A subexponential lower bound for Zadeh’s pivoting
rule for solving linear programs and games
Oliver Friedmann
Department of Computer Science,
Ludwig-Maximilians-Universität Munich, Germany.
Oliver Friedmann (LMU)
Zadeh Lower Bound
1
Linear Programming and Simplex
Linear Programming and Simplex
Oliver Friedmann (LMU)
Zadeh Lower Bound
2
Linear Programming and Simplex
The simplex algorithm, Dantzig (1947)
maximize
cT x
subject to Ax ≤ b
Oliver Friedmann (LMU)
Zadeh Lower Bound
3
Linear Programming and Simplex
Basic feasible solutions and pivoting
max
s.t.
2x1 − 2x3 − 2x5 − x6
2
2
1
3 x1 + x2 − 3 x3 − 3 x5
x3 + x4 − x6
x5 + x6
x1 , x2 , x3 , x4 , x5 , x6
=
=
=
≥
1
1
1
0
{x2 , x4 , x6 }
The corners of the polytope correspond to basic feasible
solutions: At most n, the number of equality constraints,
variables are non-zero. The non-zero variables, or basic variables,
form a basis.
Moving along an edge corresponds to pivoting: Exchange a
variable in the basis with a non-basic variable.
Oliver Friedmann (LMU)
Zadeh Lower Bound
4
Linear Programming and Simplex
Basic feasible solutions and pivoting
max −1 + 2x1 − 2x3 − x5
s.t. x2 = 1 − 13 x1 + 32 x3 + 23 x5
x4 = 2 − x3 − x5
x6 = 1 − x5
x1 , x2 , x3 , x4 , x5 , x6 ≥ 0
x5
x1
x3
{x2 , x4 , x6 }
The corners of the polytope correspond to basic feasible
solutions: At most n, the number of equality constraints,
variables are non-zero. The non-zero variables, or basic variables,
form a basis.
Moving along an edge corresponds to pivoting: Exchange a
variable in the basis with a non-basic variable.
Oliver Friedmann (LMU)
Zadeh Lower Bound
4
Linear Programming and Simplex
Basic feasible solutions and pivoting
max 5 − 6x2 + 2x3 + 3x5
s.t. x1 = 3 − 3x2 + 2x3 + 2x5
x4 = 2 − x3 − x5
x6 = 1 − x5
x1 , x2 , x3 , x4 , x5 , x6 ≥ 0
x5
{x1 , x4 , x6 }
x2
x3
The corners of the polytope correspond to basic feasible
solutions: At most n, the number of equality constraints,
variables are non-zero. The non-zero variables, or basic variables,
form a basis.
Moving along an edge corresponds to pivoting: Exchange a
variable in the basis with a non-basic variable.
Oliver Friedmann (LMU)
Zadeh Lower Bound
4
Linear Programming and Simplex
Basic feasible solutions and pivoting
max 9 − 6x2 − 2x4 + x5
s.t. x1 = 7 − 3x2 − 2x4
x3 = 2 − x4 − x5
x6 = 1 − x5
x1 , x2 , x3 , x4 , x5 , x6 ≥ 0
x5
{x1 , x3 , x6 }
x4
x2
The corners of the polytope correspond to basic feasible
solutions: At most n, the number of equality constraints,
variables are non-zero. The non-zero variables, or basic variables,
form a basis.
Moving along an edge corresponds to pivoting: Exchange a
variable in the basis with a non-basic variable.
Oliver Friedmann (LMU)
Zadeh Lower Bound
4
Linear Programming and Simplex
Basic feasible solutions and pivoting
{x1 , x3 , x5 }
max 10 − 6x2 − 2x4 − x6
s.t. x1 = 7 − 3x2 − 2x4
x3 = 1 − x4 + x6
x5 = 1 − x6
x1 , x2 , x3 , x4 , x5 , x6 ≥ 0
x2
x4
x6
The corners of the polytope correspond to basic feasible
solutions: At most n, the number of equality constraints,
variables are non-zero. The non-zero variables, or basic variables,
form a basis.
Moving along an edge corresponds to pivoting: Exchange a
variable in the basis with a non-basic variable.
Oliver Friedmann (LMU)
Zadeh Lower Bound
4
Linear Programming and Simplex
Pivoting rules
Simplex method is parameterized by a pivoting rule: the method of
chosing adjacent vertices with better objective
Oliver Friedmann (LMU)
Zadeh Lower Bound
5
Linear Programming and Simplex
Pivoting rules
Simplex method is parameterized by a pivoting rule: the method of
chosing adjacent vertices with better objective
Deterministic, oblivious rules: Almost all known natural oblivious
deterministic pivoting rules are known to be exponential. See, e.g.,
Amenta and Ziegler (1996).
Oliver Friedmann (LMU)
Zadeh Lower Bound
5
Linear Programming and Simplex
Pivoting rules
Simplex method is parameterized by a pivoting rule: the method of
chosing adjacent vertices with better objective
Deterministic, oblivious rules: Almost all known natural oblivious
deterministic pivoting rules are known to be exponential. See, e.g.,
Amenta and Ziegler (1996).
