Boolean Functions I

1
Boolean Functions Ive Known and Not Known
Memories of Peter L. Hammer
Fred Roberts Rutgers University
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Peter Hammer, RUTCOR, Boolean Functions

• Topics of the Talk
• Peters influence
• A little about the history of RUTCOR
• Some work on Boolean functions that relates to
  Peters interests

3
How I Met Peter

• I met Peter in 1977 when I heard him give a talk
  about threshold graphs.
• I think it was at the Southeastern conference in
  Boca Raton, Florida.

4
How I Met Peter

• Little did I know then what influence he would
  have on my life.
• RUTCOR didnt exist.
• Peter was at Waterloo.
• There was no Operations Research at Rutgers.
• I was supervising my first Ph.D. student.
• Endre Boros was still working on his Masters
  Thesis On Sperner Spaces.
• Barack Obama was in high school.

5
Threshold Graphs

• My first meeting with Peter led to a thesis topic
  for my first student, Shelly Leibowitz.
• Let G be a graph with n vertices v1, v2, , vn.
• G is a threshold graph iff we can associate a
  weight w(vi) with each vertex and a threshold t
  so that for all sets S of vertices
• S is an independent set ? ?w(vi) vi ? S t
  

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Threshold Graphs

• If S is a set of vertices, associate with it a
  Boolean vector so that vi is in S iff the ith
  entry is 1.
• Equivalently, G is a threshold graph iff there is
  a hyperplane that separates the characteristic
  vectors of independent sets of vertices from
  those of non-independent sets.
• Chvatal and Hammer (1978) characterized threshold
  graphs.
• That caught my attention.

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Threshold Boolean Functions

• Suppose f is a Boolean function assigning each
  0-1 vector (x1,x2,,xn) to 0 or 1.
• f is a threshold Boolean function if there exist
  weights wi and a threshold t so that
• f(x1,x2,,xn) 1 ? ?i wixi t
• Thus, the Boolean function that assigns 1 to
  characteristic vectors of independent sets is a
  threshold Boolean function.

8
Sample Leibowitz Result Guttman Scaling

• A famous study of soldiers physical reactions to
  battle during WW II (Suchman, reported in
  Stouffer, et al. 1950).
• Which of the following reactions have you had?
• Violent pounding of the heart
• Sinking feeling of the stomach
• Feeling of weakness or feeling faint
• Feeling sick to the stomach
• Cold sweat
• Shaking or trembling all over
• Losing bladder control

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Sample Leibowitz Result Guttman Scaling

• Which of the following reactions have you had?
• Violent pounding of the heart
• Sinking feeling of the stomach
• Feeling of weakness or feeling faint
• Feeling sick to the stomach
• Cold sweat
• Shaking or trembling all over
• Losing bladder control
• Suchman The reactions could be ordered so that
  if a soldier had one of them, he had all those
  below it on the list.

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Guttman Scaling

• This is an example of Guttman scaling
• (after the psychologist Louis Guttman).
• Set A of subjects and set X of items.
• Can we order the set A?X so that a subject agrees
  with all items preceding it and disagrees with
  all items following it?
• Useful in education students and test items
• Useful in opinion scaling individuals and
  opinions
• So what does this have to do with threshold
  graphs?

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Guttman Scaling

• Build a graph G.
• Vertex set is A?X.
• Edge between a in A and x in X if subject a
  agrees with item x.
• Edge between every element of A.
• Leibowitz observed Agreement defines a Guttman
  scale iff G is a threshold graph.
• This led to a large number of results about
  Guttman scales, sequences, threshold graphs, etc.

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Recruiting Peter to Rutgers

• It was not threshold graphs that led us to
  recruit Peter to Rutgers.

