A subexponential lower bound for Zadehs pivoting rule for solving - - PowerPoint PPT Presentation

A subexponential lower bound for Zadeh’s pivoting rule for solving linear programs and games

Oliver Friedmann

Department of Computer Science, Ludwig-Maximilians-Universit¨ at 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 cTx subject to Ax ≤ b

Oliver Friedmann (LMU) Zadeh Lower Bound 3

Linear Programming and Simplex

Basic feasible solutions and pivoting

max 2x1 − 2x3 − 2x5 − x6 s.t.

1 3x1 + x2 − 2 3x3 − 2 3x5 = 1

x3 + x4 − x6 = 1 x5 + x6 = 1 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

Basic feasible solutions and pivoting

max −1 + 2x1 − 2x3 − x5 s.t. x2 = 1 − 1

3x1 + 2 3x3 + 2 3x5

x4 = 2 − x3 − x5 x6 = 1 − x5 x1, x2, x3, x4, x5, x6 ≥ 0

{x2, x4, x6} x1 x3 x5

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

{x1, x4, x6} x2 x3 x5

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

{x1, x3, x6} x2 x4 x5

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 10 − 6x2 − 2x4 − x6 s.t. x1 = 7 − 3x2 − 2x4 x3 = 1 − x4 + x6 x5 = 1 − x6 x1, x2, x3, x4, x5, x6 ≥ 0

{x1, x3, x5} 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 7

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Reward: −1

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Reward: −1

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Reward: −1

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Reward: −1 − 4

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Reward: −1 − 4

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Reward: −1 − 4

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Reward: −1 − 4

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Reward: −1 − 4

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Reward: −1 − 4

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Shapley (1953), Bellman (1957): There exists an optimal history-independent choice from each state. Reward: −1 − 4

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Reward: −1 − 4

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

Reward: −1 − 4 + 6

Oliver Friedmann (LMU) Zadeh Lower Bound 8

Games and Policy Iteration

Markov decision processes (MDPs)

t 6

• 4
• 1

2 3 1 3

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

• 4
• 1

2 3 1 3

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.

t 6

• 4
• 1

2 3 1 3

2

• 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. An action is an improving switch w.r.t. π if it improves the values.

t 6

• 4
• 1

2 3 1 3

2

• 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. 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.

t 6

• 4
• 1

2 3 1 3

2

• 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. 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.

t 6

• 4
• 1

2 3 1 3

6 6

• 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. 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.

t 6

• 4
• 1

2 3 1 3

6 6 2 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. 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.

t 6

• 4
• 1

2 3 1 3

6 6 2 2

Oliver Friedmann (LMU) Zadeh Lower Bound 9

Games and Policy Iteration

MDPs and linear programming

No improving switches for optimal policy π∗: ∀i ∈ S : valπ∗(i) = max

a∈Ai ra +

• j∈S

pa,jvalπ∗(j) 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∗: ∀i ∈ S : valπ∗(i) = max

a∈Ai ra +

• j∈S

pa,jvalπ∗(j) 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: minimize

• i∈S

vi s.t. ∀i ∈ S ∀a ∈ Ai : vi ≥ ra +

• j∈S

pa,jvj

Oliver Friedmann (LMU) Zadeh Lower Bound 10

Games and Policy Iteration

Primal and dual LPs for MDPs

minimize

• i∈S

vi s.t. ∀i ∈ S ∀a ∈ Ai : vi ≥ ra +

• j∈S

pa,jvj maximize

• i∈S
• a∈Ai

raxa s.t. ∀i ∈ S :

• a∈Ai

xa = 1 +

• j∈S
• a∈Aj

pa,ixa

Oliver Friedmann (LMU) Zadeh Lower Bound 11

Games and Policy Iteration

Primal and dual LPs for MDPs

minimize

• i∈S

vi s.t. ∀i ∈ S ∀a ∈ Ai : vi ≥ ra +

• j∈S

pa,jvj maximize

• i∈S
• a∈Ai

raxa s.t. ∀i ∈ S :

• a∈Ai

xa = 1 +

• j∈S
• a∈Aj

pa,ixa Flow conservation:

x1 = 1 x2 = 6 x3 = 4 x4 = 2

x1 + x2 = 1 + x3 + x4

Oliver Friedmann (LMU) Zadeh Lower Bound 11

Games and Policy Iteration

Primal and dual LPs for MDPs

minimize

• i∈S

vi s.t. ∀i ∈ S ∀a ∈ Ai : vi ≥ ra +

• j∈S

pa,jvj maximize

• i∈S
• a∈Ai

raxa s.t. ∀i ∈ S :

• a∈Ai

xa = 1 +

• j∈S
• a∈Aj

pa,ixa Flow conservation:

x1 = 7 x2 = 0 x3 = 4 x4 = 2

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

• 4
• 1

2 3 1 3

2

• 1

x1 = 0 x2 = 1 x3 = 0 x4 = 2 x5 = 0 x6 = 1

xa is the expected number of times action a is used, summed

• ver all starting states.