Randomized rules: We (joint work with Thomas D. Hansen and
Uri Zwick) have recently shown that Random-Edge and
Random-Facet are subexponential.
Oliver Friedmann (LMU)
Zadeh Lower Bound
5
Linear Programming and Simplex
Pivoting rules
Simplex method is parameterized by a pivoting rule: the method of
chosing adjacent vertices with better objective
Deterministic, oblivious rules: Almost all known natural oblivious
deterministic pivoting rules are known to be exponential. See, e.g.,
Amenta and Ziegler (1996).
Randomized rules: We (joint work with Thomas D. Hansen and
Uri Zwick) have recently shown that Random-Edge and
Random-Facet are subexponential.
History-based rules (this talk): We show that Zadeh’s
Least-Entered rule is subexponential.
Oliver Friedmann (LMU)
Zadeh Lower Bound
5
Linear Programming and Simplex
Pivoting rules
Simplex method is parameterized by a pivoting rule: the method of
chosing adjacent vertices with better objective
Deterministic, oblivious rules: Almost all known natural oblivious
deterministic pivoting rules are known to be exponential. See, e.g.,
Amenta and Ziegler (1996).
Randomized rules: We (joint work with Thomas D. Hansen and
Uri Zwick) have recently shown that Random-Edge and
Random-Facet are subexponential.
History-based rules (this talk): We show that Zadeh’s
Least-Entered rule is subexponential.
Our results have been obtained by using a deep relation between
algorithmic game theory and linear programming.
Oliver Friedmann (LMU)
Zadeh Lower Bound
5
Linear Programming and Simplex
On the diameter of polytopes
Hirsch conjecture (1957): The diameter of any n-facet polytope in
d-dimensional Euclidean space is at most n − d.
Oliver Friedmann (LMU)
Zadeh Lower Bound
6
Linear Programming and Simplex
On the diameter of polytopes
Hirsch conjecture (1957): The diameter of any n-facet polytope in
d-dimensional Euclidean space is at most n − d.
Santos (2010): A counter-example to the Hirsch conjecture.
It remains open whether the diameter is polynomial, or even linear,
in n and d.
Oliver Friedmann (LMU)
Zadeh Lower Bound
6
Games and Policy Iteration
Games and Policy Iteration
Oliver Friedmann (LMU)
Zadeh Lower Bound
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Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Reward: −1
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Reward: −1
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Reward: −1
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Reward: −1 − 4
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Reward: −1 − 4
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Reward: −1 − 4
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Reward: −1 − 4
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Reward: −1 − 4
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Reward: −1 − 4
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
Shapley (1953),
Bellman (1957):
-4
There exists an optimal history-independent
choice from each state.
-1
Reward: −1 − 4
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Reward: −1 − 4
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Reward: −1 − 4 + 6
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Markov decision processes (MDPs)
t
6
1
3
2
3
-4
-1
Reward: −1 − 4 + 6 = 1
Oliver Friedmann (LMU)
Zadeh Lower Bound
8
Games and Policy Iteration
Policies and corresponding values
A policy π is a choice of an action
from each state.
t
6
1
3
2
3
-4
-1
Oliver Friedmann (LMU)
Zadeh Lower Bound
9
Games and Policy Iteration
Policies and corresponding values
A policy π is a choice of an action
from each state.
The value valπ (i) of a state i ∈ S for
a policy π, is the expected sum of
rewards obtained when moving
according to π, starting from i.
1
3
2
Oliver Friedmann (LMU)
Zadeh Lower Bound
t
6
2
3
0
-4
0
-1
-1
9
Games and Policy Iteration
Policies and corresponding values
A policy π is a choice of an action
from each state.
The value valπ (i) of a state i ∈ S for
a policy π, is the expected sum of
rewards obtained when moving
according to π, starting from i.
An action is an improving switch
w.r.t. π if it improves the values.
Oliver Friedmann (LMU)
Zadeh Lower Bound
t
6
1
3
2
2
3
0
-4
0
-1
-1
9
Games and Policy Iteration
Policies and corresponding values
A policy π is a choice of an action
from each state.
The value valπ (i) of a state i ∈ S for
a policy π, is the expected sum of
rewards obtained when moving
according to π, starting from i.
An action is an improving switch
w.r.t. π if it improves the values.
It suffices to check whether an action
is improving for one step w.r.t. the
current values.
Oliver Friedmann (LMU)
Zadeh Lower Bound
t
6
1
3
2
2
3
0
-4
0
-1
-1
9
Games and Policy Iteration
Policies and corresponding values
A policy π is a choice of an action
from each state.
The value valπ (i) of a state i ∈ S for
a policy π, is the expected sum of
rewards obtained when moving
according to π, starting from i.
An action is an improving switch
w.r.t. π if it improves the values.
It suffices to check whether an action
is improving for one step w.r.t. the
current values.
Oliver Friedmann (LMU)
Zadeh Lower Bound
t
6
1
3
6
2
3
6
-4
0
-1
-1
9
Games and Policy Iteration
Policies and corresponding values
A policy π is a choice of an action
from each state.
