The Minimum Test Set Problem MTS

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The Minimum Test Set Problem (MTS)

• Leen Stougie
• TU Eindhoven and CWI Amsterdam
• Joint work with
• Koen de Bontridder - Siemens Bjorni
  Halldorsson Iceland University
• Cor Hurkens TU Eindhoven Magnus
  Halldorsson Iceland University
• Ben Lageweg Ortec R.Ravi
  CMU Pittsburgh
• Jan Karel Lenstra CWI
• Jim Orlin MIT Cambridge MA

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• Set of m items 1,2,...,m
• Collection of n tests T1,T2,...,Tn
• Test Tj distinguishes items that react
• positively (1) on Tj from the items that react
• negaitively (0) on Tj
• A test is given by the items that react
  positively
• A test set is a subcollection of tests such that
  
• each pair of items is distinguished by at least
• one test in the the subcollection
• Find a test set of minimum cardinality

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Potatoes and diseases

• Potato Varieties Potato diseases
• V1 D1
  
• V2 D2
• V3 D3
• V4 D4
• V5 .
• Test Set is a set of varieties that discriminates
  between all diseases
• minimum test set V1,V4
• D1 has 1 , 1
• D2 has 1 , 0
  23 items (potato diseases)
• D3 has 0 , 0
  68 tests (potato varieties)
• D4 has 0 , 1

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IdentificationA test set gives each of a set of
individuals(items) a unique binary signature

• Binary attributes (tests)
• potato varieties
• antibodies detecting
• presence of epitopes
• (short peptide sequences)
• fault detecting tests
• fysical and chemical tests

• Individuals (items)
• potato diseases
• proteins
• faults in product
• diseases

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The Set Cover Problem (SCP)

• Set of M elements 1,2,...,M
• Collection of N sets S1,S2,...,SN
• Each set is a subset of the elements
• Set Sj covers the elements it contains
• A set cover is a subcollection of sets such that
  each
• element is covered by at least one set in the
  subcollection
• Find a set cover of minimum cardinality

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MTS and the Set Cover Problem (SCP)

• MTS
• pair of items i,j
• m items
• test T
• n tests
• Ti1,Ti2,...,Tik test set

• SCP
• element e(i,j)
• Mm(m-1)/2 elements
• set S containing all e(i,j)
• s.t. i in T and j not in T
• n sets
• Si1,Si2,...,Sik set cover

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• SCP is well studied and is the problem that
  models crew scheduling problems, workforce
  planning, class-scheduling etc.
• SCP is NP-hard
• Column generation methods solve practical SCPs

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• SCP is well studied and is the problem that
  models crew scheduling problems, workforce
  planning, class-scheduling etc.
• SCP is NP-hard
• Column generation methods solve practical SCPs
• MTS can be solved as SCP
• MTS is NP-hard (reduction from SCP)
• MTS tends to give difficult instances of SCP

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Three directions

• - Approximation algorithms
• - Exact optimization algorithms
• - Heuristics

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Approximation algorithms (1)

• Greedy algorithm
• At each iteration, given a partial test set (set
  of already selected tests), select the test that
  distinguishes most yet undistinguished item pairs
  and add to the partial test set
• Stop if all item pairs are distinguished
• Lemma Greedy has approximation ratio O(ln m)
• Lemma 2-phase Greedy has approximation ratio
  O(log k) for k the size of the largest test
• Lemma Greedy has approximation ratio 11/8 for k2

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A beautiful graph problem (1)

• MTS2 Each test contains exactly 2 items
• Item Vertex of graph, Test i,j
  Edge i,j of graph
• Example
• 7 items
• 10 tests

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A beautiful graph problem (2)

• MTS2 Each test contains exactly 2 items
• Item Vertex of graph, Test i,j
  Edge i,j of graph
• Example
• 7 items
• 10 tests
• By the red edge its two vertices are
  distinguished from all other vertices but not
  from one another

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A beautiful graph problem (3)

• MTS2 Each test contains exactly 2 items
• Item Vertex of graph, Test i,j
  Edge i,j of graph
• Example
• 7 items
• 10 tests
• By the path of two red edges its three vertices
  are distinguished from all other vertices and
  also from one another

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A beautiful graph problem (4)

• MTS2 Each test contains exactly 2 items
• Item Vertex of graph, Test i,j
  Edge i,j of graph
• Example
• 7 items
• 10 tests
• red paths form a test cover (1 isolated vertex is
  allowed)
• Graph Problem Given a graph, pack as many vertex
  disjoint paths of length 2 as possible

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Approximation algorithms (2)

• No polynomial time algorithm gives a solution
  guaranteed within o(log m) times optimal unless
  PNP (was proved for SCP in RazSafra 1997)
• No polynomial time algorithm gives a solution
  guaranteed within (1-b)ln m for any bgt0 unless NP
  is contained in DTIME(mloglogm)
  (was proved for
  SCP in Feige 1998)
• No polynomial time algorithm for the problem with
  at most 2 items per test (MTS2) gives a solution
  guaranteed within (1b) for any bgt0 unless PNP
  (MTS2 is APX-hard)

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Branch-and-Bound algorithms (1)Ingredients

• The nodes of the search tree correspond to
  partial test sets together with sets of rejected
  tests
• A partial test set defines an equivalence
  relation on the set of items
• Definition Given a partial test set, two items
  are equivalent if there is no test that
  distinguishes them
• A partial test set T gives equivalence classes of
  items

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Branch-and-Bound (2)Quality criteria

• Criterion 1 Separation criterion for test T not
  in T
• Criterion 2 Power criterion for test T not in T
• Criterion 3 Information criterion for test T not
  in T
• with

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Branch-and-Bound (3)Branching

• 2 different branching rules

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Branch-and-Bound (4)Lower bounds

• Lower bound by ideal tests
• Lower bound by power
• with F(m,n) the minimum power any set of n
  tests need to discriminate any set of m items
• ..... 2 more lower bounds

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Branch-and-Bound (5)Experimental results
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Branch-and-Bound (6)Experimental results
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Heuristics

• Halldorsson et al. applied heuristics for the
  proteomic test set problem
• We have no experience, but it is interesting to
  investigate in combination with real-life problems

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Minimum Test Set in the future

• Find some more applications
• Improve Branch and Bound algorithms
• Apply homeopathic algorithms
• Introduce possibilities for test results other
  than 0 or 1
• Construct software

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