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Recruiting Peter to Rutgers

• 1980-81, 1981-82 There was no O.R. at Rutgers,
  but there were courses in networks, graph theory,
  linear optimization, etc.
• A group of us decided we needed a graduate
  program in O.R.
• We decided we needed a leader to create and
  manage such a program.
• We convinced Dean of the Graduate School, Ken
  Wolfson

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The Recruitment

• Search Committee (with 80 confidence)
• Fred Roberts, Math
• Bob Vichnevetsky, CS
• Bill Strawderman, Stat
• Mike Grigoriadis, CS

Bob Vichnevetsky
Bill Strawderman
Mike Grigoriadis?
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The Recruitment

• The recruitment encompassed numerous locations.

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The Recruitment

• The recruitment encompassed numerous locations.

LaGuardia Airport
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The Recruitment

• The recruitment encompassed numerous locations.

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RUTCOR

• We wanted a graduate program.
• Peter had a much grander vision A CENTER FOR
  OPERATIONS RESEARCH
• RUTCOR was born.

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RUTCOR

• How do you build a center?

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RUTCOR Building a Center

• Space
• Hill Center for the Mathematical Sciences

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RUTCOR Building a Center

• Space A Building
• The true story of how the RUTCOR building came to
  have a deck

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RUTCOR Building a Center

• Courses
• Borrowing courses from Math, CS, Stat,
  Industrial Engineering, etc.
• Linear Programming from CS Dept.
• Networks and Combinatorial Optimization from CS
  Dept.
• Design and Analysis of Data Structures and
  Algorithms from Stat Dept.
• Etc.

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RUTCOR Building a Center

• Graduate Students
• The key is to recruit bright, hard-working
  graduate students

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RUTCOR Building a Center

• Graduate Students
• The key is to recruit good students
• Peters connections in Europe and elsewhere were
  critical.

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RUTCOR Building a Center

• Graduate Students
• Peters mode of operation from the beginning was
  to work closely with students and integrate them
  in the work of RUTCOR.
• And encourage them to work hard.

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RUTCOR Building a Center

• Graduate Students How to Support Them
• GA lines from others.
• Math Dept.
• Grad School
• Business School
• We started a grad program with two students in
  year 1

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RUTCOR Building a Center

• How are we Going to Pay for a Center?
• Surely the publisher will support RUTCOR
• Oh, perhaps thats not enough. I think Ill
  create a journal.

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RUTCOR Building a Center

• Recruiting a Faculty

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RUTCOR Building a Center

• How to Run a Center from Canada
• Agree to come to Rutgers, set up the center, but
  work on RUTCOR from Waterloo.
• Ask someone to be director during the first year.
• Fred Roberts as Director of RUTCOR during
  1982-83.
• Commute to Rutgers a few times each semester.
• Work hard from a distance to
• recruit students
• plan for courses
• raise funds
• develop plans for faculty

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RUTCOR Building a Center

• How to Run a Center Committees
• Universities need committees
• Executive Committee of RUTCOR
• Involve faculty from other departments
• RUTCOR fellows

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RUTCOR Building a Center

• How to Run a Center Voting Rules
• Many discussions to develop voting rules for
  RUTCOR.
• Peters research also touched upon voting rules.
• Boolean functions arise in voting.
• I dont know if the research on Boolean functions
  influenced the voting rules of RUTCOR.

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RUTCOR Building a Center

• How to Run a Center Voting Rules
• What do Boolean functions have to do with voting?

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Power of a Voter

• Think of a voting game
• Every coalition (subset of the players) is either
  strong enough to win or not.
• Represent a subset of the players as a 0-1 vector
  with the ith entry 1 iff player i is in the
  subset.
• Then the voting game corresponds to a Boolean
  function that assigns 1 to vectors corresponding
  to winning subsets (winning coalitions) and 0
  to losing subsets.
• If a winning coalition can never be contained in
  a losing one, we say we have a simple game.

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Power of a Voter

• Shapley-Shubik Power Index
• Think of a voting game
• Shapley-Shubik index measures the power
• of each player in a multi-player game.
• Consider a simple game.