The value valπ (i) of a state i ∈ S for
a policy π, is the expected sum of
rewards obtained when moving
according to π, starting from i.
An action is an improving switch
w.r.t. π if it improves the values.
It suffices to check whether an action
is improving for one step w.r.t. the
current values.
Oliver Friedmann (LMU)
Zadeh Lower Bound
t
6
1
3
6
2
3
6
-4
2
1
-1
9
Games and Policy Iteration
Policies and corresponding values
A policy π is a choice of an action
from each state.
The value valπ (i) of a state i ∈ S for
a policy π, is the expected sum of
rewards obtained when moving
according to π, starting from i.
An action is an improving switch
w.r.t. π if it improves the values.
It suffices to check whether an action
is improving for one step w.r.t. the
current values.
A policy π ∗ is optimal iff there are no
improving switches. Optimal policies
simultaneously maximize the values of
all states.
Oliver Friedmann (LMU)
Zadeh Lower Bound
t
6
1
3
6
2
3
6
-4
2
2
-1
9
Games and Policy Iteration
MDPs and linear programming
No improving switches for optimal policy π ∗ :
X
∀i ∈ S : valπ∗ (i) = max ra +
pa,j valπ∗ (j)
a∈Ai
j∈S
where Ai is the set of actions from state i, ra is the expected
reward of using action a, and pa,j is the probability of moving to
state j when using action a.
Oliver Friedmann (LMU)
Zadeh Lower Bound
10
Games and Policy Iteration
MDPs and linear programming
No improving switches for optimal policy π ∗ :
X
∀i ∈ S : valπ∗ (i) = max ra +
pa,j valπ∗ (j)
a∈Ai
j∈S
where Ai is the set of actions from state i, ra is the expected
reward of using action a, and pa,j is the probability of moving to
state j when using action a.
This can be used to formulate an LP for solving the MDP:
X
minimize
vi
i∈S
s.t. ∀i ∈ S ∀a ∈ Ai : vi ≥ ra +
X
pa,j vj
j∈S
Oliver Friedmann (LMU)
Zadeh Lower Bound
10
Games and Policy Iteration
Primal and dual LPs for MDPs
minimize
X
vi
i∈S
s.t. ∀i ∈ S ∀a ∈ Ai : vi ≥ ra +
X
pa,j vj
j∈S
maximize
XX
ra xa
i∈S a∈Ai
s.t. ∀i ∈ S :
X
a∈Ai
Oliver Friedmann (LMU)
xa = 1 +
XX
pa,i xa
j∈S a∈Aj
Zadeh Lower Bound
11
Games and Policy Iteration
Primal and dual LPs for MDPs
Flow conservation:
minimize
X
vi
i∈S
x1 = 1 x2 = 6
s.t. ∀i ∈ S ∀a ∈ Ai : vi ≥ ra +
X
pa,j vj
j∈S
maximize
XX
ra xa
x3 = 4 x4 = 2
i∈S a∈Ai
s.t. ∀i ∈ S :
X
a∈Ai
Oliver Friedmann (LMU)
xa = 1 +
XX
pa,i xa
j∈S a∈Aj
Zadeh Lower Bound
x1 + x2 = 1 + x3 + x4
11
Games and Policy Iteration
Primal and dual LPs for MDPs
Flow conservation:
minimize
X
vi
i∈S
x1 = 7 x2 = 0
s.t. ∀i ∈ S ∀a ∈ Ai : vi ≥ ra +
X
pa,j vj
j∈S
maximize
XX
ra xa
x3 = 4 x4 = 2
i∈S a∈Ai
s.t. ∀i ∈ S :
X
a∈Ai
xa = 1 +
XX
pa,i xa
j∈S a∈Aj
x1 + x2 = 1 + x3 + x4
Every basic feasible solution corresponds to a policy π.
Oliver Friedmann (LMU)
Zadeh Lower Bound
11
Games and Policy Iteration
Variables of the primal LP
t
6
1
3
x2 = 1
2
3
2
0
x4 = 2
xa is the expected number of
times action a is used, summed
over all starting states.
x1 = 0
-4
x3 = 0
0
x5 = 0
-1
x6 = 1
Oliver Friedmann (LMU)
-1
Zadeh Lower Bound
12
Games and Policy Iteration
Variables of the primal LP
t
6
1
3
x2 = 1
2
3
2
0
x4 = 2
x1 = 0
-4
x3 = 0
0
xa is the expected number of
times action a is used, summed
over all starting states.
