A note on the NP–hardness of the consecutive block minimization problem

Haddadi, Salim

doi:10.1111/1475-3995.00387

Cited by 18 publications

(14 citation statements)

References 3 publications

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“…Thus, solving CBM is equivalent to minimizing the number of holes. CBM is NP-hard even when restricted to binary matrices having two 1s per row (Haddadi 2002). Haddadi and Layouni (2008) converted CBM to traveling salesman problem (TSP) satisfying the triangle inequality.…”

Section: Preprocessingmentioning

confidence: 99%

Benders decomposition for set covering problems

Haddadi

2015

J Comb Optim

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This paper describes Benders decomposition approaches to optimally solve set covering problems " almost" satisfying the consecutive ones property. The decompositions are based on the fact that set covering problems with consecutive ones property have totally unimodular coefficient matrices. Given a binary matrix, a totally unimodular matrix is enforced by filling up every row with ones between its first and its last non-zero entries. The resulting mistake is handled by introducing additional integer variables whose number depends on the reordering of the columns of the given matrix. This leads us to consider the consecutive block minimization problem. Two cutting plane algorithms are proposed and run on a large set of benchmark instances. The results obtained show that the cutting plane algorithms outperform an existing tree search method designed exclusively for such instances.

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Section: Preprocessingmentioning

confidence: 99%

Benders decomposition for set covering problems

Haddadi

2015

J Comb Optim

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“…The first node corresponding to each item is pointed from a header table and each FP Tree node contains a link to the next node corresponding to the same item. Table 1, each node contains an item:frequency pair, and dotted arrows represent node links Example 1: Considering the transaction database in Table 1, the FList contains items in the order (3,2,4,5,1). Column 3 of Table 1 presents items in each transaction ordered according to the FList, and Figure 1 presents the corresponding FP Tree.…”

Section: Trie-based Representationsmentioning

confidence: 99%

“…Counting the number of 1-bits, we obtain the itemset support count of 2. The resulting bit vector also identifies the transactions (i.e., 4,6) that contain the query itemset.…”

Section: Bitmap-based Representationsmentioning

confidence: 99%

“…Unfortunately, reorganizing transactions to achieve such an optimal ordering in general is same as the consecutive block minimization problem (or CBMP) which was proven NP-complete in 70's by Kou [3]. More recently, even a fairly restricted version of this problem which limits the number of 1's in each row to 2, called 2CBMP [4], was also proven NP-hard.…”

Section: Increasing Bitmap Compressibility By Reordering Transactionsmentioning

confidence: 99%

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Optimizing Frequency Queries for Data Mining Applications

Malik

Kender

2007

Seventh IEEE International Conference on Data Mining (ICDM 2007)

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“…It has been shown that deciding if the columns of M can be ordered in such a way that every row contains at most k contigs is NP-complete even if k = 2 [2]. Also finding an ordering of the columns that minimizes the number of gaps in M is NP-complete even if each row of M has at most two ones [3].…”

Section: Introductionmentioning

confidence: 99%

On the Gapped Consecutive-Ones Property

Chauve

Maňuch

Patterson

2009

Electronic Notes in Discrete Mathematics

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Abstract. Motivated by problems of comparative genomics and paleogenomics, we introduce the Gapped Consecutive-Ones Property Problem (k,δ)-C1P: given a binary matrix M and two integers k and δ, can the columns of M be permuted such that each row contains at most k sequences of 1's and no two consecutive sequences of 1's are separated by a gap of more than δ 0's. The classical C1P problem, which is known to be polynomial, is equivalent to the (1,0)-C1P Problem. We show that the (2,δ)-C1P Problem is NP-complete for δ ≥ 2. We conjecture that the (k, δ)-C1P Problem is NPcomplete for k ≥ 2, δ ≥ 1, (k, δ) = (2, 1). We also show that the (k,δ)-C1P problem can be reduced to a graph bandwidth problem parameterized by a function of k, δ and of the maximum number s of 1's in a row of M , and hence is polytime solvable if all three parameters are constant.

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A note on the NP–hardness of the consecutive block minimization problem

Abstract: We show that consecutive block minimization problem is NP-hard even when restricted to binary matrices that have two ones per row.

Cited by 18 publications

References 3 publications

Benders decomposition for set covering problems

Benders decomposition for set covering problems

Optimizing Frequency Queries for Data Mining Applications

On the Gapped Consecutive-Ones Property

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