random matching and traveling salesman problems johan wästlund chalmers university of technology...
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Random matching and traveling salesman problems
Johan Wästlund
Chalmers University of Technology
Sweden
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Mean field model of distance
The edges of a complete graph on n vertices are given i. i. d. nonnegative costs
Exponential(1) distribution.
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Mean field model of distance
We are interested in the cost of the minimum matching, minimum traveling salesman tour etc, for large n.
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Matching
Set of edges giving a pairing of all points
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Traveling salesman
Tour visiting all points
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Walkup’s theorem
Theorem (Walkup 1979): The expected cost of the minimum matching is bounded
Bipartite model
n
RL
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Walkup’s theorem
= cost of the minimum assignment. Modify the graph model: Multiple edges with costs
given by a Poisson process This obviously doesn’t change the minimum
assignment
nA
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Walkup’s theorem
Give each edge a random direction
Choose the five cheapest edges from each vertex.
We show that whp this set contains a perfect matching
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Hall’s criterion
An edge set contains a perfect matching iff for every subset S of L,
SS )(
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Hall’s criterion
If Hall’s criterion holds, an incomplete matching can always be extended.
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Hall’s criterion
If Hall’s criterion fails for S, then it also fails for
S
T
(S)
RST c )(
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Hall’s criterion
Here we can take |S| + |T| = n+1 If Hall’s criterion fails, then it fails for some S (in
L or in R) with
2
2n
S
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Walkup’s theorem
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Walkup’s theorem
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Walkup’s theorem
The directed edges from a given vertex have costs from a rate n/2 Poisson process
The 5 cheapest edges have expected costs 2/n, 4/n, 6/n, 8/n, 10/n.
The average cost in this set is 6/n, and there are n edges in a perfect matching
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Walkup’s theorem
If Hall’s criterion holds, there is a perfect matching of expected cost at most 6.
What about the cases of failure?
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Walkup’s theorem
Randomly color the edges Red p Blue 1-p Take the 5 cheapest blue edges from each
vertex. If Hall’s criterion holds, this gives a matching of cost 6/(1-p)
Otherwise the red edges 1-1, 2-2 etc give a matching of cost n/p.
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Walkup’s theorem
Total expected cost
Take p = 1/n for instance. For large n, the expected cost is < 6 + o(1) This completes the proof.
p
n
nO
p
5
1
1
6
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Walkup’s theorem
Actually
but we return to this…
2
1
9
1
4
11)(
nAE n
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Walkup’s theorem
Walkup’s theorem obviously carries over to the complete graph (for even n)
The method also works for the TSP, minimum spanning tree, and other related problems
Natural conjecture: E(cost) converges in probability to some constant.
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Statistical physics
The typical edge in the optimum solution has cost of order 1/n, and the number of edges in a solution is of order n.
Analogous to spin systems of statistical physics
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Disordered Systems
Spin glasses AuFe random alloy Fe atoms interact
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Statistical physics
Each particle essentially interacts only with its close neighbors
Macroscopic observables (magnetic field) arise as sums of many small terms, and are essentially independent of individual particles
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Statistical physics
Convergence in probability to a constant?
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Statistical Physics / C-S Spin configuration Hamiltonian Ground state energy Temperature Gibbs measure Thermodynamic limit
Feasible solution Cost of solution Cost of minimal
solution Artificial parameter T Gibbs measure n→∞
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Statistical physics
Replica-cavity method of statistical mechanics has given spectacular predictions for random optimization problems
M. Mézard, G. Parisi, W. Krauth, 1980’s Limit of /12 for minimum matching on the
complete graph (Aldous 2000) Limit 2.0415… for the TSP (Wästlund 2006)
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Non-rigorous derivation of the /12 limit Matching problem on Kn for large n. In principle, this requires even n, but we shall
consider a relaxation Let the edges be exponential of mean n, so
that the sequence of ordered edge costs from a given vertex is approximately a Poisson process of rate 1.
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Non-rigorous derivation of the /12 limit The total cost of the minimum matching is of
order n. Introduce a punishment c>0 for not using a
particular vertex. This makes the problem well-defined also for
odd n. For fixed c, let n tend to infinity. As c tends to infinity, we expect to recover
the behavior of the original problem.
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Non-rigorous derivation of the /12 limit For large n, suppose that the problem
behaves in the same way for n-1 vertices. Choose an arbitrary vertex to be the root What does the graph look like locally around
the root? When only edges of cost <2c are considered,
the graph becomes locally tree-like
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Non-rigorous derivation of the /12 limit Non-rigorous replica-cavity method Aldous derived equivalent equations with the
Poisson-Weighted Infinite Tree (PWIT)
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Non-rigorous derivation of the /12 limit Let X be the difference in cost between the
original problem and that with the root removed.
If the root is not matched, then X = c. Otherwise X = i – Xi, where Xi is distributed like X, and i is the cost of the i:th edge from the root.
The Xi’s are assumed to be independent.
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Non-rigorous derivation of the /12 limit
It remains to do some calculations.