We have:
X
X
ra xπa
valπ (i) =
i∈S
a∈π
x5 = 0
-1
x6 = 1
Oliver Friedmann (LMU)
-1
Zadeh Lower Bound
12
Games and Policy Iteration
From MDP to LP
t
6
1
3
max −1 + 2x1 − 2x3 − x5
s.t. x2 = 1 − 13 x1 + 32 x3 + 23 x5
x4 = 2 − x3 − x5
x6 = 1 − x5
x1 , x2 , x3 , x4 , x5 , x6 ≥ 0
x2 = 1
2
3
2
0
x4 = 2
x1 = 0
-4
x3 = 0
0
x5 = 0
x5
-1
x6 = 1
x1
-1
x3
Oliver Friedmann (LMU)
Zadeh Lower Bound
{x2 , x4 , x6 }
13
Games and Policy Iteration
From MDP to LP
t
6
1
3
max 5 − 6x2 + 2x3 + 3x5
s.t. x1 = 3 − 3x2 + 2x3 + 2x5
x4 = 2 − x3 − x5
x6 = 1 − x5
x1 , x2 , x3 , x4 , x5 , x6 ≥ 0
x2 = 0
2
3
6
6
x4 = 2
x1 = 3
-4
x3 = 0
x5
0
{x1 , x4 , x6 }
x2
x5 = 0
-1
x6 = 1
Oliver Friedmann (LMU)
-1
Zadeh Lower Bound
x3
13
Games and Policy Iteration
From MDP to LP
t
6
1
3
max 9 − 6x2 − 2x4 + x5
s.t. x1 = 7 − 3x2 − 2x4
x3 = 2 − x4 − x5
x6 = 1 − x5
x1 , x2 , x3 , x4 , x5 , x6 ≥ 0
x2 = 0
2
3
6
6
x4 = 0
x1 = 7
-4
x3 = 2
x5
2
{x1 , x3 , x6 }
x4
x5 = 0
1
x6 = 1
Oliver Friedmann (LMU)
-1
x2
Zadeh Lower Bound
13
Games and Policy Iteration
From MDP to LP
t
6
1
3
max 10 − 6x2 − 2x4 − x6
s.t. x1 = 7 − 3x2 − 2x4
x3 = 1 − x4 + x6
x5 = 1 − x6
x1 , x2 , x3 , x4 , x5 , x6 ≥ 0
x2 = 0
2
3
6
6
x4 = 0
{x1 , x3 , x5 }
x1 = 7
-4
x3 = 1
x2
2
x4
x6
x5 = 1
2
x6 = 0
Oliver Friedmann (LMU)
-1
Zadeh Lower Bound
13
Games and Policy Iteration
Diameter
Question: theoretically possible to have polynomially many iterations?
Let G be a Markov decision process and n be the number of nodes.
Definition: the diameter of G is the least number of iterations required
to solve G
Small Diameter Theorem
The diameter of G is less or equal to n.
Oliver Friedmann (LMU)
Zadeh Lower Bound
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Lower Bound for Zadeh’s Rule
Lower Bound for Zadeh’s Rule
Oliver Friedmann (LMU)
Zadeh Lower Bound
15
Lower Bound for Zadeh’s Rule
Lower bound construction
We define a family of lower bound MDPs Gn such that the
Least-Entered pivoting rule will simulate an n-bit binary
counter.
Oliver Friedmann (LMU)
Zadeh Lower Bound
16
Lower Bound for Zadeh’s Rule
Lower bound construction
We define a family of lower bound MDPs Gn such that the
Least-Entered pivoting rule will simulate an n-bit binary
counter.
We make use of exponentially growing rewards (and penalties): To
get a higher reward the MDP is willing to sacrifice everything that
has been built up so far.
Oliver Friedmann (LMU)
Zadeh Lower Bound
16
Lower Bound for Zadeh’s Rule
Lower bound construction
We define a family of lower bound MDPs Gn such that the
Least-Entered pivoting rule will simulate an n-bit binary
counter.
We make use of exponentially growing rewards (and penalties): To
get a higher reward the MDP is willing to sacrifice everything that
has been built up so far.
Notation: Integer priority p corresponds to reward (−N )p , where
N = 7n + 1.
... < 5 < 3 < 1 < 2 < 4 < 6 < ...
5
Oliver Friedmann (LMU)
for
Zadeh Lower Bound
(−N )5
16
Lower Bound for Zadeh’s Rule
Background
The use of priorities is inspired by parity games.
Oliver Friedmann (LMU)
Zadeh Lower Bound
17
Lower Bound for Zadeh’s Rule
Background
The use of priorities is inspired by parity games.
Friedmann (2009): The strategy iteration algorithm may require
exponentially many iterations to solve parity games.
Fearnley (2010): The strategy iteration algorithm may require
exponentially many iterations to solve MDPs.
Oliver Friedmann (LMU)
Zadeh Lower Bound
17
Lower Bound for Zadeh’s Rule
Background
The use of priorities is inspired by parity games.
Friedmann (2009): The strategy iteration algorithm may require
exponentially many iterations to solve parity games.
Fearnley (2010): The strategy iteration algorithm may require
exponentially many iterations to solve MDPs.
We also first proved a lower bound for parity games and then
transferred the result to MDPs and linear programs.