Oliver Friedmann (LMU) Zadeh Lower Bound 12

Games and Policy Iteration

Variables of the primal LP

t 6

• 4
• 1

2 3 1 3

2

• 1

x1 = 0 x2 = 1 x3 = 0 x4 = 2 x5 = 0 x6 = 1

xa is the expected number of times action a is used, summed

• ver all starting states.

We have:

• i∈S

valπ(i) =

• a∈π

raxπ

a

Oliver Friedmann (LMU) Zadeh Lower Bound 12

Games and Policy Iteration

From MDP to LP

t 6

• 4
• 1

2 3 1 3

2

• 1

x1 = 0 x2 = 1 x3 = 0 x4 = 2 x5 = 0 x6 = 1

max −1 + 2x1 − 2x3 − x5 s.t. x2 = 1 − 1

3x1 + 2 3x3 + 2 3x5

x4 = 2 − x3 − x5 x6 = 1 − x5 x1, x2, x3, x4, x5, x6 ≥ 0

{x2, x4, x6} x1 x3 x5

Oliver Friedmann (LMU) Zadeh Lower Bound 13

Games and Policy Iteration

From MDP to LP

t 6

• 4
• 1

2 3 1 3

6 6

• 1

x1 = 3 x2 = 0 x3 = 0 x4 = 2 x5 = 0 x6 = 1

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

{x1, x4, x6} x2 x3 x5

Oliver Friedmann (LMU) Zadeh Lower Bound 13

Games and Policy Iteration

From MDP to LP

t 6

• 4
• 1

2 3 1 3

6 6 2 1 x1 = 7 x2 = 0 x3 = 2 x4 = 0 x5 = 0 x6 = 1

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

{x1, x3, x6} x2 x4 x5

Oliver Friedmann (LMU) Zadeh Lower Bound 13

Games and Policy Iteration

From MDP to LP

t 6

• 4
• 1

2 3 1 3

6 6 2 2 x1 = 7 x2 = 0 x3 = 1 x4 = 0 x5 = 1 x6 = 0

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

{x1, x3, x5} x2 x4 x6

Oliver Friedmann (LMU) 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 14

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 for (−N)5

Oliver Friedmann (LMU) Zadeh Lower Bound 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 17

Lower Bound for Zadeh’s Rule

Related game-theoretic settings

Abstract LP-type problems Concrete Linear programming Turn-based stochastic games 21/2 players Mean payoff games 2 players Markov decision problems 11/2 players Parity games 2 players Deterministic MDPs 1 player

Oliver Friedmann (LMU) Zadeh Lower Bound 18

Lower Bound for Zadeh’s Rule

Related game-theoretic settings

Abstract LP-type problems Concrete Linear programming Turn-based stochastic games 21/2 players Mean payoff games 2 players Markov decision problems 11/2 players Parity games 2 players Deterministic MDPs 1 player ∈ NP ∩ coNP ∈ P

Oliver Friedmann (LMU) 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 19

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 20

Lower Bound for Zadeh’s Rule

Binary Counting

T y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

TPrinciple: If a bit can be set, then all bits can be set. y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1

1 1 1

T Tie-Breaking: We decide to set the first bit. y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 R

1 1 1 1 1 1

T Set the second bit and reset the first bit. y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1

1 1 1 1

T Set the first bit again. y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1