Martin Shubik
Lloyd Shapley
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Power of a Voter

• Shapley-Shubik Power Index
• Consider a coalition forming at random, one
  player at a time.
• A player i is pivotal if addition of i throws
  coalition from losing to winning.
• Shapley-Shubik index of i probability i is
  pivotal if an order of players is chosen at
  random.
• Power measure applying to more general games than
  voting games is called Shapley Value.

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Power of a Voter

• Example Shareholders of Company
• Shareholder 1 holds 3 shares.
• Shareholders 2, 3, 4, 5, 6, 7 hold 1 share each.
• A majority of shares are needed to make a
  decision.
• Coalition 1,4,6 is winning.
• Coalition 2,3,4,5,6 is winning.
• Shareholder 1 is pivotal if he is 3rd, 4th, or
  5th.
• So shareholder 1s Shapley value is 3/7.
• Sum of Shapley values is 1 (since they are
  probabilities)
• Thus, each other shareholder has Shapley value
• (4/7)/6 2/21

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Power of a Voter

• Example Government of Australia
• The previous game actually arises in the
  government of Australia.
• There are six states and the federal government.
• A coalition wins (can pass a law) if it consists
  of at least five states or at least two states
  and the federal government.
• This is equivalent to the above voting game.

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Power of a Voter

• Example United Nations Security Council

• 15 member nations
• 5 permanent members
• China, France,
• Russia, UK, US
• 10 non-permanent
• Permanent members
• have veto power
• Coalition wins iff it has all 5 permanent members
• and at least 4 of the 10 non-permanent members.

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Power of a Voter

• Example United Nations Security Council

• What is the power of each
• Member of the Security
• Council?
• Fix non-permanent member i.
• i is pivotal in permutations in
• which all permanent members
• precede i and exactly 3 non-
• permanent members do.
• How many such permutations are there?

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Power of a Voter

• Example United Nations Security Council

• Choose 3 non-permanent members preceding i.
• Order all 8 members preceding i.
• Order remaining 6 non-permanent members.
• Thus the number of such permutations is
• C(9,3) x 8! x 6! 9!/3!6! x 8! x 6! 9!8!/3!
• The probability that i is pivotal power of
  non-permanent member
• 9!8!/3!15! .001865
• The power of a permanent member
• 1 10 x .001865/5 .1963.
• Permanent members have 100 times power of
  non-permanent members.

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Power of a Voter

• There are a variety of other power indices used
  in game theory and political science (Banzhaf
  index, Coleman index, )
• Need calculate them for huge games
• Mostly computationally intractable

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Power of a Voter Allocation/Sharing Costs and
Revenues

• Shapley-Shubik power index and more general
  Shapley value have been used to allocate costs to
  different users in shared projects.
• Allocating runway fees in airports
• Allocating highway fees to trucks of different
  sizes
• Universities sharing library facilities
• Fair allocation of telephone calling charges
  among users sharing complex phone systems
  (Cornells experiment)

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Power of a Voter Allocating/Sharing Costs and
Revenues

• Multicasting
• Applications in multicasting.
• Unicast routing Each packet sent from a source
  is delivered to a single receiver.
• Sending it to multiple sites Send multiple
  copies and waste bandwidth.
• In multicast routing Use a directed tree
• connecting source to all receivers.
• At branch points, a packet is duplicated as
• necessary.

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Multicasting
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Power of a Voter Allocating/Sharing Costs and
Revenues

• Multicasting
• Multicast routing Use a directed tree connecting
  source to all receivers.
• At branch points, a packet is duplicated as
  necessary.
• Bandwidth is not directly attributable to a
  single receiver.
• How to distribute costs among receivers?
• One idea Use Shapley value.

