Solving the Set Covering Problem based on a new Clustering Heuristic

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Solving the Set Covering Problem based on a new
Clustering Heuristic

• Nikolaos Mastrogiannis
• Ioannis Giannikos
• Basilis Boutsinas

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The Set Covering Problem

• The Set Covering Problem (SCP) is the problem of
  covering the rows of an m-row, n-column, zero-one
  matrix ( ) by a subset of the columns at
  minimum cost. Defining
• if column j is in the
  solution (with cost )
• otherwise
• the SCP is Minimize
  subject to

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Clustering Heuristics General Description

• Given the set covering problem as stated above, a
  clustering heuristic must specify
• A method for partitioning the column-set into
  homogeneous clusters formed by similar columns.
• A rule for selecting a best column for each
  cluster.
• If the set J of all the selected columns forms a
  cover, then a prime cover PJ is extracted from J.
  Else, the current partition is modified and the
  process is repeated.
• The proposed Clustering Heuristic, is based on
  the general principles of
• k-means clustering algorithm (MacQueen 1967)
• k-modes clustering algorithm (Huang 1998)
• ELECTRE methods and especially ELECTRE I
  multicriteria method (Roy, 1968)

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The Clustering Heuristic Introduction (1)

• The Clustering Heuristic consists of the
  following steps
• Select k initial centroids, one for each cluster.
• Assign each column to the proper cluster
  according to
• - the distance of the column to be clustered
  from the best of the centroids, for each of the
  rows (attributes) that describe both the column
  and the centroid
• - the importance of the best of the centroids
  in terms of its rows (attributes) weights
• - the possible veto that some of the rows
  (attributes) might oppose in the result of the
  assignment process
• Update the centroid of each cluster after
  each assignment

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The Clustering Heuristic Introduction (2)

• After all columns have been assigned to clusters,
  retest the dissimilarity of columns against the
  current centroids according to Step 2. If a
  column is found nearest to another cluster
  rather than its current one, re-assign the column
  to that cluster and update the centroids of both
  clusters.
• Repeat 3 until no column has changed clusters
  after a full cycle test of the whole dataset.

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The Clustering Heuristic Introduction (3)

• After partitioning the set of columns into
  homogeneous clusters, the best column for each
  cluster is chosen according to the Chvatals
  selector
• When the best column for each cluster is
  identified, we solve the SCP as an integer
  programming problem in MS Excel Solver in order
  to extract a prime cover from the above
  partition. Else we modify the partition (by
  changing the number of clusters) and we repeat
  the process.

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Description of the Clustering Heuristic (1)

• Step 1 The initial centroids Kt, t 1, k are
  selected among the columns to be clustered.
• Step 2.1 For every row (attribute) l1,m common
  between column yj , j1, n and each of the
  centroids Kt, t 1, k we calculate the distance,
• 
  , where
• and , are the relative
  frequencies of values Ktl and yjl
• If , where ql is an
  indifference threshold, then row (attribute) l
  belong to the concordance coalition, that is, it
  supports the proposition column yj is similar to
  centroids Kt on row (attribute) l. Otherwise,
  row (attribute) l, belongs to the discordance
  coalition.
• Observation The indifference threshold ql is a
  variable automatically valued according to the
  number of the discrete non-zero distances
  .

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Description of the Clustering Heuristic (2)

• Step 2.2 In order to choose the best of the
  centroids Kt, t 1, k and then assign the column
  to be clustered to its cluster, we calculate the
  concordance index (CIt, i1, k) and threshold
  (CTt, i1, k), as follows
• 
  ,
• where wc, wf, wp are weights of rows (attributes)
  and bonus is a parameter valued automatically,
  set to enforce those clusters (1, k) that contain
  as much zero differences as possible in every
  row.
• We sort in an descending order, the concordance
  index (CIt, i1, k) and its corresponding
  threshold (CTt, i1, k) for every cluster.