1 1 1 1 1 2

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 R R

1 1 1 1 1 1 1 1 2

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1

1 1 1 1 2

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1

1 1 1 1 1 1 3

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1 R

1 1 1 1 1 2 1 1 3

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1

1 1 1 1 1 2 3

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1 1

1 1 1 1 1 2 1 1 4

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 R R R

1 1 1 1 1 1 1 1 2 1 1 4

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1

1 1 1 1 2 4

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1

1 1 1 1 2 1 1 5

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1 R

1 1 1 1 1 1 3 1 1 5

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1

1 1 1 1 1 1 3 5

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1 1

1 1 1 1 1 1 3 1 1 6

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1 R R

1 1 1 1 1 2 1 1 3 1 1 6

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1

1 1 1 1 1 2 3 6

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1 1

1 1 1 1 1 2 3 1 1 7

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1 1 R

1 1 1 1 1 2 1 1 4 1 1 7

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1 1

1 1 1 1 1 2 1 1 4 7

T Continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting

1 1 1 1

1 1 1 1 1 2 1 1 4 1 1 8

T Problem: Occurrence record unbalanced! y

Oliver Friedmann (LMU) Zadeh Lower Bound 21

Lower Bound for Zadeh’s Rule

Binary Counting (... again!)

TLet’s do it again - watch the occurrence record this time! y

Oliver Friedmann (LMU) Zadeh Lower Bound 22

Lower Bound for Zadeh’s Rule

Binary Counting (... again!)

T Everything okay so far... y

Oliver Friedmann (LMU) Zadeh Lower Bound 22

Lower Bound for Zadeh’s Rule

Binary Counting (... again!)

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!)

1 R

1 1 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!)

1

1 1 1 1

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

TReplace gadget by two-bit, conjunctive structure. y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

T Gadget is set iff both edges are going in. y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1 1 1 1 1 1 1

T Set one improving edge of every gadget. y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1

1 1 1 1 1 1 1 1 1 1 1

T Set other improving edge of first gadget. y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1

1 1 1 1 1 1 1 1

T Other gadgets have updated to their old setting. y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

T Set one improving edge of every gadget again. y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 R

1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1

T Set other improving edge of second gadget. y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1

1 1 1 1 1 1 1 1 2 1 1

T Reset all other gadgets. y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1

1 1 2 1 1 2 1 1 1 1 2 1 1 2

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1

1 1 2 1 1 2 1 1 1 1 2 1 1 1 2 2

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1

1 2 1 2 1 1 1 1 2 1 1 1 2 2

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1

1 2 2 1 2 2 1 1 1 1 2 1 1 1 2 2

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 R R

1 2 2 1 1 1 2 3 1 1 1 1 2 1 1 1 2 2

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1

2 2 1 1 1 2 3 1 2 2 2

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1

2 2 1 1 1 2 3 1 2 2 2 2

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1

1 3 2 1 1 1 2 3 1 2 2 1 3 2

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1

1 3 2 1 1 1 2 3 1 2 2 1 1 1 3 3

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1

3 2 1 1 1 2 3 2 2 1 1 1 3 3

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1

1 3 3 1 1 1 2 3 1 2 3 1 1 1 3 3

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1 R

1 3 3 1 1 1 2 3 1 1 1 3 3 1 1 1 3 3

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1

3 3 1 1 1 2 3 1 1 1 3 3 3 3

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1

1 3 4 1 1 1 2 3 1 1 1 3 3 1 3 4

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1 1

1 3 4 1 1 1 2 3 1 1 1 3 3 1 1 1 4 4

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1 1

3 4 1 1 1 2 3 1 1 1 3 3 1 1 1 4 4

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 1 1

1 4 4 1 1 1 2 3 1 1 1 3 3 1 1 1 4 4

T Everything okay so far, continue... y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1 R R R

1 1 1 4 5 1 1 1 2 3 1 1 1 3 3 1 1 1 4 4

T Reset all three lower gadgets. y

Oliver Friedmann (LMU) Zadeh Lower Bound 23

Lower Bound for Zadeh’s Rule

Binary Counting with conjunctive bits

1

1 1 1 4 5 2 3 3 3 4 4

T 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

1

1 1 1 4 5 1 3 3 3 3 4 4

T 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

1

1 1 1 4 5 1 3 3 1 4 3 4 4

T 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

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

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 1 1 1 1 1 1 1 1 1 1 1 1 1 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

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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

1 1 1 1 1 1 1 1 1 1 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

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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 R

1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 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

1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 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

