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23
FlowBased Propagators for the SEQUENCE and Related Global Constraints
, 2008
"... We propose new filtering algorithms for the SEQUENCE constraint and some extensions of the SEQUENCE constraint based on network flows. We enforce domain consistency on the SEQUENCE constraint in O(n 2) time down a branch of the search tree. This improves upon the best existing domain consistency al ..."
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Cited by 16 (3 self)
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We propose new filtering algorithms for the SEQUENCE constraint and some extensions of the SEQUENCE constraint based on network flows. We enforce domain consistency on the SEQUENCE constraint in O(n 2) time down a branch of the search tree. This improves upon the best existing domain consistency algorithm by a factor of O(log n). The flows used in these algorithms are derived from a linear program. Some of them differ from the flows used to propagate global constraints like GCC since the domains of the variables are encoded as costs on the edges rather than capacities. Such flows are efficient for maintaining bounds consistency over large domains and may be useful for other global constraints.
MINIMUM CONVEX COST DYNAMIC NETWORK FLOWS
, 1984
"... This paper presents and solves in polynomial time the minimum convex cost dynamic network flow problem, an infinite horizon integer programming problem which involves network flows evolving over time. The model is a finite network in which each arc has an associated transit time for flow to pass thr ..."
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Cited by 14 (5 self)
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This paper presents and solves in polynomial time the minimum convex cost dynamic network flow problem, an infinite horizon integer programming problem which involves network flows evolving over time. The model is a finite network in which each arc has an associated transit time for flow to pass through it. An integral amount of flow is to be sent through arcs of the network in each period over an infinite horizon so as to satisfy conservation of flow from some fixed period on. Furthermore, the net amount of flow "in transit" is assumed fixed over the infinite horizon. The objective is to minimize the average convex cost per period of sending flow. This problem is a generalization of the minimum convex cost network flow problem, the maximum throughput dynamic network flow problem, and the minimum costtotime ratio circuit problem. Furthermore, the model has applications in periodic production and transshipment, airplane scheduling, cyclic capacity scheduling, and cyclic staffing. To solve the dynamic network flow problem, one first obtains an optimal continuousvalued flow which repeats every period. The fractional variables of this solution may be rounded in such a way as to obtain an optimal integral flow which repeats every q periods, where q is the least common denominator of the fractional parts of the continuous flow. Furthermore, this integral flow is also an optimal continuous flow.
Approximation and FixedParameter Algorithms for Consecutive Ones Submatrix Problems
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
"... We develop an algorithmically useful refinement of a forbidden submatrix characterization of 0/1matrices fulfilling the Consecutive Ones Property (C1P). This characterization finds applications in new polynomialtime approximation algorithms and fixedparameter tractability results for the NPhard ..."
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Cited by 13 (0 self)
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We develop an algorithmically useful refinement of a forbidden submatrix characterization of 0/1matrices fulfilling the Consecutive Ones Property (C1P). This characterization finds applications in new polynomialtime approximation algorithms and fixedparameter tractability results for the NPhard problem to delete a minimum number of rows or columns from a 0/1matrix such that the remaining submatrix has the C1P.
Set covering with almost consecutive ones property
 OF DISCRETE ALGORITHMS
, 2003
"... In this paper we consider set covering problems with a coefficient matrix almost having the consecutive ones property, i.e., in many rows of the coefficient matrix, the ones appear consecutively. If this property holds for all rows it is well known that the set covering problem can be solved efficie ..."
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Cited by 12 (0 self)
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In this paper we consider set covering problems with a coefficient matrix almost having the consecutive ones property, i.e., in many rows of the coefficient matrix, the ones appear consecutively. If this property holds for all rows it is well known that the set covering problem can be solved efficiently. For our case of almost consecutive ones we present a reformulation exploiting the consecutive ones structure to develop bounds and a branching scheme. Our approach has been tested on realworld data as well as on theoretical problem instances.
Algorithmic Aspects of the ConsecutiveOnes Property
, 2009
"... We survey the consecutiveones property of binary matrices. Herein, a binary matrix has the consecutiveones property (C1P) if there is a permutation of its columns that places the 1s consecutively in every row. We provide an overview over connections to graph theory, characterizations, recognition ..."
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Cited by 11 (1 self)
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We survey the consecutiveones property of binary matrices. Herein, a binary matrix has the consecutiveones property (C1P) if there is a permutation of its columns that places the 1s consecutively in every row. We provide an overview over connections to graph theory, characterizations, recognition algorithms, and applications such as integer linear programming and solving Set Cover.
