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Physics of Algorithms, Santa Fe 2009 Replica Symmetry and Combinatorial Optimization Johan W astlund Physics of Algorithms, Santa Fe 2009 Johan W astlund Replica Symmetry and Combinatorial Optimization Physics of Algorithms, Santa Fe

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Physics of Algorithms, Santa Fe 2009

Replica Symmetry and Combinatorial Optimization

Johan W¨ astlund Physics of Algorithms, Santa Fe 2009

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Comments (like this one) have been added in order to make sense

  • f some of the slides. Some of those comments represent what I

said, or might have said, in the talk. The talk is based on the paper arXiv:0908.1920.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Mean Field Model of Distance

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Mean Field Model of Distance

i.i.d edge lengths, say uniform [0, 1]

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Mean Field Model of Distance

i.i.d edge lengths, say uniform [0, 1] Quenched disorder

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Optimization problems

Minimize total length:

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Optimization problems

Minimize total length: Spanning Tree

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Optimization problems

Minimize total length: Spanning Tree Matching

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Optimization problems

Minimize total length: Spanning Tree Matching Traveling Salesman

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Optimization problems

Minimize total length: Spanning Tree Matching Traveling Salesman 2-factor

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Optimization problems

Minimize total length: Spanning Tree Matching Traveling Salesman 2-factor Edge Cover

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Replica/cavity method

Replica symmetric prediction of the length of the optimum solution for large N. (Sketch of the background with a certain bias towards people in the audience...)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Replica/cavity method

Replica symmetric prediction of the length of the optimum solution for large N. (Sketch of the background with a certain bias towards people in the audience...) M´ ezard-Parisi (1985–87), Matching, TSP

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Replica/cavity method

Replica symmetric prediction of the length of the optimum solution for large N. (Sketch of the background with a certain bias towards people in the audience...) M´ ezard-Parisi (1985–87), Matching, TSP Krauth-M´ ezard (1989), TSP

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Replica/cavity method

Replica symmetric prediction of the length of the optimum solution for large N. (Sketch of the background with a certain bias towards people in the audience...) M´ ezard-Parisi (1985–87), Matching, TSP Krauth-M´ ezard (1989), TSP Predictions tested by N. Sourlas, A. Percus, O. Martin,

  • S. Boettcher,...

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Replica/cavity method

Replica symmetric prediction of the length of the optimum solution for large N. (Sketch of the background with a certain bias towards people in the audience...) M´ ezard-Parisi (1985–87), Matching, TSP Krauth-M´ ezard (1989), TSP Predictions tested by N. Sourlas, A. Percus, O. Martin,

  • S. Boettcher,...

Success of BP, D. Shah, M. Bayati,...

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Giorgio Parisi in Les Houches 1986, having calculated the π2/12 limit for minimum matching.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Limits in pseudo-dimension 1

Limit costs for uniform [0, 1] edge lengths (no normalization needed!)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Limits in pseudo-dimension 1

Limit costs for uniform [0, 1] edge lengths (no normalization needed!) Matching

π2 12 ≈ 0.822 M´

ezard-Parisi, Aldous

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Limits in pseudo-dimension 1

Limit costs for uniform [0, 1] edge lengths (no normalization needed!) Matching

π2 12 ≈ 0.822 M´

ezard-Parisi, Aldous Spanning tree ζ(3) ≈ 1.202 Frieze

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Limits in pseudo-dimension 1

Limit costs for uniform [0, 1] edge lengths (no normalization needed!) Matching

π2 12 ≈ 0.822 M´

ezard-Parisi, Aldous Spanning tree ζ(3) ≈ 1.202 Frieze TSP/2-factor ∞

0 ydx, (1 + x/2)e−x + (1 + y/2)e−y = 1

≈ 2.0415 Krauth-M´ ezard-Parisi, W¨ astlund (TSP ∼ 2-factor proved by Frieze)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Limits in pseudo-dimension 1

Limit costs for uniform [0, 1] edge lengths (no normalization needed!) Matching

π2 12 ≈ 0.822 M´

ezard-Parisi, Aldous Spanning tree ζ(3) ≈ 1.202 Frieze TSP/2-factor ∞

0 ydx, (1 + x/2)e−x + (1 + y/2)e−y = 1

≈ 2.0415 Krauth-M´ ezard-Parisi, W¨ astlund (TSP ∼ 2-factor proved by Frieze) Edge cover

