Olivier Hudry (INFRES-MIC2 Télécom ParisTech) A Branch and Bound Algorithm to Compute a Median Permutation Algorithms & Permutations 2012, Paris. http://igm.univ-mlv.fr/AlgoB/algoperm2012/

1. A branch and bound method
to compute a median permutation
Irène Charon, Olivier Hudry
Télécom ParisTech
olivier.hudry@telecom-paristech.fr

2. A permutation problem in voting theory
 Given a profile Π = (σ1, σ2, …, σm) of m permutations
(i.e. linear orders) σi (1 ≤ i ≤ m) on a set X of n = |X|
elements, how to aggregate them into a unique
permutation which summarizes Π as accurately as
possible?
 In voting theory (Condorcet, 1784): we want to rank n
candidates from the rankings provided by m voters.

3. Example
 X = {a, b, c, d, e, f}, m = 5
voter 1: σ1 = a > b > c > f > d > e
voter 2: σ2 = a > c > f > b > d > e
voter 3: σ3 = e > d > a > f > b > c
voter 4: σ4 = b > c > d > e > f > a
voter 5: σ5 = c > f > b > e > a > d.

4. A combinatorial optimization problem
 Symmetric difference distance d between R and R′ :
d(R, R′ ) = |{(x, y)Œ X2 with [xRy and not xR′ y]
∈
or [not xRy and xR′ y]}|.
 Let Σ be the set of all the permutations defined on X.
Then, for Π = (σ1, σ2, …, σm):
m
Minimize ρΠ(σ) = ∑ d (σ , σ i ) for σ ŒΣ
∈
i= 1
(cf. J.-P. Barthélemy, B. Monjardet, 1981)

5.  d(R, R′ ) measures the number of disagreements
between R and R′.
 ρΠ(σ) (= remoteness of σ from Π) measures the total
number of disagreements between σ and Π.
 σ* minimizing ρΠ over Σ is called a median
permutation (or a median linear order) of Π.
 Theorem (J.J. Bartholdi III et alii, 1989;
O. Hudry, 1989; C. Dwork et alii, 2001):
The computation of σ* is NP-hard.

6. A 0-1 linear programming problem
 σ = (σxy)(x, y)∈Œ with σxy = 1 if σ ranks x better than y (x >σ y)
X2
and σxy = 0 otherwise.
 mxy = m – 2|{i: 1 ≤ i ≤ m and x >σi y}| = –myx
 Then: ρΠ(σ) = C + ∑ mxy σ xy
( x, y )∈ X 2
with :
∀ x ŒX, σxx = 1
∈ (reflexivity)
∀ (x, y) ŒX2, x ≠ y, σxy + σyx = 1 (antisymmetry)
∈
∀ (x, y, z) ∈Œ3, σxy + σyz – σxz ≤ 1 (transitivity)
X
∀ (x, y) ŒX2, σxy Œ{0, 1}
∈ ∈ (binarity)

7. Lagrangean relaxation
 Relaxation of the transitivity constraints:
∀ (x, y, z) ŒX3, σxy + σyz – σxz ≤ 1
∈
 Lagrangean function L for σ = (σxy)(x, y)∈X2 with σxy ∈ {0, 1}, σxx
= 1, σxy + σyx = 1, and Λ = (λxyz)(x, y, z)∈X3 with λxyz ≥ 0:
∑ λ xyz ( σ xy + σ yz − σ xz − 1)
L(σ, Λ) = ρΠ(σ) +
( x, y , z )∈ X 3
∑ a xy (Λ )σ xy − ∑ λ xyz
= C +x, y )∈ X 2
( ( x, y , z )∈ X 3
with a xy (Λ ) = mxy + ∑ (λ xyz + λ zxy − λ xzy )
z∈ X

8. Lagrangean relaxation (end)
 Dual function for Λ = (λxyz)(x, y, z)∈X3 with λxyz ≥ 0:
D(Λ) = min{L(σ, Λ) with σ ∈ A}
with A = {reflexive and antisymmetric relations defined on X}.
 Dual problem: maximize D(Λ) for Λ ≥ 0.
 The maximum of D gives a lower bound of the minimum of ρΠ.
 Computation of D(Λ) for a given Λ:
if axy ≥ 0, set σxy = 0, and σxy = 1 otherwise.
 Resolution of the dual problem by subgradient methods.

9. The components of the BB algorithm
 Initial bound: found by a metaheuristic (a self-tuned noising method; I.
Charon and O. Hudry, 1993, 2009)
 Evaluation function: provided by the Lagrangean relaxation.
 Branching rule (J.-P. Barthélemy, A. Guénoche, O. Hudry, 1989;
I. Charon, A. Guénoche, O. Hudry, F. Woirgard, 1996):
The root of the BB-tree contains all the permutations defined on X.
A node of the BB-tree contains the permutations sharing a given beginning
section S (i.e. a permutation of a subset of X):
S(xj1, xj2, …, xjp) = xj1 >σ xj2 >σ … >σ xjp.
The branching principle consists in expanding this beginning section:
S(xj1, xj2, …, xjp, x) = xj1 >σ xj2 >σ … xjp >σ x
with x ∉{xj1, xj2, …, xjp}.