1 1 2 1 1 2 1 1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2

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 1

1 1 2 1 1 2 1 1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 1 2 2

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 1

1 2 1 2 1 1 1 1 2 1 2 1 2 1 2 1 2 1 1 1 2 2

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 1

1 2 2 1 2 2 1 1 1 1 2 1 2 2 1 2 2 1 2 2 1 2 2 1 1 1 2 2

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 R R

1 2 2 1 1 1 2 3 1 1 1 1 2 1 2 2 1 2 2 1 2 2 1 2 2 1 1 1 2 2

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

2 2 1 1 1 2 3 1 2 2 2 2 2 2 2 2 2 2 2

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

1 3 2 1 1 1 2 3 1 2 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2

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

1 3 2 1 1 1 2 3 1 1 1 2 3 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2

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 1

1 3 2 1 1 1 2 3 1 1 1 2 3 1 1 1 3 3 1 3 2 1 3 2 1 3 2 1 3 2

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 1

3 2 1 1 1 2 3 2 3 1 1 1 3 3 3 2 3 2 3 2 3 2

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 1

1 3 3 1 1 1 2 3 1 3 3 1 1 1 3 3 1 3 3 1 3 3 1 3 3 1 3 3

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 1 R

1 3 3 1 1 1 2 3 1 3 3 1 1 1 3 3 1 3 3 1 3 3 1 1 1 4 3 1 3 3

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 1

3 3 1 1 1 2 3 3 3 3 3 3 3 3 3 1 1 1 4 3 3 3

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 1

1 3 4 1 1 1 2 3 1 3 4 1 3 4 1 3 4 1 3 4 1 1 1 4 3 1 3 4

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 1 1

1 3 4 1 1 1 2 3 1 3 4 1 3 4 1 3 4 1 3 4 1 1 1 4 3 1 1 1 4 4

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 1 1

3 4 1 1 1 2 3 3 4 3 4 3 4 3 4 1 1 1 4 3 1 1 1 4 4

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 1 1

1 4 4 1 1 1 2 3 1 4 4 1 4 4 1 4 4 1 4 4 1 1 1 4 3 1 1 1 4 4

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 R R R

1 1 1 4 5 1 1 1 2 3 1 4 4 1 4 4 1 4 4 1 4 4 1 1 1 4 3 1 1 1 4 4

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

1 1 1 4 5 2 3 4 4 4 4 4 4 4 4 4 3 4 4

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 a x y . . . ε

1−ε 2 1−ε 2

Oliver Friedmann (LMU) Zadeh Lower Bound 25

Lower Bound for Zadeh’s Rule

Bit Gadget

b a x y . . . ε

1−ε 2 1−ε 2

Bit unset Improving to go in valσ(a) = ε · valσ(b)

• ≈0

+ (1 − ε)

≈1

·valσ(x) ≈ valσ(x)

Oliver Friedmann (LMU) Zadeh Lower Bound 25

Lower Bound for Zadeh’s Rule

Bit Gadget

b a x y . . . ε

1−ε 2 1−ε 2

Bit still unset (only one edge going in) Still improving to go in with the other edge valσ(a) = 2ε 1 + ε · valσ(b)

• ≈0

+ 1 − ε 1 + ε

≈1

·valσ(x) ≈ valσ(x)

Oliver Friedmann (LMU) Zadeh Lower Bound 25

Lower Bound for Zadeh’s Rule

Bit Gadget

b a x y . . . ε

1−ε 2 1−ε 2

y has now better valuation than x Gadget could close, but also

• pen completely again

Oliver Friedmann (LMU) Zadeh Lower Bound 25

Lower Bound for Zadeh’s Rule

Bit Gadget

b a x y . . . ε

1−ε 2 1−ε 2

Bit unset No improvements

Oliver Friedmann (LMU) Zadeh Lower Bound 25

Lower Bound for Zadeh’s Rule

Bit Gadget

b a x y . . . ε

1−ε 2 1−ε 2

Bit unset Improving to go in

Oliver Friedmann (LMU) Zadeh Lower Bound 25

Lower Bound for Zadeh’s Rule

Bit Gadget

b a x y . . . ε

1−ε 2 1−ε 2

Bit set Now b can be “observed” from a valσ(a) = valσ(b)

Oliver Friedmann (LMU) Zadeh Lower Bound 25

Lower Bound for Zadeh’s Rule

Full Construction

s t k[1;n] b0

i,1

ki 2i+7 t k[1;n] c0

i

7 c1

i

7 b1

i,1

k[1;n] t A0

i

A1

i

t k[1;n] b0

i,0

d0

i

6 d1

i

6 b1

i,0

t h0

i

2i+8 s s h1

i

2i+8 k[i+2;n] ki+1 ε ε

1−ε 2 1−ε 2 1−ε 2 1−ε 2

kn+1 2n+9 t

Oliver Friedmann (LMU) Zadeh Lower Bound 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