1 2 min(x2 + e−x) ≈ 0.728 Not yet proved, but

the bipartite case is done in joint work with M. Hessler

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Pseudo-dimension d

Pseudo-dimension d means P(l < r) ∝ rd as r → 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Pseudo-dimension d

Pseudo-dimension d means P(l < r) ∝ rd as r → 0 For general d take lij = (N · Xij)1/d where Xij is exponential(1). This normalization is different from a moment ago! Now unit of length = distance at which expected number of neighbors equals 1.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Pseudo-dimension d

Pseudo-dimension d means P(l < r) ∝ rd as r → 0 For general d take lij = (N · Xij)1/d where Xij is exponential(1). This normalization is different from a moment ago! Now unit of length = distance at which expected number of neighbors equals 1. Theorem For d ≥ 1, Cost[Matching] N/2

p

− → βM(d)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Pseudo-dimension d

Pseudo-dimension d means P(l < r) ∝ rd as r → 0 For general d take lij = (N · Xij)1/d where Xij is exponential(1). This normalization is different from a moment ago! Now unit of length = distance at which expected number of neighbors equals 1. Theorem For d ≥ 1, Cost[Matching] N/2

p

− → βM(d) Cost[TSP] N

p

− → βTSP(d)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Pseudo-dimension d

Pseudo-dimension d means P(l < r) ∝ rd as r → 0 For general d take lij = (N · Xij)1/d where Xij is exponential(1). This normalization is different from a moment ago! Now unit of length = distance at which expected number of neighbors equals 1. Theorem For d ≥ 1, Cost[Matching] N/2

p

− → βM(d) Cost[TSP] N

p

− → βTSP(d) Replica symmetric predictions of βM(d) and βTSP(d) are correct

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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The right hand side is the average length of an edge in the solution, measured in the length unit at which the expected number of neighbors is 1. What is new is that the theorem holds also for d > 1, but I will not say so much about the parameter d in the rest of the talk.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Graph Exploration

2-person zero-sum game: θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Graph Exploration

2-person zero-sum game: Alice and Bob take turns choosing edges of a self-avoiding walk θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Graph Exploration

2-person zero-sum game: Alice and Bob take turns choosing edges of a self-avoiding walk They pay the length of the chosen edge to the

  • pponent,

θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Graph Exploration

2-person zero-sum game: Alice and Bob take turns choosing edges of a self-avoiding walk They pay the length of the chosen edge to the

  • pponent,
  • r terminate by paying

θ/2 to the opponent θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Graph Exploration

2-person zero-sum game: Alice and Bob take turns choosing edges of a self-avoiding walk They pay the length of the chosen edge to the

  • pponent,
  • r terminate by paying

θ/2 to the opponent θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Graph Exploration

2-person zero-sum game: Alice and Bob take turns choosing edges of a self-avoiding walk They pay the length of the chosen edge to the

  • pponent,
  • r terminate by paying

θ/2 to the opponent θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Graph Exploration

2-person zero-sum game: Alice and Bob take turns choosing edges of a self-avoiding walk They pay the length of the chosen edge to the

  • pponent,
  • r terminate by paying

θ/2 to the opponent θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Graph Exploration

2-person zero-sum game: Alice and Bob take turns choosing edges of a self-avoiding walk They pay the length of the chosen edge to the

  • pponent,
  • r terminate by paying

θ/2 to the opponent θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Graph Exploration

2-person zero-sum game: Alice and Bob take turns choosing edges of a self-avoiding walk They pay the length of the chosen edge to the

  • pponent,
  • r terminate by paying

θ/2 to the opponent θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Graph Exploration

2-person zero-sum game: Alice and Bob take turns choosing edges of a self-avoiding walk They pay the length of the chosen edge to the

  • pponent,
  • r terminate by paying

θ/2 to the opponent θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Graph Exploration

2-person zero-sum game: Alice and Bob take turns choosing edges of a self-avoiding walk They pay the length of the chosen edge to the