10. Shape of the BB-tree
x1 xj xn–1 xn
x 2 … 1 …
… …
… … … …
xj > xj > …>x
1 2 jp – 1
… …
xj > xj > …>x
1 2 jp – 1 > x jp
xj > xj > …>x
1 2 jp – 1 > x j p > x h 1
… x j > x j > …> x
jp – 1 >x jp > h n – p
x
1 2
… …

11. Other components to prune the BB-tree
 Hamiltonian permutations.
* We may summarize a profile Π of permutations by a tournament
T (weighted by –mxy > 0): there is an arc (x, y) if a majority of
voters prefer x to y (we assume that there is no tie).
* We say that a permutation σ is Hamiltonian if it induces a
Hamiltonian path in T.
* Theorem (R. Remage and W.A. Thompson, 1966): a median
permutation is Hamiltonian.
→ xj1 >σ xj2 >σ … >σ xjp is expanded into xj1 >σ … >σ xjp >σ x only
if a majority of voters prefer xjp to x.

12. Example
 X = {a, b, c, d, e, f}
1
σ1 = a > b > c > f > d > e a b
1
σ2 = a > c > f > b > d > e 1 1
σ3 = e > d > a > f > b > c f 1 1 c
3
σ4 = b > c > d > e > f > a 1 3
σ5 = c > f > b > e > a > d. 1 3 3 3
1
e d
Here, a > c > f > b > d > e 1
is a median permutation and
induces a Hamiltonian path.

13. Other components to prune the BB-tree
 We compute the variation of ρΠ when, from a permutation σ
beginning with S = xj1 >σ xj2 >σ … >σ xjp, we take an interval
xjh >σ … >σ xjp (1 ≤ h ≤ p) and we shift it at the end of σ,
after the elements of X – S (= OS = « out of section »):
σ = xj1 >σ xj2 >σ … xjh–1 >σ xjh >σ … >σ xjp >σ (OS)
becomes
σ′ = xj1 >σ′ xj2 >σ′ … xjh–1 >σ′ (OS) >σ′ xjh >σ′ … >σ′ xjp.
If ρΠ decreases, we do not keep the node associated with S.
OSmoves will count this kind of cuts.

14. Other components to prune the BB-tree (end)
 When we deal with a new beginning section
S = xj1 >σ xj2 >σ … xjh–1 >σ xjh >σ … >σ xjp >σ x,
we consider the beginning sections that we can get by moving,
inside S, an “interval” of S including x, i.e., the beginning
sections with the following shape:
xjh >σ′ … >σ′ xjp >σ′ x >σ′ xj1 >σ′ xj2 >σ′ … xjh–1.
If ρΠ decreases, we do not keep the node associated with S.
Smoves will count this kind of cuts.

15. An experimental result on the efficiency of
the branch and bound components
 Numbers of cuts for an instance on 39 candidates
n u m b er o f n o d es
80000
n u m b e r o f s e p a r a t e d n o d e s : 2 1 ,3 2 3
70000 n u m b e r o f c u t n o d e s : 6 8 7 ,2 3 0
60000 h a m : 2 8 8 ,3 8 8
O S m o v e s : 2 0 7 ,0 2 2
50000
S m o v e s : 9 3 ,2 9 6
40000 l e x : 2 ,3 7 7
B e g S e c : 3 ,2 1 2
30000
r e l a x : 9 2 ,9 3 5
20000
10000
d e p th -le v e l
0
1 2 3 4 5 6 7 8 9 10 11 12 13 1 4 ... 2 8

16. CPU times for m ∈ {3, 4, 100, 101}
 CPU times in seconds (Rk: order = n).
a v erag e C .P .U . tim e
1000
3 : 0 .8 0
4 : 0 .6 0
100
1 0 0 : 0 .6 7
1 0 1 : 0 .6 8
10
1
o rd e r
0 ,1
15 20 25 30 35 40 45 50 55

17. Number of median permutations
versus number of Hamiltonian permutations
 Let M(n) and H(n) denote respectively the maximum number of
median permutations or of Hamiltonian permutations for
instances on n candidates.
 If n is even with n ≥ 2: M(n) = n!
 If n is odd: M(n) ≤ H(n).
 Theorem (N. Alon, 1990): H(n) ≤ (c × n1.5 × n!)/2n where c is a
constant.
 Theorem (I. Charon, O. Hudry, 2000): for n = 3k,
30.75(n – 1)/n2 ≤ M(n).

18. Thank you for your attention!
