Boolean Functions I’ve Known
and Not Known: Memories of
Peter L. Hammer
Fred Roberts
Rutgers University
1
Peter Hammer, RUTCOR, Boolean
Functions
Topics of the Talk
• Peter’s influence
• A little about the history of RUTCOR
• Some work on Boolean functions that relates to
Peter’s interests
2
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.
3
How I Met Peter
• Little did I know then what influence he would
have on my life.
• RUTCOR didn’t 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.
4
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
5
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.
6
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.
7
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
8
– Losing bladder control
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
9
those below it on the list.
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?
10
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.
11
Recruiting Peter to Rutgers
It was not threshold graphs that led us to recruit
Peter to Rutgers.
12
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
13
The Recruitment
• Search Committee (with 80% confidence)
– Fred Roberts, Math
– Bob Vichnevetsky, CS
– Bill Strawderman, Stat
– Mike Grigoriadis, CS
Bob Vichnevetsky
14
Mike Grigoriadis?
Bill Strawderman
The Recruitment
• The recruitment encompassed numerous
locations.
15
The Recruitment
• The recruitment encompassed numerous
locations.
LaGuardia Airport
16
The Recruitment
• The recruitment encompassed numerous
locations.
17
RUTCOR
• We wanted a graduate program.
• Peter had a much grander vision: A CENTER
FOR OPERATIONS RESEARCH
• RUTCOR was born.
18
RUTCOR
How do you build a center?
19
RUTCOR: Building a Center
Space
Hill Center for the Mathematical Sciences
20
RUTCOR: Building a Center
Space: A Building
The true story of how the RUTCOR
building came to have a deck
21
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.
22
RUTCOR: Building a Center
Graduate Students
• The key is to recruit bright, hardworking graduate students
23
RUTCOR: Building a Center
Graduate Students
• The key is to recruit good students
• Peter’s connections in Europe and
elsewhere were critical.
24
RUTCOR: Building a Center
Graduate Students
• Peter’s 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.
25
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
26
RUTCOR: Building a Center
How are we Going to Pay for a Center?
• “Surely the publisher will support
RUTCOR”
• “Oh, perhaps that’s not enough. I think
I’ll create a journal.”
27
RUTCOR: Building a Center
Recruiting a Faculty
28
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
29
– develop plans for faculty
RUTCOR: Building a Center
How to Run a Center: Committees
•
•
•
•
Universities need committees
Executive Committee of RUTCOR
Involve faculty from other departments
RUTCOR “fellows”
30
RUTCOR: Building a Center
How to Run a Center: Voting Rules
• Many discussions to develop voting rules for
RUTCOR.
• Peter’s research also touched upon voting rules.
• Boolean functions arise in voting.
• I don’t know if the research on Boolean functions
influenced the voting rules of RUTCOR.
31
RUTCOR: Building a Center
How to Run a Center: Voting Rules
What do Boolean functions have to do with voting?
32
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.
33
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
34
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.
35
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 1’s 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
36
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.
37
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.
38
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 nonpermanent members do.
•How many such permutations are there?
39
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 nonpermanent 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 40
non-permanent members.
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
41
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 (Cornell’s
experiment)
42
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.
43
Multicasting
44
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.
45
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.
46
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
47
Algorithms for
Container Inspection at Ports
•My recent work has gotten me more heavily
into Boolean functions.
•I’m glad that RUTCOR faculty and students
have gotten involved.
48
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,
…)
49
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
50
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
51
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”
52
Sequential Decision Making
Problem
• Stream of containers arrives at a port
− Similar analysis for inspection prior to departure
• The Decision Maker’s 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
53
and methods
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
54
Sequential Decision Making
Problem
For 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
55
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).
56
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=
57
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
58
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
59
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
60
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
61
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
62
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 > 4 types of tests/sensors
brute force
63
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.
64
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
65
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
66
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.
67
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)
68
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
69
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
70
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.
71
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.
72
73