  • pponent,
  • r terminate by paying

θ/2 to the opponent Edges longer than θ are irrelevant! θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Diluted Matching Problem

Optimization: θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Diluted Matching Problem

Optimization: Partial matching θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Diluted Matching Problem

Optimization: Partial matching Cost = total length of edges + θ/2 for each unmatched vertex θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Diluted Matching Problem

Optimization: Partial matching Cost = total length of edges + θ/2 for each unmatched vertex Feasible solutions exist also for odd N θ ≥ 0

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Solution to Graph Exploration

Fix θ and edge costs

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Solution to Graph Exploration

Fix θ and edge costs M(G) = cost of diluted matching problem

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Solution to Graph Exploration

Fix θ and edge costs M(G) = cost of diluted matching problem f (G, v) = Bob’s payoff under optimal play from v

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Solution to Graph Exploration

Fix θ and edge costs M(G) = cost of diluted matching problem f (G, v) = Bob’s payoff under optimal play from v Lemma f (G, v) = M(G) − M(G − v)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Solution to Graph Exploration

Lemma f (G, v) = M(G) − M(G − v)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Solution to Graph Exploration

Lemma f (G, v) = M(G) − M(G − v) Proof. f (G, v) = min(θ/2, li − f (G − v, vi))

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Solution to Graph Exploration

Lemma f (G, v) = M(G) − M(G − v) Proof. f (G, v) = min(θ/2, li − f (G − v, vi)) M(G) = min(θ/2 + M(G − v), li + M(G − v − vi))

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Solution to Graph Exploration

Lemma f (G, v) = M(G) − M(G − v) Proof. f (G, v) = min(θ/2, li − f (G − v, vi)) M(G) = min(θ/2 + M(G − v), li + M(G − v − vi)) M(G) − M(G − v) = min(θ/2, li − (M(G − v) − M(G − v − vi)))

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Solution to Graph Exploration

Lemma f (G, v) = M(G) − M(G − v) Proof. f (G, v) = min(θ/2, li − f (G − v, vi)) M(G) = min(θ/2 + M(G − v), li + M(G − v − vi)) M(G) − M(G − v) = min(θ/2, li − (M(G − v) − M(G − v − vi))) f (G, v) and M(G) − M(G − v) satisfy the same recursion.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Solution to Graph Exploration

Alice’s and Bob’s optimal strategies are given by the optimum diluted matchings on G and G − v respectively

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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PWIT-approximation

Poisson Weighted Infinite Tree (Aldous)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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PWIT-approximation

Poisson Weighted Infinite Tree (Aldous)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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PWIT-approximation

Poisson Weighted Infinite Tree (Aldous)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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PWIT-approximation

Poisson Weighted Infinite Tree (Aldous) θ-cluster = component of the root after edges of cost more than θ have been deleted

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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PWIT-approximation

The PWIT is a local weak limit of the mean field model:

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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PWIT-approximation

The PWIT is a local weak limit of the mean field model: Lemma Fix positive integer k. Then there exists a coupling of the PWIT and rooted KN such that P(isomorphic (k, θ)-neighborhoods) ≥ 1 − (2 + θ)k N1/3 Has to be modified slightly for d > 1.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Graph Exploration on the PWIT

Study Graph Exploration

  • n the PWIT!

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Graph Exploration on the PWIT

Study Graph Exploration

  • n the PWIT!

What if the θ-cluster is infinite???

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Graph Exploration on the PWIT

Study Graph Exploration

  • n the PWIT!

What if the θ-cluster is infinite??? Optimistic (Pessimistic) k-look-ahead values f k

A

and f k

B Look k moves

ahead and assume the

  • pponent will pay θ/2

and terminate immediately if the game goes on beyond k moves

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Graph Exploration on the PWIT

Study Graph Exploration

  • n the PWIT!