Oliver Friedmann (LMU)
Zadeh Lower Bound
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Lower Bound for Zadeh’s Rule
Related game-theoretic settings
LP-type problems
Abstract
Concrete
Turn-based stochastic games
21/2 players
Mean payoff games
2 players
Parity games
2 players
Oliver Friedmann (LMU)
Linear programming
Markov decision problems
11/2 players
Deterministic MDPs
1 player
Zadeh Lower Bound
18
Lower Bound for Zadeh’s Rule
Related game-theoretic settings
LP-type problems
Abstract
Concrete
Turn-based stochastic games
21/2 players
Mean payoff games
2 players
Markov decision problems
11/2 players
∈ NP ∩ coNP
Parity games
2 players
Oliver Friedmann (LMU)
Linear programming
∈P
Deterministic MDPs
1 player
Zadeh Lower Bound
18
Lower Bound for Zadeh’s Rule
Zadeh’s pivoting rule
Zadeh’s Least-Entered rule
Perform single switch that has been applied least often.
(taken from David Avis’ paper)
Oliver Friedmann (LMU)
Zadeh Lower Bound
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Lower Bound for Zadeh’s Rule
Tie-Breaking Rule
Tie-Breaking Rule = method of selecting a switch in case of a tie
(w.r.t. the occurrence record)
Proof of Small Diameter Theorem implies:
Corollary
There is a tie-breaking rule s.t. Zadeh’s rule requires linearly many
iterations in the worst-case.
Consequence: lower bound construction is equipped with particular
tie-breaking rule
Oliver Friedmann (LMU)
Zadeh Lower Bound
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Lower Bound for Zadeh’s Rule
Binary Counting
0
0
0
0
0
0
0
0
0
0
0
0
T
Oliver Friedmann (LMU)
0
0
0
0
y
Zadeh Lower Bound
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Lower Bound for Zadeh’s Rule
Binary Counting
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
TPrinciple: If a bit can be set, then all bits can be set.
Oliver Friedmann (LMU)
Zadeh Lower Bound
y
21
Lower Bound for Zadeh’s Rule
Binary Counting
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
T Tie-Breaking: We decide to set the first bit.
Oliver Friedmann (LMU)
Zadeh Lower Bound
1
y
21
Lower Bound for Zadeh’s Rule
Binary Counting
0
0
0
1
0
0
1
0
0
0
1
1
R
1
1
T Set the second bit and reset the first bit.
Oliver Friedmann (LMU)
Zadeh Lower Bound
1
y
21
Lower Bound for Zadeh’s Rule
Binary Counting
0
0
0
1
0
0
1
0
0
0
T
Oliver Friedmann (LMU)
1
0
1
1
Set the first bit again.
Zadeh Lower Bound
0
0
y
21
Lower Bound for Zadeh’s Rule
Binary Counting
0
0
0
1
0
0
1
0
0
0
1
T
Oliver Friedmann (LMU)
1
1
2
1
1
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
0
0
1
R
0
1
1
0
1
1
1
T
Oliver Friedmann (LMU)
1
R
2
1
1
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
0
0
1
0
0
1
0
0
1
1
0
T
Oliver Friedmann (LMU)
1
0
2
0
0
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
0
0
1
0
0
1
0
0
1
1
0
T
Oliver Friedmann (LMU)
1
1
3
1
1
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
0
0
1
1
0
1
1
0
1
1
1
T
Oliver Friedmann (LMU)
2
R
3
1
1
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
0
0
1
1
0
1
1
0
1
1
1
T
Oliver Friedmann (LMU)
2
0
3
0
0
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
0
0
1
1
0
1
1
0
1
1
1
T
Oliver Friedmann (LMU)
2
1
4
1
1
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
1
1
R
R
1
1
1
1
1
1
1
T
Oliver Friedmann (LMU)
2
R
4
1
1
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
1
1
0
0
1
0
0
1
1
0
0
T
Oliver Friedmann (LMU)
2
0
4
0
0
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
1
1
0
0
1
0
0
1
1
0
0
T
Oliver Friedmann (LMU)
2
1
5
1
1
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
1
1
0
1
1
0
1
1
1
0
1
T
Oliver Friedmann (LMU)
3
R
5
1
1
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
1
1
0
1
1
0
1
1
1
0
1
T
Oliver Friedmann (LMU)
3
0
5
0
0
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
1
1
0
1
1
0
1
1
1
0
1
T
Oliver Friedmann (LMU)
3
1
6
1
1
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
1
1
1
R
1
1
1
1
2
1
1
T
Oliver Friedmann (LMU)
3
R
6
1
1
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
1
1
1
0
1
1
0
1
2
1
0
T
Oliver Friedmann (LMU)
3
0
6
0
0
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
1
1
1
0
1
1
0
1
2
1
0
T
Oliver Friedmann (LMU)
3
1
7
1
1
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
1
1
1
1
1
1
1
1
2
1
1
T
Oliver Friedmann (LMU)
4
R
7
1
1
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
1
1
1
1
1
1
1
1
2
1
1
T
Oliver Friedmann (LMU)
4
0
7
0
0
Continue... y
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting
1
1
1
1
1
1
1
2
1
1
T
4
1
1
8
1
1
Problem: Occurrence record unbalanced! y
Oliver Friedmann (LMU)
Zadeh Lower Bound
21
Lower Bound for Zadeh’s Rule
Binary Counting (... again!)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
TLet’s do it again - watch the occurrence record this time!
