Generalized Universal Covering Space

In this paper we study the existence problem of a generalized universal covering space of a given digital space, which can be essentially used in classifying digital spaces. To be specific, we show a method to establish a generalized universal covering space of a simple closed k-curve and prove that a digital wedge need not have a generalized universal covering space.     

Solution techniques for the Large Set Covering Problem

Given a finite set E and a family F={E₁,…,E_m} of subsets of E such that F covers E, the famous unicost set covering problem (USCP) is to determine the smallest possible subset of F that also covers E. We study in this paper a variant, called the Large Set Covering Problem (LSCP), which differs from the USCP in that E and the subsets E_i are not given in extension because they are very large sets that are possibly infinite. We propose three exact algorithms for solving the LSCP. Two of them determine minimal covers, while the third one produces minimum covers. Heuristic versions of these algorithms are also proposed and analysed. We then give several procedures for the computation of a lower bound on the minimum size of a cover. We finally present algorithms for finding the largest possible subset of F that does not cover E. We also show that a particular case of the LSCP is to determine irreducible infeasible sets in inconsistent constraint satisfaction problems. All concepts presented in the paper are illustrated on the k-colouring problem which is formulated as a constraint satisfaction problem.     

The Robust Set Covering Problem with interval data

We study the Set Covering Problem with uncertain costs. For each cost coefficient, only an interval estimate is known, and it is assumed that each coefficient can take on any value from the corresponding uncertainty interval, regardless of the values taken by other coefficients. It is required to find a robust deviation (also called minmax regret) solution. For this strongly NP-hard problem, we present and compare computationally three exact algorithms, where two of them are based on Benders decomposition and one uses Benders cuts in the context of a Branch-and-Cut approach, and several heuristic methods, including a scenario-based heuristic, a Genetic Algorithm, and a Hybrid Algorithm that uses a version of Benders decomposition within a Genetic Algorithm framework.     

A Genetic Algorithm for the Set Covering Problem

In this paper, the set covering problem (SCP) is considered. Several algorithms have been suggested in the literature for solving it. We propose a new algorithm for solving the SCP which is based on the genetic technique. This algorithm has been implemented and tested on various standard and randomly generated test problems. Preliminary results are encouraging, and are better than the existing heuristics for the problem.     

Maximally Disjoint Solutions of the Set Covering Problem

This paper is concerned with finding two solutions of a set covering problem that have a minimum number of variables in common. We show that this problem is NP-complete, even in the case where we are only interested in completely disjoint solutions. We describe three heuristic methods based on the standard greedy algorithm for set covering problems. Two of these algorithms find the solutions sequentially, while the third finds them simultaneously. A local search method for reducing the overlap of the two given solutions is then described. This method involves the solution of a reduced set covering problem. Finally, extensive computational tests are given demonstrating the nature of these algorithms. These tests are carried out both on randomly generated problems and on problems found in the literature.     

源项反演问题的条件稳定性

对于一维扩散方程的源项反演问题,探讨了反问题数据的相容性并应用积分恒等式方法建立了非线性源项反演的一种稳定性.     

关于条件信任函数

本文首先基于理论和实践两方面的考虑,给出了建立条件信任函数公式的准则;然后依据给定的准则.讨论了已有的几类条件信任函数公式的优点和弊病;最后我们给出了一种新的条件信任函数公式,其具备所期望的所有条件.同时为验证条件信任函数的性质的需要,我们还首次建立了几类有限集上信任函数的实例.     

A Branch-and-price Algorithm for the Multi-Vehicle Covering Tour Problem

In this paper, we propose a Branch-and-price (BP) algorithm and a Column Generation Heuristic (CGH) for the Multi-Vehicle Covering Tour Problem (m-CTP). Specific dominance and extension pruning rules are introduced to accelerate the resolution of the pricing problems. To the best of our knowledge, this is the first work dedicated to the exact resolution of m-CTP. The algorithm managed to solve about 30% of the instances in our test bed, within a 4 hour CPU time limit. Our preliminary computational experiments suggest that both the lower bounds provided by the formulation behind BP and the CGH upper bounds are of good quality.     

Frobenius Problem and the Covering Radius of a Lattice

Let and let be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as are non-negative integers. The condition that implies that such a number exists. The general problem of determining the Frobenius number given N and is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.