What if the θ-cluster is infinite??? Optimistic (Pessimistic) k-look-ahead values f k

A

and f k

B Look k moves

ahead and assume the

  • pponent will pay θ/2

and terminate immediately if the game goes on beyond k moves Infinite look-ahead values fA and fB (when k → ∞)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Graph Exploration on the PWIT

Theorem Almost surely fA = fB Sketch of proof.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Graph Exploration on the PWIT

Theorem Almost surely fA = fB Sketch of proof. Let both Alice and Bob be infinite look-ahead optimistic players

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Graph Exploration on the PWIT

Theorem Almost surely fA = fB Sketch of proof. Let both Alice and Bob be infinite look-ahead optimistic players If they do not agree on the value of the game, play has to go on forever

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Graph Exploration on the PWIT

Theorem Almost surely fA = fB Sketch of proof. Let both Alice and Bob be infinite look-ahead optimistic players If they do not agree on the value of the game, play has to go on forever Reasonable lines of play do not percolate

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Graph Exploration on the PWIT

Theorem Almost surely fA = fB Sketch of proof. Let both Alice and Bob be infinite look-ahead optimistic players If they do not agree on the value of the game, play has to go on forever Reasonable lines of play do not percolate Contradiction!

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Replica Symmetry

fA = fB means Replica Symmetry holds: To find how to match v it suffices to look at a neighborhood of size independent of N KN✬

✫ ✩ ✪

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Replica Symmetry

fA = fB means Replica Symmetry holds: To find how to match v it suffices to look at a neighborhood of size independent of N KN✬

✫ ✩ ✪ qv

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Replica Symmetry

fA = fB means Replica Symmetry holds: To find how to match v it suffices to look at a neighborhood of size independent of N KN✬

✫ ✩ ✪ qv ✒✑ ✓✏

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Replica Symmetry

fA = fB means Replica Symmetry holds: To find how to match v it suffices to look at a neighborhood of size independent of N KN✬

✫ ✩ ✪ qv ✒✑ ✓✏

Cost[Diluted Matching] N/2

p

− → βM(d, θ)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Results

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Results

Proof of convergence of Cost[Matching] N/2 involves θ → ∞ (nontrivial)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Results

Proof of convergence of Cost[Matching] N/2 involves θ → ∞ (nontrivial) βM(1) = π2/6 (Already mentioned, proved by Aldous in 2001)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Results

Proof of convergence of Cost[Matching] N/2 involves θ → ∞ (nontrivial) βM(1) = π2/6 (Already mentioned, proved by Aldous in 2001) βM(2) ≈ 1.14351809919776

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Results

Proof of convergence of Cost[Matching] N/2 involves θ → ∞ (nontrivial) βM(1) = π2/6 (Already mentioned, proved by Aldous in 2001) βM(2) ≈ 1.14351809919776 βTSP(2) ≈ 1.285153753372032 But how do we get results for the TSP? Let me explain by analogy to...

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Refusal Chess

80a0Z0s0Z 7o0Z0j0Z0 60ZRZ0Z0Z 5Z0O0Z0Zq 40L0ZpZpO 3ZPZ0O0Z0 2PZ0Z0ZBZ 1Z0Z0Z0J0 a b c d e f g h

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In Refusal Chess, when a player makes a move, the opponent can either accept (and play on) or refuse, in case the move is taken back and the player has to choose another move. A player has the right to refuse once per move. For the chess players: A player is in check or checkmate if they would be in ordinary chess (so the rules are kind of illogical; for instance you cannot leave your king threatened just because you can refuse your opponent to capture it in the next move). If a player has only one legal move, the opponent cannot refuse it.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Refusal Chess

80Z0Z0s0Z 7o0Z0j0Z0 60ZRZ0Z0Z 5Z0O0Z0Zq 40L0ZpZpO 3ZPZ0O0Z0 2PZ0Z0ZBa 1Z0Z0Z0J0 a b c d e f g h

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Refusal Chess

80a0Z0s0Z 7o0Z0j0Z0 60ZRZ0Z0Z 5Z0O0Z0Zq 40L0ZpZpO 3ZPZ0O0Z0 2PZ0Z0ZBZ 1Z0Z0Z0J0 a b c d e f g h

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Refusal Chess

80a0Z0s0Z 7o0Z0j0Z0 60ZRZ0Z0Z 5Z0O0Z0Z0 40L0ZpZpl 3ZPZ0O0Z0 2PZ0Z0ZBZ 1Z0Z0Z0J0 a b c d e f g h

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Refusal Chess

80L0Z0s0Z 7o0Z0j0Z0 60ZRZ0Z0Z 5Z0O0Z0Z0 40Z0ZpZpl 3ZPZ0O0Z0 2PZ0Z0ZBZ 1Z0Z0Z0J0 a b c d e f g h

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

This would be a bad move in ordinary chess since Black can simply take back with the rook. But in refusal chess it is not so clear...

