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46
Improved algorithms for optimal winner determination in combinatorial auctions and generalizations
, 2000
"... Combinatorial auctions can be used to reach efficient resource and task allocations in multiagent systems where the items are complementary. Determining the winners is NPcomplete and inapproximable, but it was recently shown that optimal search algorithms do very well on average. This paper present ..."
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Cited by 513 (53 self)
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Combinatorial auctions can be used to reach efficient resource and task allocations in multiagent systems where the items are complementary. Determining the winners is NPcomplete and inapproximable, but it was recently shown that optimal search algorithms do very well on average. This paper presents a more sophisticated search algorithm for optimal (and anytime) winner determination, including structural improvements that reduce search tree size, faster data structures, and optimizations at search nodes based on driving toward, identifying and solving tractable special cases. We also uncover a more general tractable special case, and design algorithms for solving it as well as for solving known tractable special cases substantially faster. We generalize combinatorial auctions to multiple units of each item, to reserve prices on singletons as well as combinations, and to combinatorial exchanges  all allowing for substitutability. Finally, we present algorithms for determining the winners in these generalizations.
LexBFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing
, 2000
"... ..."
private communication
"... A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic BreadthFirst Search (LBFS) and Maximum Cardinality Search (M ..."
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Cited by 56 (4 self)
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A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic BreadthFirst Search (LBFS) and Maximum Cardinality Search (MCS), Corneil and Krueger propose in 2008 the socalled Maximal Neighborhood Search (MNS) and show that one sweep of MNS is enough to recognize chordal graphs. We develop the MNS properties of rigid interval graphs and characterize this graph class in several different ways. This allows us obtain several linear time multisweep MNS algorithms for recognizing rigid interval graphs and unit interval graphs, generalizing a corresponding 3sweep LBFS algorithm for unit interval graph recognition designed by Corneil in 2004. For unit interval graphs, we even present a new linear time 2sweep MNS certifying recognition algorithm. Submitted:
PC trees and circularones arrangements
 Theoretical Computer Science
"... A 01 matrix has the consecutiveones property if its columns can be ordered so that the ones in every row are consecutive. It has the circularones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all cons ..."
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Cited by 34 (4 self)
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A 01 matrix has the consecutiveones property if its columns can be ordered so that the ones in every row are consecutive. It has the circularones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all consecutiveones orderings of the columns of a matrix that has the consecutiveones property. We give an analogous structure, called a PC tree, for representing all circularones orderings of the columns of a matrix that has the circularones property. No such representation has been given previously. In contrast to PQ trees, PC trees are unrooted. We obtain a much simpler algorithm for computing PQ trees that those that were previously available, by adding a zero column, x, to a matrix, computing the PC tree, and then picking the PC tree up by x to root it. 1
Certifying algorithms for recognizing interval graphs and permutation graphs
 SIAM J. COMPUT
, 2006
"... A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give lineartime certifying algorithms for recognition o ..."
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Cited by 31 (7 self)
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A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give lineartime certifying algorithms for recognition of interval graphs and permutation graphs, and for a few other related problems. Previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class. We show that our certificates of nonmembership can be authenticated in O(V) time.
A simple test for the consecutive ones property
 Journal of Algorithms
, 1992
"... A (0,1)matrix satisfies the consecutive ones property if there exists a column permutation such that the ones in each row of the resulting matrix are consecutive. Booth and Lueker [1976] designed a linear time testing algorithm for this property based on a data structure called "PQtrees" ..."
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Cited by 28 (2 self)
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A (0,1)matrix satisfies the consecutive ones property if there exists a column permutation such that the ones in each row of the resulting matrix are consecutive. Booth and Lueker [1976] designed a linear time testing algorithm for this property based on a data structure called "PQtrees". This procedure is quite complicated and the linear time amortized analysis is also rather involved. We developed an offline linear time test for the consecutive ones property without using PQtrees and the corresponding template matching, which is considerably simpler. A simplification of the consecutive ones test will immediately simplify algorithms (and computer codes) for interval graph and planar graph recognition. Our approach is based on a decomposition technique that separates the rows into prime subsets, each of which admits essentially a unique column ordering that realizes the consecutive ones property. The success of this approach is based on finding a good "row ordering " to be tested iteratively. 1.