Oliver Friedmann (LMU)
Zadeh Lower Bound
y
22
Lower Bound for Zadeh’s Rule
Binary Counting (... again!)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
T Everything okay so far... y
Oliver Friedmann (LMU)
Zadeh Lower Bound
22
Lower Bound for Zadeh’s Rule
Binary Counting (... again!)
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
T Everything okay so far... y
Oliver Friedmann (LMU)
Zadeh Lower Bound
22
Lower Bound for Zadeh’s Rule
Binary Counting (... again!)
0
0
0
1
0
0
1
0
0
0
1
1
R
1
1
1
T Everything okay so far... y
Oliver Friedmann (LMU)
Zadeh Lower Bound
22
Lower Bound for Zadeh’s Rule
Binary Counting (... again!)
0
0
0
1
0
0
1
0
0
0
1
1
0
1
0
0
T Problem: We have to set one of the higher bits now! y
Oliver Friedmann (LMU)
Zadeh Lower Bound
22
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
TReplace gadget by two-bit, conjunctive structure.
Oliver Friedmann (LMU)
0
0
0
0
0
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
T Gadget is set iff both edges are going in.
Oliver Friedmann (LMU)
0
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
1
0
0
0
1
1
1
0
1
0
0
0
1
0
0
0
1
1
0
0
0
0
0
T Set one improving edge of every gadget.
Oliver Friedmann (LMU)
0
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
1
0
0
0
1
1
1
1
0
0
0
T
0
0
1
0
0
1
1
1
1
1
0
0
Set other improving edge of first gadget.
Oliver Friedmann (LMU)
1
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
1
0
0
0
0
0
0
0
0
T
1
0
0
1
0
0
0
1
1
1
1
1
1
0
0
Other gadgets have updated to their old setting.
Oliver Friedmann (LMU)
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
1
0
0
0
0
0
0
0
1
T
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1
Set one improving edge of every gadget again.
Oliver Friedmann (LMU)
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
1
0
0
1
0
0
1
1
0
0
1
T
1
1
2
1
1
R
1
1
1
1
1
1
1
Set other improving edge of second gadget.
Oliver Friedmann (LMU)
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
1
0
0
1
0
0
1
1
0
0
1
1
0
T
Oliver Friedmann (LMU)
2
1
0
0
1
0
1
0
0
1
1
Reset all other gadgets.
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
2
0
0
1
1
1
1
2
0
0
1
1
0
T
Oliver Friedmann (LMU)
2
1
0
0
2
1
1
0
0
1
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
2
0
0
1
1
1
1
2
0
0
1
1
0
T
Oliver Friedmann (LMU)
2
1
0
1
2
1
2
1
1
1
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
2
0
0
1
0
0
1
2
0
0
1
1
0
T
Oliver Friedmann (LMU)
2
1
0
1
2
1
2
1
1
1
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
2
0
0
1
0
0
1
2
0
0
2
2
1
T
Oliver Friedmann (LMU)
2
1
1
1
2
1
2
1
1
1
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
2
0
1
R
0
1
1
3
0
1
2
2
1
T
Oliver Friedmann (LMU)
2
1
1
R
2
1
2
1
1
1
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
2
0
1
0
0
1
0
3
0
1
2
2
0
T
Oliver Friedmann (LMU)
2
0
1
0
2
0
2
0
0
1
0
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
2
0
1
0
0
1
0
3
0
1
2
2
0
T
Oliver Friedmann (LMU)
2
0
1
0
2
0
2
0
0
2
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
2
0
1
0
0
1
0
3
0
1
3
2
1
T
Oliver Friedmann (LMU)
2
0
1
0
2
0
3
1
0
2
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
2
0
1
0
0
1
0
3
0
1
3
2
1
T
Oliver Friedmann (LMU)
2
0
1
1
3
1
3
1
1
2
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
2
0
1
0
0
1
0
3
0
1
3
2
0
T
Oliver Friedmann (LMU)
2
0
1
1
3
1
3
1
1
2
0
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
3
0
1
0
1
1
1
3
0
1
3
2
0
T
Oliver Friedmann (LMU)
3
0
1
1
3
1
3
1
1
2
0
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
3
0
1
1
1
1
1
3
0
1
3
2
0
T
Oliver Friedmann (LMU)
3
1
1
R
3
1
3
1
1
3
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
3
0
1
1
0
1
1
3
0
1
3
2
0
T
Oliver Friedmann (LMU)
3
1
1
0
3
0
3
0
0
3
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
4
0
1
1
1
1
1
3
0
1
3
2
0
T
Oliver Friedmann (LMU)
3
1
1
0
4
1
3
0
0
3
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
4
0
1
1
1
1
1
3
0
1
3
2
0
T
Oliver Friedmann (LMU)
3
1
1
1
4
1
4
1
1
3
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
4
0
1
1
0
1
1
3
0
1
3
2
0
T
Oliver Friedmann (LMU)
3
1
1
1
4
1
4
1
1
3
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
4
0
1
1
0
1
1
3
0
1
4
2
1
T
Oliver Friedmann (LMU)
3
1
1
1
4
1
4
1
1
3
1
Everything okay so far, continue...
