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17
Rounding algorithms for covering problems
, 1998
"... In the last 25 years approximation algorithms for discrete optimization problems have been in the center of research in the fields of mathematical programming and computer science. Recent results from computer science have identified barriers to the degree of approximability of discrete optimization ..."
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Cited by 17 (2 self)
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In the last 25 years approximation algorithms for discrete optimization problems have been in the center of research in the fields of mathematical programming and computer science. Recent results from computer science have identified barriers to the degree of approximability of discrete optimization problems unless P NP. As a result, as far as negative results are concerned a unifying picture is emerging. On the other hand, as far as particular approximation algorithms for different problems are concerned, the picture is not very clear. Different algorithms work for different problems and the insights gained from a successful analysis of a particular problem rarely transfer to another. Our goal in this paper is to present a framework for the approximation of a class of integer programming problems (covering problems) through generic heuristics all based on rounding (deterministic using primal and dual information or randomized but with nonlinear rounding functions) of the optimal solution of a linear programming (LP) relaxation. We apply these generic heuristics to obtain in a systematic way many known as well as new results for the set covering, facility location, general covering, network design and cut covering problems.
From Valid Inequalities to Heuristics: A Unified View of Primaldual Approximation Algorithms in Covering Problems
 OPERATIONS RESEARCH
, 1998
"... In recent years approximation algorithms based on primaldual methods have been successfully applied to a broad class of discrete optimization problems. In this paper, we propose a generic primaldual framework to design and analyze approximation algorithms for integer programming problems of the co ..."
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Cited by 16 (0 self)
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In recent years approximation algorithms based on primaldual methods have been successfully applied to a broad class of discrete optimization problems. In this paper, we propose a generic primaldual framework to design and analyze approximation algorithms for integer programming problems of the covering type that uses valid inequalities in its design. The worstcase bound of the proposed algorithm is related to a fundamental relationship (called strength) between the set of valid inequalities and the set of minimal solutions to the covering problems. In this way, we can construct an approximation algorithm simply by constructing the required valid inequalities. We apply the proposed algorithm to several problems, such as covering problems related to totally balanced matrices, cyclic scheduling, vertex cover, general set covering, intersections of polymatroids, and several network design problems attaining (in most cases) the best worstcase bound known in the literature.
Distributed and parallel algorithms for weighted vertex cover and other covering problems
 IN: PROC. 28TH PODC
, 2009
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Cyclical scheduling and multishift scheduling: complexity and approximation algorithms. Discrete Optimization
, 2006
"... We consider the multiple shift scheduling problem modelled as a covering problem. Such problems are characterized by a constraint matrix that has in every column blocks of consecutive ones, each corresponding to a shift. We focus on three type of shift scheduling problems classified according to the ..."
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Cited by 10 (1 self)
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We consider the multiple shift scheduling problem modelled as a covering problem. Such problems are characterized by a constraint matrix that has in every column blocks of consecutive ones, each corresponding to a shift. We focus on three type of shift scheduling problems classified according to the column structure in the constraint matrix: consecutive ones columns, cyclical ones columns and k consecutive blocks columns. In particular the complexity of the cyclical scheduling problem, where the matrix satisfies the cyclical 1’s property in each column was noted recently by Hochbaum and Tucker to be open. They further showed that the unit demand case is polynomially solvable. Here we extend this result to the uniform requirements case, and provide a 2approximation algorithm for the nonuniform case. We also establish that the cyclical scheduling problem’s complexity is equivalent to that of the exact matching problem – a problem the complexity status of which is known to be randomized polynomial, RP. We then investigate the three types of shift scheduling problems and show that while the consecutive ones version is polynomial and the kblock columns is NPhard (for k ≥ 2), For the kblocks problem we give a simple kapproximation algorithm, which is the first approximation algorithm determined for the problem. 1
Approximability of sparse integer programs
 In Proc. 17th ESA
, 2009
"... The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs {min cx: Ax ≥ b,0 ≤ x ≤ d} where A has at most k nonzeroes per row, we give a kapproximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥ 2 and ..."
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Cited by 9 (1 self)
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The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs {min cx: Ax ≥ b,0 ≤ x ≤ d} where A has at most k nonzeroes per row, we give a kapproximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥ 2 and ǫ> 0, if P = NP this ratio cannot be improved to k − 1 − ǫ, and under the unique games conjecture this ratio cannot be improved to k − ǫ. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsackcover inequalities. Second, for packing integer programs {max cx: Ax ≤ b,0 ≤ x ≤ d} where A has at most k nonzeroes per column, we give a 2 k k 2approximation algorithm. This is the first polynomialtime approximation algorithm for this problem with approximation ratio depending only on k, for any k> 1. Our approach starts from iterated LP relaxation, and then uses probabilistic and greedy methods to recover a feasible solution. Note added after publication: This version includes subsequent developments: a O(k 2) approximation for the latter problem using the iterated rounding framework, and several literature reference updates including a O(k)approximation for the same problem by Bansal et al.
