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

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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

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

### 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 }

Individuals (items)

potato diseases

proteins

faults in product

diseases

Binary attributes (tests)

potato varieties

antibodies detecting

presence of epitopes

(short peptide sequences)

fault detecting tests

fysical and chemical tests

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

MTS

pair of items i,j

m items

test T

n tests

Ti1,Ti2,...,Tik test set

SCP

element e(i,j)

M=m(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,...,Sikset cover

MTS and the Set Cover Problem (SCP)
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 SCP’s

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 SCP’s

MTS can be solved as SCP

MTS is NP-hard (reductionfrom SCP)

MTS tends to give difficult instances of SCP

Three directions

- Approximation algorithms

- Exact optimization algorithms

- Heuristics

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)forkthe size of the largest test

Lemma: Greedy has approximation ratio 11/8 for k=2

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

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

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

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

Approximation algorithms (2)
• No polynomial time algorithm gives a solution guaranteed within o(log m) times optimal unless P=NP (was proved for SCP in [Raz&Safra 1997])
• No polynomial time algorithm gives a solution guaranteed within (1-b)ln m for any b>0 unless NP iscontained in DTIME(m^{loglogm}) (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 (1+b) for any b>0 unless P=NP (MTS2 is APX-hard)
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 setdefines an equivalence relation on the set of items

Definition: Given a partial test set, two items are equivalentif there is no test that distinguishes them

A partial test set T gives equivalence classes of items

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

Branch-and-Bound (3)Branching

2 different branching rules

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

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

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