Multicover Problem Research Articles

Approximating the online set multicover problems via randomized winnowing

In this paper, we consider the weighted online set k -multicover problem. In this problem, we have a universe V of elements, a family S of subsets of V with a positive real cost for every S ∈ S , and a “coverage factor” (positive integer) k . A subset { i 0 , i 1 , … } ⊆ V of elements are presented online in an arbitrary order. When each element i p is presented, we are also told the collection of all (at least k ) sets S i p ⊆ S and their costs to which i p belongs and we need to select additional sets from S i p if necessary such that our collection of selected sets contains at least k sets that contain the element i p . The goal is to minimize the total cost of the selected sets. 1 1 Our algorithm and competitive ratio bounds can be extended to the case when a set can be selected at most a prespecified number of times instead of just once; we do not report these extensions for simplicity and also because they have no relevance to the biological applications that motivated our work. In this paper, we describe a new randomized algorithm for the online multicover problem based on a randomized version of the winnowing approach of [N. Littlestone, Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm, Machine Learning 2 (1988) 285–318]. This algorithm generalizes and improves some earlier results in [N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, J. Naor, A general approach to online network optimization problems, in: Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms, 2004, pp. 570–579; N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, J. Naor, The online set cover problem, in: Proceedings of the 35th Annual ACM Symposium on the Theory of Computing, 2003, pp. 100–105]. We also discuss lower bounds on competitive ratios for deterministic algorithms for general k based on the approaches in [N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, J. Naor, The online set cover problem, in: Proceedings of the 35th Annual ACM Symposium on the Theory of Computing, 2003, pp. 100–105].

Randomized approximation algorithms for set multicover problems with applications to reverse engineering of protein and gene networks

In this paper we investigate the computational complexity of a combinatorial problem that arises in the reverse engineering of protein and gene networks. Our contributions are as follows: • We abstract a combinatorial version of the problem and observe that this is “equivalent” to the set multicover problem when the “coverage” factor k is a function of the number of elements n of the universe. An important special case for our application is the case in which k = n - 1 . • We observe that the standard greedy algorithm produces an approximation ratio of Ω ( log n ) even if k is “large” i.e k = n - c for some constant c > 0 . • Let 1 < a < n denote the maximum number of elements in any given set in our set multicover problem. Then, we show that a non-trivial analysis of a simple randomized polynomial-time approximation algorithm for this problem yields an expected approximation ratio E [ r ( a , k ) ] that is an increasing function of a / k . The behavior of E [ r ( a , k ) ] is roughly as follows: it is about ln ( a / k ) when a / k is at least about e 2 ≈ 7.39 , and for smaller values of a / k it decreases towards 1 as a linear function of a / k with lim a / k → 0 E [ r ( a , k ) ] = 1 . Our randomized algorithm is a cascade of a deterministic and a randomized rounding step parameterized by a quantity β followed by a greedy solution for the remaining problem. We also comment about the impossibility of a significantly faster convergence of E [ r ( a , k ) ] towards 1 for any polynomial-time approximation algorithm.

El problema del multicubrimiento: una aplicación para la selección de paradas en la red de transporte de la Ciudad de México

En este artículo se desarrolla un algoritmo heurístico y su correspondiente implementación para resolver un problema de muestreo en la red de rutas de transporte urbano de la Ciudad de México. El problema consiste en la selección de al menos 2 puntos (paradas de la ruta) en cada una de las 236 rutas en el estudio con un total de 8390 paradas. El problema anterior, se plantea como un problema de multicubrimiento (multicover problem) con 236 restricciones y 8390 variables binarias. Este problema es un problema NP-duro, por lo que se implementó un algoritmo heurístico para obtener los puntos de muestreo.

Randomized approximation of bounded multicovering problems

The problem of finding approximate solutions for a subclass of multicovering problems denoted byILP(k, b) is considered. The problem involves findingxź{0,1}n that minimizes źjxj subject to the constraintAxźb, whereA is a 0---1m×n matrix with at mostk ones per row,b is an integer vector, andb is the smallest entry inb. This subclass includes, for example, the Bounded Set Cover problem whenb=1, and the Vertex Cover problem whenk=2 andb=1. An approximation ratio ofkźb+1 is achievable by known deterministic algorithms. A new randomized approximation algorithm is presented, with an approximation ratio of (kźb+1)(1ź(c/m)1/(kźb+1)) for a small constantc>0. The analysis of this algorithm relies on the use of a new bound on the sum of independent Bernoulli random variables, that is of interest in its own right.

The multicovering problem

The multicovering problem is: Min CX subject to AX ⩾ b, X ϵ {0,}, where A is a matrix whose elements are all zero or one and b is a vector of positive integers: this problem has many applications in scheduling, location and business areas. A variety of industrial, commercial and military contexts give rise to large problems of this kind, frequently at or beyond the limits of current computational tractability. Our research goal is the development of a special-purpose algorithm capable of solving large problems of this kind. Conceptual and computational inputs to the algorithm arise from three sources: from the generalization of computational tools developed for more specific problems (in particular the set covering problem), from the adaptation of general integer programming and combinatorial optimization methods, and from the development within this paper of special techniques designed to take advantage of the structure of the problem studied. Following a description of the algorithm we give some computational results for randomly generated problems.

Heuristic solutions and confidence intervals for the multicovering problem

The multicovering problem can be expressed as: Minimize CX subject to AX ⩾, b, X ϵ {0, 1}, where A is a zero-one matrix and b is a vector of positive integers. This mathematical model has many applications to scheduling and location problems. Large examples of such problems arise in industrial, commercial and military settings, and their size frequently exceeds the limits of computational tractability. For this important problem, we examine a variety of simple heuristic approaches which can be applied when optimal solutions are not available. The approximate solutions thus generated are used to construct confidence intervals for the unknown value of the optimal solution. A large-scale computational study for randomly generated problmes suggests that these intervals are both very narrow and very likely to contain the optimal solution value. A study of 10 very large real-world problems further supports the success of our methodology and the quality of the approximate solutions found.

A fast approximation algorithm for the multicovering problem

The multicovering problem is: MIN cx subject to Ax≥ b, xϵ{0, 1} n , where A is a matrix whose elements are all zero or one and b is a vector of positive integers. We present a fast heuristic for this important class of problems and analyze its worst-case performance: the ratio of the heuristic value to the optimum does not exceed the maximum row sum of the matrix A. The heuristic algorithm also provides a feasible dual solution vector that is useful in generating lower bounds on the value of the optimum.