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Description of the Clustering Heuristic (3)

• If , where
  b 1,, k denotes the k positions of the k
  centroids and the corresponding indices and
  thresholds in the descending order stated above,
  the column j is clustered to cluster b.
  Otherwise, the process is repeated until the
  column is clustered.
• Observation1 Parameter is
  the only parameter defined by the user. If m1 is
  0.7, this means that for each column, the best of
  the centroids incorporates 70 of the strength of
  the weights of the attributes that belong to both
  the concordance and the discordance coalition.
• Observation2 The weighting of the rows l1, m in
  the proposed algorithm is based on the density of
  1s in matrix A. Thus

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Description of the Clustering Heuristic (4)

• Step 2.3 This step confirms or rejects the
  allocation of a column to a cluster
• If j, is the column to be clustered, t, is the
  cluster assigned to the column according to step
  2.2 and l, are all the rows that belong the
  discordance coalition, then
• If
  , for every row l that belongs to
• the discordance coalition, that is
  , then the clustering is confirmed, the
  centroid of the cluster is updated and we proceed
  to the next column. Otherwise, we return to step
  2.2.
• Observation Parameter Ul is called veto
  threshold, it is automatically valued, and for
  every row l, Veto threshold (Ul) gt Indifference
  threshold (ql).

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Description of the Clustering Heuristic (5)

• Step 3 We retest the dissimilarity of columns
  against the current centroids according to Step
  2, we re-assign every column needed to the proper
  cluster and update the centroids of both
  clusters.
• Step 4 We repeat Step 3 until no column has
  changed clusters after a full cycle test.
• Finding a new centroid for cluster k
• Column xj is a centroid for the zero-one
  matrix ( ) if it minimizes
• If is the number of columns with a
  discrete value on row l, and
• is the relative frequency of appearance of
  on the set of columns, then DIS is minimized iff
  
  for for every l1,,m

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Computational Experimentation

• The algorithm was tested using datasets from the
  OR Library of J.E.Beasley.
• The datasets include problems of size 50 x 500
  and 200 x 1000.
• In 80 of the tested datasets, the optimal
  solution was found using in the
  50 x 500 problems and in the 200
  x 1000 problems.
• Final SCP was solved using Premium Solver V.8 and
  in particular the XPRESS-MP Solver Engine

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Conclusions

• The new clustering heuristic
• combines three different scientific fields (set
  covering, data mining, multicriteria analysis).
• takes into consideration the weight of each row
  (attribute) in the clustering process rather than
  considering each row equivalently weighted.
• calculates these weights according to the density
  of 1s to the set.
• analyzes the dataset in details through the
  pairwise comparisons of each column with each
  centroid for each row.
• takes into consideration the possible objections
  of the minority of the rows (attributes) for the
  clustering process.
• presents covering results as well as processing
  time which are very promising.

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Bibliography

• Beasley, J.E (1987), An algorithm for set
  covering problem, European Journal of Operational
  Research, 31, pp.85-93
• Huang, Z. (1998), Extensions to the k-means
  Algorithm for Clustering Large Data Sets with
  categorical Values, Data Mining and Knowledge
  Discovery, vol.2, no.3, pp.283-304.
• Huang, Z., Ng, M.K., Rong, H. Li, Z. (2005),
  Automated Variable Weighting in k-means type
  clustering, IEEE Transactions on Pattern Analysis
  and Machine Intelligence, vol.27, no.5,
  pp.657-668.
• MacQueen, J.B. (1967), Some methods for
  classification and analysis of multivariate
  observations, In proceedings of the 5th Berkeley
  Symposium on Mathematical Statistics and
  Probability, pp.281-297
• Roy, B. (1968) Classement et choix en présence de
  points de vue muptiples  La méthode ELECTRE.
  R.I.R.O., 8, 57-75.
• Roy, B. (1991) The outranking approach and the
  foundations of ELECTRE methods. Theory and
  Decision, 31, 49-73.

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Thank you for your attention