Approximate Set Covering in Uniform Hypergraphs
"... The weighted set covering problem, restricted to the class of runiform hypergraphs, is considered. We propose a new approach, based on a recent result of Aharoni, Holzman and Krivelevich about the ratio of integer and fractional covering numbers in kcolorable runiform hypergraphs. This approach, ..."
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Cited by 8 (2 self)
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The weighted set covering problem, restricted to the class of runiform hypergraphs, is considered. We propose a new approach, based on a recent result of Aharoni, Holzman and Krivelevich about the ratio of integer and fractional covering numbers in kcolorable runiform hypergraphs. This approach, applied to hypergraphs of maximal degree bounded by , yields an algorithm with approximation ratio r(1 c= 1 r 1 ). Next, we combine this approach with an adaptation of the local ratio theorem of BarYehuda and Even for hypergraphs and present a general framework of approximation algorithms, based on subhypergraph exclusion. An application of this scheme is described, providing an algorithm with approximation ratio r(1 c=n r 1 r ) for hypergraphs on n vertices. We discuss also the limitations of this approach.
Greedy ΔApproximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost
 ALGORITHMICA
, 2012
"... This paper describes a simple greedy Δapproximation algorithm for any covering problem whose objective function is submodular and nondecreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most Δ var ..."
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Cited by 6 (1 self)
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This paper describes a simple greedy Δapproximation algorithm for any covering problem whose objective function is submodular and nondecreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most Δ variables of the problem. (A simple example is VERTEX COVER, with Δ = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems.
Distributed Algorithms for Covering, Packing and Maximum Weighted Matching
"... This paper gives polylogarithmicround, distributed δapproximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodularcost Covering). The approximation ratio δ is the maximum number of variables in any constraint. Special cases include Covering Mix ..."
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Cited by 4 (1 self)
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This paper gives polylogarithmicround, distributed δapproximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodularcost Covering). The approximation ratio δ is the maximum number of variables in any constraint. Special cases include Covering Mixed Integer Linear Programs (CMIP), and Weighted Vertex Cover (with δ = 2). Via duality, the paper also gives polylogarithmicround, distributed δapproximation algorithms for Fractional Packing linear programs (where δ is the maximum number of constraints in which any variable occurs), and for Max Weighted cMatching in hypergraphs (where δ is the maximum size of any of the hyperedges; for graphs δ = 2). The paper also gives parallel (RNC) 2approximation algorithms for CMIP with two variables per constraint and Weighted Vertex Cover. The algorithms are randomized. All of the approximation ratios exactly match those of comparable centralized algorithms.
On an Integer Multicommodity Flow Problem from the Airplane Industry
, 1997
"... Here we discuss a new integer multicommodity flow problem in which the commodities can not be shipped independently. The problem emerges in the routing of airplane parts from production sites to assembly sites. The parts are of such size that they have to be carried on dedicated trailers. Each type ..."
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Cited by 3 (0 self)
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Here we discuss a new integer multicommodity flow problem in which the commodities can not be shipped independently. The problem emerges in the routing of airplane parts from production sites to assembly sites. The parts are of such size that they have to be carried on dedicated trailers. Each type of part has its own type of trailer. A part is loaded on its trailer after it is produced, carried on its trailer to its assembly site, and then the trailer has to be recycled. The transport of the parts is done with huge specially built transportation aircrafts. For reasons of stability such aircrafts can only carry some prespecified combinations of parts on trailers and empty trailers. We consider the problem of finding a feasible transportation plan that minimises the total flying time of the transportation aircrafts. For this purpose we develop both optimisation and approximation algorithms. 1 Introduction In this paper we discuss a routing problem from the airplane industry, where the...
General Terms
"... The paper presents distributed and parallel δapproximation algorithms for covering problems, where δ is the maximum number of variables on which any constraint depends (for example, δ = 2 for vertex cover). Specific results include the following. • For weighted vertex cover, the first distributed 2 ..."
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The paper presents distributed and parallel δapproximation algorithms for covering problems, where δ is the maximum number of variables on which any constraint depends (for example, δ = 2 for vertex cover). Specific results include the following. • For weighted vertex cover, the first distributed 2approximation algorithm taking O(logn) rounds and the first parallel 2approximation algorithm in RNC. The algorithms generalize to covering mixed integer linear programs (CMIP) with two variables per constraint (δ = 2). • For any covering problem with monotone constraints and submodular cost, a distributed δapproximation algorithm taking O(log2 C) rounds, where C  is the number of constraints. (Special cases include CMIP, facility location, and probabilistic (twostage) variants of these problems.)