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
5
1
R
R
1
1
1
3
1
1
4
2
1
T
Oliver Friedmann (LMU)
3
1
1
R
4
1
4
1
1
3
1
Reset all three lower gadgets.
Zadeh Lower Bound
y
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
5
1
0
0
1
0
0
3
1
0
4
T
2
1
3
0
0
0
4
0
4
0
0
3
0
Occurrence record of gadget #3 is pretty low... y
Oliver Friedmann (LMU)
Zadeh Lower Bound
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
5
1
0
0
1
0
0
3
1
0
4
T
3
1
3
0
1
0
4
0
4
0
0
3
0
Occurrence record of gadget #3 is pretty low... y
Oliver Friedmann (LMU)
Zadeh Lower Bound
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive bits
5
1
0
0
1
0
0
3
1
0
4
T
3
1
3
0
1
0
4
0
4
0
0
4
1
Problem: We have to set gadget #2 or #3 now! y
Oliver Friedmann (LMU)
Zadeh Lower Bound
23
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
TReplace gadget by two conjunctive structures. y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
0
0
1
1
0
0
0
0
0
0
0
0
0
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
0
1
0
0
1
0
0
0
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
0
0
1
0
1
0
1
0
0
0
0
0
0
0
1
0
1
0
1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
0
0
0
1
0
0
1
0
0
0
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
0
0
1
0
1
0
1
0
1
1
1
1
1
1
1
0
1
0
1
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
1
1
1
0
1
1
0
1
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
1
0
0
1
0
0
1
0
1
0
1
1
1
0
0
1
1
1
0
0
1
1
2
1
1
1
1
1
1
1
0
1
1
1
1
1
1
0
0
0
1
R
1
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
1
0
0
1
0
0
1
0
1
0
1
0
1
0
0
1
0
1
0
0
1
0
2
1
0
1
0
1
0
1
0
1
0
0
1
1
1
0
0
0
1
0
1
0
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
2
0
0
1
1
1
1
0
2
0
1
0
2
1
0
1
0
2
1
0
1
0
2
1
0
2
1
1
0
2
1
1
0
0
1
1
2
1
0
0
1
0
1
0
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
2
0
0
1
1
1
1
0
2
0
1
0
2
1
0
1
0
2
1
0
1
0
2
1
0
2
1
1
0
2
1
2
1
0
1
1
2
1
1
0
1
1
1
0
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
2
0
0
1
0
0
1
0
2
0
1
0
2
0
0
1
0
2
0
0
1
0
2
1
0
2
0
1
0
2
1
2
1
0
1
1
2
0
1
0
1
1
1
0
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
2
0
0
1
0
0
1
0
2
0
2
1
2
0
0
2
1
2
0
0
2
1
2
1
1
2
0
2
1
2
1
2
1
0
1
1
2
0
1
0
2
1
2
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
1
2
0
3
1
2
1
2
1
2
0
2
0
0
1
0
1
2
1
1
1
2
0
0
2
1
R
2
0
2
1
2
1
2
1
0
1
0
2
R
1
2
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
1
2
0
3
1
2
0
2
1
2
0
2
0
0
1
0
0
2
0
1
0
2
0
0
2
0
0
2
0
2
0
2
0
2
0
0
0
0
2
0
0
2
0
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
1
2
0
3
1
3
1
2
1
2
0
2
0
0
1
0
1
2
0
2
1
2
0
0
3
1
0
2
0
3
1
2
0
3
1
0
0
0
3
0
0
3
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
1
2
0
3
1
3
1
2
1
2
0
2
0
0
1
0
1
3
1
2
1
2
0
0
3
1
0
2
0
3
1
2
0
3
1
0
1
0
3
0
0
3
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
1
2
0
3
1
3
1
2
1
2
0
2
0
0
1
0
1
3
1
2
1
2
0
0
3
1
1
3
1
3
1
2
0
3
1
1
1
0
3
0
0
3
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
1
2
0
3
1
3
0
2
1
2
0
2
0
0
1
0
0
3
0
2
0
2
0
0
3
0
1
3
1
3
1
2
0
3
0
1
0
0
3
0
0
3
0
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
0
1
3
1
3
1
3
0
2
1
3
1
3
1
0
1
0
0
3
0
3
1
3
1
0
3
0
1
3
1
3
1
3
1
3
0
1
0
0
3
0
0
3
0
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
3
0
1
1
1
0
3
1
3
0
3
1
0
0
3
2
1
3
1
0
3
1
3
1
3
1
3
0
1
3
1
3
1
1
3
R
0
0
0
3
1
0
4
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