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Allocating/Sharing Costs and Revenues

• Feigenbaum, Papadimitriou, Shenker (2001) no
  feasible implementation for Shapley value in
  multicasting.
• Note Shapley value is uniquely characterized by
  four simple axioms.
• Sometimes we state axioms as general principles
  we want a solution concept to have.
• Jain and Vazirani (1998) polynomial time
  computable cost-sharing algorithm
• Satisfying some important axioms
• Calculating cost of optimum multicast tree within
  factor of two of optimal.

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Voting Games at RUTCOR

• Boolean functions I have not known
• As far as I know, Peter never used the
  Shapley-Shubik index or the Banzhaf or Coleman
  indices to
• Settle votes at RUTCOR
• Allocate costs and revenues at RUTCOR

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Algorithms for Container Inspection at Ports

• My recent work has gotten me more heavily
• into Boolean functions.
• Im glad that RUTCOR faculty and students
• have gotten involved.

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Sequential Decision Making

• Sequential decision making problems arise in many
  areas
• Communication networks (testing connectivity,
  paging cellular customers, sequencing tasks, )
• Manufacturing (testing machines, fault
  diagnosis, routing customer service calls, )
• Medicine (diagnosing patients, sequencing
  treatments, )

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Sequential Decision Making

• These problems are especially relevant in
  homeland security inspection contexts.
• Sheer size of modern decision problems in
  homeland security makes classical methods
  impractical quickly
• We seek new methods that will scale to address
  modern applications

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Container Inspection Algorithms

• This work has gotten me and our students to
  interesting places.
• Thanks to Capt. David Scott, US Coast Guard
  Captain of Port, Delaware Bay, we were taken on a
  tour of the port of Philadelphia

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Container Inspection Algorithms

• Goal Find ways to intercept illicit
• nuclear materials and weapons
• destined for the U.S. via the
• maritime transportation system
• Goal inspect all containers arriving at ports

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Sequential Decision Making Problem

• Stream of containers arrives at a port
• Similar analysis for inspection prior to
  departure
• The Decision Makers Problem
• Which to inspect?
• Which inspections next based on previous results?
• Approach
• decision logics
• combinatorial optimization methods
• Builds on ideas of Stroud
• and Saeger at Los Alamos
• Need for new models
• and methods

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Sequential Decision Making Problem

• Containers arriving to be classified into
  categories.
• Simple case 0 ok, 1 suspicious
• Inspection scheme specifies which inspections
  are to be made based on previous observations

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Sequential Decision Making ProblemFor Container
Inspection

• Containers have attributes, each
• in a number of states
• Sample attributes
• Levels of certain kinds of chemicals or
  biological materials
• Levels of radiation
• Whether or not there are items of a certain kind
  in the cargo list
• Whether cargo was picked up in a certain
• port

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Sequential Decision Making Problem

• Simplest Case Attributes are in state 0 or 1
  (absent or present)
• Then Container is a binary string like 011001
• So Classification is a decision function F that
  assigns each binary string to a category.

011001
F(011001)
If attributes 2, 3, and 6 are present, assign
container to category F(011001).
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Sequential Decision Making Problem

• If there are two categories, 0 and 1 (safe or
  suspicious), the decision function F is a
  Boolean function.
• Example
• F(000) F(111) 1, F(abc) 0 otherwise
• This classifies a container as positive iff it
  has none of the attributes or all of them.

1
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Binary Decision Tree Approach

• Sensors (or other tests) measure
  presence/absence of attributes so 0 or 1
• Use two outcome categories 0, 1 (safe or
  suspicious)
• Binary Decision Tree
• Nodes are sensors or categories
• Two arcs exit from each sensor node, labeled left
  and right.
• Take the right arc when sensor says the attribute
  is present, left arc otherwise

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Binary Decision Tree Approach

• Reach category 1 from
• the root by
• a0 L to a1 R a2 R 1 or
• a0 R a2 R1
• Container classified in category 1 iff it has
• a1 and a2 and not a0 or
• a0 and a2 and possibly a1.