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Refusal Chess

80L0Z0s0Z 7o0Z0j0Z0 60ZRZ0Z0Z 5Z0O0Z0Z0 40Z0ZpZpZ 3ZPZ0O0Z0 2PZ0Z0ZBZ 1Z0Z0l0J0 a b c d e f g h

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Black accepts White’s move and plays on. Now White is in trouble, since refusing this move will allow Black to capture White’s queen. But let us not analyze this particular position further. The original position is taken from a spectacular finish by Jonathan Yedidia, one of the participants of the conference and former chess

  • pro. He played 32 — Bh2+! and White resigned in view of 33.

Kh1 Qxh4!, after which 34. Qb7+ (or any other move) is countered with a deadly discovered check.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

My tour is better than yours!

Alice and Bob play “My tour is better than yours!” Bob has this edge in his tour

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

My tour is better than yours!

Alice and Bob play “My tour is better than yours!” Alice says: “Good for you, but I have this edge in my tour!”

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

My tour is better than yours!

Alice and Bob play “My tour is better than yours!” Bob says: “Well, so do I”, effectively cancelling Alice’s move (this is the difference from Graph Exploration)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

My tour is better than yours!

Alice and Bob play “My tour is better than yours!” “Allright”, says Alice, “but I have this edge and you don’t!”

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

My tour is better than yours!

Alice and Bob play “My tour is better than yours!” Bob has already admitted having two edges from that vertex, so he cannot cancel this move.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

My tour is better than yours!

Alice and Bob play “My tour is better than yours!” Similarly, Alice can refuse one of Bob’s moves by claiming that she also has this edge in her tour.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

My tour is better than yours!

Alice and Bob play “My tour is better than yours!” But if she does...

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

My tour is better than yours!

Alice and Bob play “My tour is better than yours!” ...she will have to accept Bob’s second move, and so on...

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Here I am sweeping a number of details under the rug, but the structure of the game is the same as in Refusal Chess (you always play your second best move; we can call the game “Refusal Exploration”). The results for this game and the conclusions for the TSP are analogous to the results for matching.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

The end is near...

Future work

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

The end is near...

Future work 0 < d < 1

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

The end is near...

Future work 0 < d < 1 Games for other optimization problems (edge cover?)

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

The end is near...

Future work 0 < d < 1 Games for other optimization problems (edge cover?) Efficiency of Belief Propagation?

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

The end is near...

Future work 0 < d < 1 Games for other optimization problems (edge cover?) Efficiency of Belief Propagation? Computer analysis of games

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

The end is near...

Future work 0 < d < 1 Games for other optimization problems (edge cover?) Efficiency of Belief Propagation? Computer analysis of games RS holds for Chess but not Go???

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

Refusal Chess Problem

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Mate in 2: (a) Ordinary chess (b) Refusal chess.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization

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Physics of Algorithms, Santa Fe 2009

For the chess players: This is a two-mover with one solution for

  • rdinary chess and a different solution for Refusal Chess. In
  • rdinary chess, the solution is 1. Ra8, with the variations 1. —

Rxf2, 2. f8Q, and 1. — Kxd7, 2. f8N. In Refusal Chess the two keys are 1. f8R+ forbidding Kxf8, and

  • 1. f8B+ again forbidding Kxf8. On 1. f8R+ Rf7 the mating moves

are 2. Qxf7 and 2. Rgxf7, while on 1. f8B+ Kd8, the mating moves are 2. Qb6 and 2. Ra8. So the problem is a so-called Allumwandlung: The white pawn promotes to queen and knight in ordinary chess, and to rook and bishop in refusal chess. Notice that in Refusal, 1. f8Q+ doesn’t work since White then cannot refuse 1. — Kxf8.

Johan W¨ astlund Replica Symmetry and Combinatorial Optimization