3
0
1
0
1
0
3
1
3
0
3
0
0
0
3
2
1
3
0
0
3
0
3
0
3
0
3
0
0
3
0
3
1
1
3
0
0
0
0
3
1
0
4
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
4
0
1
1
1
0
3
1
3
0
4
1
0
0
4
2
1
4
1
0
4
1
3
0
4
1
3
0
0
3
0
3
1
1
3
0
1
0
0
3
1
0
4
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
4
0
1
1
1
0
3
1
3
0
4
1
0
0
4
2
1
4
1
0
4
1
3
0
4
1
4
1
0
3
0
3
1
1
3
1
1
0
0
3
1
1
4
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
4
0
1
0
1
0
3
1
3
0
4
0
0
0
4
2
1
4
0
0
4
0
3
0
4
1
4
1
0
3
0
3
1
1
3
1
0
0
0
3
1
1
4
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
4
0
1
0
1
0
3
1
4
1
4
0
0
1
4
2
1
4
0
1
4
0
4
1
4
1
4
1
0
4
1
3
1
1
4
1
0
0
0
4
1
1
4
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
1
5
1
4
1
4
0
1
R
R
3
1
4
0
2
1
4
1
4
0
3
1
0
1
0
0
4
1
1
4
0
4
1
4
1
4
1
0
1
4
R
1
4
1
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Binary Counting with conjunctive representatives
1
5
1
4
1
4
0
1
0
0
3
0
4
0
2
0
4
0
4
0
3
0
0
0
0
0
4
0
0
4
0
4
0
4
0
4
0
0
0
4
0
0
4
0
T Only one representative subgadget is active. The upper
representative is active iff the next higher bit is not set y
Oliver Friedmann (LMU)
Zadeh Lower Bound
24
Lower Bound for Zadeh’s Rule
Bit Gadget
b
ε
x
1−ε
2
1−ε
2
..
.
y
a
Oliver Friedmann (LMU)
Zadeh Lower Bound
25
Lower Bound for Zadeh’s Rule
Bit Gadget
b
ε
x
1−ε
2
Bit unset
..
.
1−ε
2
Improving to go in
y
a
valσ (a) = ε · valσ (b) + (1 − ε) ·valσ (x) ≈ valσ (x)
| {z } | {z }
≈0
Oliver Friedmann (LMU)
≈1
Zadeh Lower Bound
25
Lower Bound for Zadeh’s Rule
Bit Gadget
b
ε
x
1−ε
2
..
.
1−ε
2
y
Bit still unset (only one edge
going in)
Still improving to go in with
the other edge
a
valσ (a) =
2ε
1−ε
· valσ (b) +
·valσ (x) ≈ valσ (x)
1+ε
1+ε
|
{z
} | {z }
≈0
Oliver Friedmann (LMU)
≈1
Zadeh Lower Bound
25
Lower Bound for Zadeh’s Rule
Bit Gadget
b
ε
x
1−ε
2
1−ε
2
..
.
y
y has now better valuation
than x
Gadget could close, but also
open completely again
a
Oliver Friedmann (LMU)
Zadeh Lower Bound
25
Lower Bound for Zadeh’s Rule
Bit Gadget
b
ε
x
1−ε
2
Bit unset
1−ε
2
..
.
No improvements
y
a
Oliver Friedmann (LMU)
Zadeh Lower Bound
25
Lower Bound for Zadeh’s Rule
Bit Gadget
b
ε
x
1−ε
2
Bit unset
1−ε
2
..
.
Improving to go in
y
a
Oliver Friedmann (LMU)
Zadeh Lower Bound
25
Lower Bound for Zadeh’s Rule
Bit Gadget
b
ε
x
1−ε
2
Bit set
1−ε
2
..
.
Now b can be “observed” from
a
y
a
valσ (a) = valσ (b)
Oliver Friedmann (LMU)
Zadeh Lower Bound
25
Lower Bound for Zadeh’s Rule
Full Construction
t
kn+1
2n+9
k[1;n]
k[i+2;n]
ki+1
t
h0i
2i+8
h1i
2i+8
s
d0i
6
d1i
6
b0i,0
1−ε
2
ε
ε
A0i
b1i,0
k[1;n]
1−ε
2
b0i,1
b1i,1
c0i
7
k[1;n]
t
ki
2i+7
Oliver Friedmann (LMU)
1−ε
2
A1i
1−ε
2
t
s
Zadeh Lower Bound
c1i
7
t
k[1;n]
t
s
26
Concluding Remarks
Concluding Remarks
Oliver Friedmann (LMU)
Zadeh Lower Bound
27
Concluding Remarks
Open problems
Obtain lower bounds for related history-based pivoting rules
Least-recently considered: subexponential lower bound
Least-recently basic, Least-recently entered, Least basic iterations:
work in progress
Polytime algorithm for two-player games and the like
Strongly polytime algorithm for LPs (and MDPs)
Resolving the Hirsch conjecture
Find game-theoretic model with unresolved diameter bounds
Oliver Friedmann (LMU)
Zadeh Lower Bound
28
Concluding Remarks
The slide usually called “the end”.
Thank you for listening!
Oliver Friedmann (LMU)
Zadeh Lower Bound
29