We have
where Xi is distributed like X
),,min( ii XcX
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Non-rigorous derivation of the /12 limit Let )(exp)(exp)()( ufdttFuXPuF
u
X
-u
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Non-rigorous derivation of the /12 limit Then if u>-c,
)()()(' ufeuFuf
)()(' ufeuf
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Non-rigorous derivation of the /12 limit
)()()()( )(')(')(')(' ufufufuf edu
deufe
du
deufufuf
Hence )()( ufuf ee is constant
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Non-rigorous derivation of the /12 limit
The constant depends on c and holds when
–c<u<c
f(-u)
f(u)
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Non-rigorous derivation of the /12 limit From definition, exp(-f(c)) = P(X=c) =
proportion of vertices that are not matched, and exp(-f(-c)) = exp(0) = 1
e-f(u) + e-f(-u) = 2 – proportion of vertices that are matched = 1 when c = infinity.
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Non-rigorous derivation of the /12 limit
1 ee
yx
6
2Area
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Non-rigorous derivation of the /12 limit What about the cost of the minimum
matching?
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Non-rigorous derivation of the /12 limit
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Non-rigorous derivation of the /12 limit
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Non-rigorous derivation of the /12 limit Hence J = area under the curve when f(u) is
plotted against f(-u)! Expected cost = n/2 times this area In the original setting = ½ times the area
= /12.
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nK
L
nL
K
K-L matching
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K-L matching
Similarly, the K-L matching problem leads to the equations:
)](min[d
iiYKX )](min[
d
iiXLY
• has rate K and has rate L• min[K] stands for K:th smallest
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Shown by Parisi (2006) that this system has an essentially unique solution
The ground state energy is given by
where x and y satisfy an explicit equation
For K = L = 2 (equivalent to the TSP), this equation is
K-L matching
0
ydx
12
12
1
eeyx yx
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The exponential bipartite assignment problem
n
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The exponential bipartite assignment problem Exact formula conjectured by Parisi (1998)
Suggests proof by induction Researchers in discrete math, combinatorics and
graph theory became interested Generalizations…
2
1
9
1
4
11)(
nCE n
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Generalizations
by Coppersmith & Sorkin to incomplete matchings
Remarkable paper by M. Buck, C. Chan & D. Robbins (2000)
Introduces weighted vertices Extremely close to proving Parisi’s
conjecture!
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Incomplete matchings
nm
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Weighted assignment problems Weights 1,…,m, 1,…, n on vertices
Edge cost exponential of rate ij
Conjectured formula for the expected cost of minimum assignment
Formula for the probability that a vertex participates in solution (trivial for less general setting!)
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The Buck-Chan-Robbins urn process Balls are drawn with probabilities proportional
to weight
1 2
3
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Proofs of the conjectures
Two independent proofs of the Parisi and Coppersmith-Sorkin conjectures were announced on March 17, 2003 (Nair, Prabhakar, Sharma and Linusson, Wästlund)
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Rigorous method
Relax by introducing an extra vertex Let the weight of the extra vertex go to zero Example: Assignment problem with
1=…=m=1, 1=…=n=1, and m+1 =
p = P(extra vertex participates) p/n = P(edge (m+1,n) participates)
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Rigorous method
p/n = P(edge (m+1,n) participates) When →0, this is
Hence
By Buck-Chan-Robbins urn theorem,
1,,1,,1,,1,,,1 nmknmknmknmknm CCCClP
n
pCCE nmknmk
01,,1,, lim
11
kmmmp
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Rigorous method
Hence
Inductively this establishes the Coppersmith-Sorkin formula
nkmnmmn
CECE nmknmk )1(
1
)1(
111,,1,,
)1(
1
)1(
1
)1(
1
)1(
11,,
knmnkmnmnmmn
CE nmk
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Rigorous results
Much simpler proofs of Parisi, Coppersmith-Sorkin, Buck-Chan-Robbins formulas
Exact results for higher moments Exact results and limits for optimization
problems on the complete graph
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The 2-dimensional urn process
2-dimensional time until k balls have been drawn
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Limit shape as n→∞
Matching:
TSP/2-factor:
1 ee
yx
12
12
1
eeyx yx
041548.2)(2
1
0
dxxy
6)(
2
0
dxxy
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Mean field TSP
If the edge costs are i.i.d and satisfy P(l<t)/t→1 as t→0 (pseudodimension 1), then as n →∞,
A. Frieze proved that whp a 2-factor can be patched to a tour at small cost
...041548.2* LLp
n
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Further exact formulas
nnn
n kn
kkC1
3
2
1
2
1
4 /11
4/12/15var
2
1)3(4)2(4
nO
n
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LP-relaxation of matching in the complete graph Kn
12
1
9
1
4
11)(
2
2
nCE n
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Future work
Explain why the cavity method gives the same equation as the limit shape in the urn process
Establish more detailed cavity predictions Use proof method of Nair-Prabhakar-Sharma
in more general settings
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Thank you!