Simple Linear Time Recognition of Unit Interval Graphs
, 1998
"... We present a linear time algorithm for unit interval graph recognition. The algorithm is simple and based on BreadthFirst Search. It is also direct  it does not first recognize the graph as an interval graph. Given a graph G, the algorithm produces an ordering of the vertices of the graph whe ..."
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Cited by 27 (1 self)
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We present a linear time algorithm for unit interval graph recognition. The algorithm is simple and based on BreadthFirst Search. It is also direct  it does not first recognize the graph as an interval graph. Given a graph G, the algorithm produces an ordering of the vertices of the graph whenever G is a unit interval graph. This order corresponds to the order of the intervals of some unit interval model for G when arranged according to the increasing order of their left end coordinates. BreadthFirst Search can also be used to construct a unit interval model for a unit interval graph on n vertices; in this model each endpoint is rational, with denominator n. Keywords: graph algorithms, interval graphs, BreadthFirst Search, unit interval graphs, proper interval graphs, design of algorithms. 1 Introduction A graph G is an interval graph if its vertices can be put in a one to one correspondence with a family of intervals I on the real line such that two vertices in G are a...
Combinatorial auctions with structured item graphs
 In Proceedings of the Twenty First National Conference on Artificial Intelligence (AAAI 2004
, 2004
"... Combinatorial auctions (CAs) are important mechanisms for allocating interrelated items. Unfortunately, winner determination is NPcomplete unless there is special structure. We study the setting where there is a graph (with some desired property), with the items as vertices, and every bid bids on a ..."
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Cited by 25 (10 self)
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Combinatorial auctions (CAs) are important mechanisms for allocating interrelated items. Unfortunately, winner determination is NPcomplete unless there is special structure. We study the setting where there is a graph (with some desired property), with the items as vertices, and every bid bids on a connected set of items. Two computational problems arise: 1) clearing the auction when given the item graph, and 2) constructing an item graph (if one exists) with the desired property. 1 was previously solved for the case of a tree or a cycle, and 2 for the case of a line graph or a cycle. We generalize the first result by showing that given an item graph with bounded treewidth, the clearing problem can be solved in polynomial time (and every CA instance has some treewidth; the complexity is exponential in only that parameter). We then give an algorithm for constructing an item tree (treewidth 1) if such a tree exists, thus closing a recognized open problem. We show why this algorithm does not work for treewidth greater than 1, but leave open whether item graphs of (say) treewidth 2 can be constructed in polynomial time. We show that finding the item graph with the fewest edges is NPcomplete (even when a graph of treewidth 2 exists). Finally, we study how the results change if a bid is allowed to have more than one connected component. Even for line graphs, we show that clearing is hard even with 2 components, and constructing the line graph is hard even with 5.
PARTITION REFINEMENT TECHNIQUES: AN INTERESTING ALGORITHMIC TOOL KIT
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
, 1999
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A New Exact Algorithm for General Orthogonal dDimensional Knapsack Problems
, 1997
"... The ddimensional orthogonal knapsack problem (OKP) has a wide range of practical applications, including packing, cutting, and scheduling. We present a new approach to this problem, using a graphtheoretical characterization of feasible packings. This characterization allows us to deal with classes ..."
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Cited by 21 (10 self)
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The ddimensional orthogonal knapsack problem (OKP) has a wide range of practical applications, including packing, cutting, and scheduling. We present a new approach to this problem, using a graphtheoretical characterization of feasible packings. This characterization allows us to deal with classes of packings that share a certain combinatorical structure, instead of single ones. Combining the use of this structure with other heuristics, we develop a twolevel tree search algorithm for finding exact solutions for the ddimensional OKP. Computational results are reported, including optimal solutions for all twodimensional test problems from recent literature.