Figure 2
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Binary Decision Tree Approach

• This binary decision tree corresponds to the same
  decision function
• Container classified in category 1 iff it has
• a1 and a2 and not a0 or
• a0 and a2 and possibly a1
• However, it has one less observation node ai.
• So, it is more efficient if all observations are
  equally costly and equally likely.

Figure 3
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Binary Decision Tree Approach

• How do we find a low-cost or least-cost binary
  decision tree corresponding to a Boolean
  function?
• Costs
• Inspection costs (use of tree nodes)
• Delay costs
• Fixed equipment costs
• False positive, false negative

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Binary Decision Tree Approach

• The problem of finding the least cost binary
  decision tree is very hard (NP-complete).
• When the number of potential tests is small, can
  try to solve it by trying all possible binary
  decision trees.
• But, even for n 4, not practical. (n 4 at
  Port of Long Beach-Los Angeles)

Port of Long Beach
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Binary Decision Tree Approach

• The number of alternative binary decision trees
  makes this brute force approach impractical.
• Methods so far have been limited to
• Special Boolean functions complete, monotone
• Special binary trees
• Limit on number of types of tests/sensors
• Even here, the brute force methods become
  infeasible for n gt 4 types of tests/sensors

brute force
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Binary Decision Tree Approach

• Stroud-Saeger adopted a cost function that
  depends on
• The expected number of tests before classifying a
  container (expected cost of utilizing the tree)
• The cost of a false positive
• The cost of a false negative
• Using this cost function, they found the least
  cost binary decision trees by brute force.

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Binary Decision Tree Approach Sensitivity
Analysis

• We did a sensitivity analysis on the
  Stroud-Saeger results.
• We varied three key parameters
• A priori probability of a bad container
• Cost of a false positive
• Cost of a false negative

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Sensitivity Analysis

• Cost of false negative was varied between 25
  million and 10 billion dollars
• (Low and high estimates of direct and indirect
  costs incurred due to a false negative.)
• Cost of false positive was varied between 180
  and 720
• (Cost incurred due to false positive
• (4 men (3 -6 hrs) (15 30 /hr))
• Probability of a bad container was varied
  between 1/10,000,000 and 1/100,000

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Sensitivity Analysis

• Tens of thousands of experimental runs.
• Surprising results
• With three types of tests, only three trees ever
  came out as least expensive.
• Similar results with four types of tests.
• Need an explanation of why.

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Binary Decision Tree Approach

• We have extended work of Stroud-Saeger in many
  ways, scaling to larger numbers of sensors
• E.g. Defined notions of complete and monotonic
  for trees (as opposed to Boolean functions)
• Developed heuristic search algorithms through an
  enlarged tree space of complete, monotonic
  trees
• This has allowed us to
• Speed up search
• Attain at least as much accuracy
• Allow more types of tests (sensors)

69
Binary Decision Tree Approach

• Much more work to do
• Explain why results so insensitive to changes in
  key parameter values
• Understand why certain binary decision trees are
  so good.
• Complications we will consider
• Different cost functions
• Role of different models for sensor errors
  thresholds (this work already begun)
• Modeling delays due to queues
• Simulation of port operations

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Container Inspection

• Collaborators on this Work
• Saket Anand
• David Madigan
• Richard Mammone
• Sushil Mittal
• Saumitr Pathak
• Los Alamos National Laboratory
• Rick Picard
• Kevin Saeger
• Phil Stroud

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Alternative Approaches to Container Inspection

• Endre Boros and Paul Kantor have taken quite a
  different approach.
• They use game theory to decide how to allocate a
  limited budget between container screening and
  actual container unpacking.
• Their screening power index summarizes the
  cost-effectiveness of different screening tests.
• The approach can yield increases of as much as
  100 in inspection with virtually no increase in
  cost.

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Container Inspection

• I can only think that Peter would have liked this
  problem.
• It uses Boolean functions in a serious way.
• It is a blend of practical Operations Research
  and theoretical O.R.
• The work has engaged both students and faculty.

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