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
Exact algorithms and applications for Treelike Weighted Set Cover
 JOURNAL OF DISCRETE ALGORITHMS
, 2006
"... We introduce an NPcomplete special case of the Weighted Set Cover problem and show its fixedparameter tractability with respect to the maximum subset size, a parameter that appears to be small in relevant applications. More precisely, in this practically relevant variant we require that the given ..."
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Cited by 10 (5 self)
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We introduce an NPcomplete special case of the Weighted Set Cover problem and show its fixedparameter tractability with respect to the maximum subset size, a parameter that appears to be small in relevant applications. More precisely, in this practically relevant variant we require that the given collection C of subsets of a some base set S should be “treelike.” That is, the subsets in C can be organized in a tree T such that every subset onetoone corresponds to a tree node and, for each element s of S, the nodes corresponding to the subsets containing s induce a subtree of T. This is equivalent to the problem of finding a minimum edge cover in an edgeweighted acyclic hypergraph. Our main result is an algorithm running in O(3 k ·mn) time where k denotes the maximum subset size, n: = S, and m: = C. The algorithm also implies a fixedparameter tractability result for the NPcomplete Multicut in Trees problem, complementing previous approximation results. Our results find applications in computational biology in phylogenomics and for saving memory in tree decomposition based graph algorithms.
Tree decompositions of graphs: Saving memory in dynamic programming
 CTW 2004: CologneTwente Workshop on Graphs and Combinatorial Optimization, Villa Vigoni (CO
, 2004
"... We propose a simple and effective heuristic to save memory in dynamic programming on tree decompositions when solving graph optimization problems. The introduced “anchor technique ” is based on a treelike set covering problem. We substantiate our findings by experimental results. Our strategy has n ..."
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Cited by 9 (2 self)
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We propose a simple and effective heuristic to save memory in dynamic programming on tree decompositions when solving graph optimization problems. The introduced “anchor technique ” is based on a treelike set covering problem. We substantiate our findings by experimental results. Our strategy has negligible computational overhead concerning running time but achieves memory savings for nice tree decompositions and path decompositions between 60 % and 98%.
Complexity and exact algorithms for multicut
 In: SOFSEM
"... Abstract. The Multicut problem is defined as: given an undirected graph and a collection of pairs of terminal vertices, find a minimum set of edges or vertices whose removal disconnects each pair. We mainly focus on the case of removing vertices, where we distinguish between allowing or disallowing ..."
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Cited by 8 (0 self)
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Abstract. The Multicut problem is defined as: given an undirected graph and a collection of pairs of terminal vertices, find a minimum set of edges or vertices whose removal disconnects each pair. We mainly focus on the case of removing vertices, where we distinguish between allowing or disallowing the removal of terminal vertices. Complementing and refining previous results from the literature, we provide several NPcompleteness and (fixedparameter) tractability results for restricted classes of graphs such as trees, interval graphs, and graphs of bounded treewidth. 1
Optimizing over consecutive 1’s and circular 1’s constraints
, 2005
"... We consider packing and covering optimization problems over constraints in consecutive and circular 1’s. Such problems arise in the context of shift scheduling, and in problems related to interval graphs. Previous approaches to this problem depended on solving several minimum cost network flow probl ..."
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Cited by 4 (0 self)
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We consider packing and covering optimization problems over constraints in consecutive and circular 1’s. Such problems arise in the context of shift scheduling, and in problems related to interval graphs. Previous approaches to this problem depended on solving several minimum cost network flow problems. We devise here substantially more efficient and strongly polynomial algorithms based on parametric shortest paths approaches. The objective function in the covering and packing problems is to either minimize or maximize the number of sets that satisfy the constraints. The various problems studied are classified according to whether the constraints are all consecutive 1’s or if there are also circular 1’s constraints, and according to whether the constraints are all of covering type; all of packing type, or mixed. The running time of our algorithm for a pure covering all consecutive 1’s constraints problem on n variables and m constraints is O(m + n). For the pure packing problem with consecutive 1’s constraints we present an O(m + n log n) time algorithm. For the “mixed ” case with both covering and packing consecutive 1’s constraints we present an O(mn) time algorithm. An O(mn + n 2 log n)time algorithm is presented for the case where the constraints are circular (consecutive 1’s constraint is also circular) of pure type – either all covering constraints or all packing constraints. Finally, we show an O(n min{mn, n 2 log n + m log 2 n}) time algorithm for the most general problem of mixed covering and packing case where the constraints are circular. All our algorithms are strongly polynomial and improve on the nonstrongly polynomial parametric minimum cost network flow or the (strongly polynomial) linear programming known approaches.