1. Hiroshi Hirai University of Tokyo hirai@mist.i.u-tokyo.ac.jp
Computing degree of determinant via
discrete convex optimization
over Euclidean building
Hiroshi Hirai
University of Tokyo
Workshop: Recent Development in Optimization 2
GRIPS, Roppongi, Tokyo, 2018/10/13

2. Contents combinatorial optimization v.s. linear algebra
non-commutative
combinatorial optimization v.s. linear algebra
Submodularity + Discrete convexity
Background: Edmonds problem and recent development
Motivation + contribution of this work

3. Edmonds Problem Can we compute the rank of linear symbolic matrix
𝐴= 𝐴 0 + 𝐴 1 𝑥 1 + 𝐴 2 𝑥 2 +…+ 𝐴 𝑚 𝑥 𝑚
in polynomial time ?
　 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚 : variables
　 𝐴 0 , 𝐴 1 ,…, 𝐴 𝑚 : 𝑛×𝑛 matrices over field 𝕂　
　𝐴: matrix over 𝕂[ 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚 ] ↪
𝕂( 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚 )

4. Algebraic Interpretation of Bipartite Matching
Motivation
Algebraic Interpretation of Bipartite Matching 1 2 3 4 1 1
𝐴= 𝑥 12 𝑥 𝑥 23 𝑥 𝑥 𝑥 41 1 2 2 2 3 3 3 4 4 4
= 𝑖𝑗∈𝐸 𝐸 𝑖𝑗 𝑥 𝑖𝑗
𝐺=(𝑈,𝑉;𝐸)
# maximum matching of 𝐺 =rank 𝐴
∵det 𝐴= 𝑀 ± 𝑖𝑗 ∈𝑀 𝑥 𝑖𝑗
min-max formula ( König-Egerváry )
polynomial time algorithm

5. Open：Deterministic polynomial time rank computation
𝐴= 𝑘=1 𝑚 𝑎 𝑘 𝑏 𝑘 𝑇 𝑥 𝑘
Linear matroid intersection
𝐴= 𝑘=1 𝑚 (𝑎 𝑘 𝑏 𝑘 𝑇 − 𝑏 𝑘 𝑎 𝑘 𝑇 ) 𝑥 𝑘
Linear matroid matching
∃ min-max theorem + polynomial time algorithm
Edmonds 1970, Lovász 1981
Open：Deterministic polynomial time rank computation
of general 𝐴= 𝐴 0 + 𝑘=1 𝑚 𝐴 𝑘 𝑥 𝑘
Randomized polynomial time algorithm (Lovász 1979)
Connection to circuit complexity (Kabanets-Impagliazzo 2004)

6. Edmonds Problem Non-commutative nc-
Ivanyos-Qiao-Subrahmanyam 2015
nc-
Can we compute the rank of linear symbolic matrix
𝐴= 𝐴 0 + 𝐴 1 𝑥 1 + 𝐴 2 𝑥 2 +…+ 𝐴 𝑚 𝑥 𝑚
in polynomial time ?
𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚 : variables
𝐴 0 , 𝐴 1 ,…, 𝐴 𝑚 : matrices over field 𝕂
𝑥 𝑖 𝑥 𝑗 ≠ 𝑥 𝑗 𝑥 𝑖
rank ≤
nc-rank
𝐴: free ring 𝕂 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚
free skew field 𝕂( 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚 ) ↪
Amitsur 1966

7. What is free skew field 𝕂( 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚 )?
skew field = ring s.t. ∀ nonzero element has inverse
𝕂( 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚 )
:= all rational expressions of +,−,×,÷, 𝑥 𝑖 , 𝑎∈𝕂 modulo ~ ∋
𝑝= 𝑥 2 𝑥 1 𝑥 2 −1 + 𝑥 3 −3 −1
𝑝 ~ 𝑞⇔𝑝 𝑋 1 , 𝑋 2 ,…, 𝑋 𝑚 = 𝑞 𝑋 1 , 𝑋 2 ,…, 𝑋 𝑚
(∀𝑑, ∀ 𝑋 𝑖 ∈Ma t 𝑑×𝑑 (𝕂) )
It is much difficult to handle elements in 𝕂 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚
than 𝕂( 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚 ), but ...

8. nc-rank in P !! Garg-Gurvits-Oliveira-Wigderson 2015 (FOCS’16): 𝕂=ℚ
Ivanyos-Qiao-Subrahmanyam 2015 (ITCS’17): 𝕂, arbitrary
Min-max theorem: Fortin-Reutenauer 2004
treatable
in 𝕂 !
nc-rank 𝐴=2𝑛 − Max. 𝑟+𝑠
s.t.
(∀𝑖)
𝑃,𝑄: nonsingular over 𝕂 ∗ 𝑟 𝑠
𝑃 𝐴 𝑖 𝑄=
rank = nc-rank
bipartite matching
linear matroid intersection
rank < nc-rank
non-bipartite matching
linear matroid matching
min-max theorem

9. König-Egerváry from Fortin-Reutenauer
𝐺:bipartite graph
2𝑛 − Max. 𝑟+𝑠 𝑖 𝑗 1 ∗ 𝑟 𝑠
s.t. 𝑃 𝑄=
(∀𝑒=𝑖𝑗∈𝐸)
permutation matrices
𝑃,𝑄: nonsingular over 𝕂
=2𝑛−#maximum stable set
=# minimun vertex cover

10. Algorithms for nc-rank
Garg-Gurvits-Oliveira-Wigderson 2015 (FOCS’16):
Gurvits’ operator scaling
Ivanyos-Qiao-Subrahmanyam 2015 (ITCS’17):
Wong sequence --- vector-space analogue of augmenting path
Hamada-Hirai 2017:
Submodularity + convex optimization on CAT(0)-space
They are beyond Euclidean convex optimization

11. Submodularity View Max. 𝑟+𝑠 s.t. 𝑃 𝐴 𝑖 𝑄= 𝑃,𝑄: nonsingular
Hamada-Hirai 2017
Max. 𝑟+𝑠 ∗ ∗
s.t.
𝑃 𝐴 𝑖 𝑄= ∗
(∀𝑖) 𝑟 ∗ 𝑠
𝑃,𝑄: nonsingular
Submodular optimization
on the modular lattice
of vector subspaces
Max. dim 𝑋+ dim 𝑌
s. t. 𝐴 𝑖 𝑋,𝑌 =0
𝑋,𝑌⊆ 𝕂 𝑛
vector subspaces
where 𝐴 𝑖 𝑥,𝑦 ≔ 𝑥 𝑇 𝐴 𝑖 𝑦 ∀𝑖 =

12. Motivation of this work
~ capture “weighted” combinatorial optimization problems
from such “non-commutative” linear algebra 1 2 3 4
𝑐 12
𝑐 43
Ex: Weighted bipartite matching
Max. 𝑐 𝑀 ≔ 𝑖𝑗∈𝑀 𝑐 𝑖𝑗
s.t. 𝑀: perfect matching
𝑐 𝑖𝑗 ∈ ℤ + : edge-weight

13. Algebraic Interpretation of Weighted Matching1
𝑐 12 1 1 2 3 4
𝐴(𝑡)= 𝑡 𝑐 12 𝑥 𝑡 𝑐 21 𝑥 𝑡 𝑐 14 𝑥 𝑡 𝑐 23 𝑥 𝑡 𝑐 32 𝑥 32 𝑡 𝑐 41 𝑥 𝑡 𝑐 43 𝑥 43 𝑡 𝑐 44 𝑥 44 1 2 2 2 3 3 3 4
𝑐 43 4 4
= 𝑖𝑗∈𝐸 𝑡 𝑐 𝑖𝑗 𝐸 𝑖𝑗 𝑥 𝑖𝑗
Max. weight of perfect matching = deg det 𝐴(𝑡)
∵deg det 𝐴=deg 𝑀 ± 𝑡 𝑐(𝑀) 𝑖𝑗 ∈𝑀 𝑥 𝑖𝑗 = max 𝑀 𝑐(𝑀)

14. Weighted Edmonds Problem
Can we compute deg det of
𝐴(𝑡)= 𝐴 0 (𝑡)+ 𝐴 1 (𝑡) 𝑥 1 +…+ 𝐴 𝑚 (𝑡) 𝑥 𝑚
in polynomial time ?
　 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚 : variable
　 𝐴 0 (𝑡), 𝐴 1 (𝑡),…, 𝐴 𝑚 (𝑡): matrices over 𝕂[𝑡] 　
　𝐴: matrix over 𝕂[ 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚 ,𝑡]
Goal: develop a non-commutative version

15. Contribution Formulate weighted non-commutative Edmonds problem
by using Dieudonné determinant Det.
Establish a min-max theorem for deg Det
　 with view from Discrete Convex Analysis beyond ℤ 𝑛
～ L-convex function on Euclidean building
Algorithm: SDA
　 ～ 𝑂(𝑛𝑑) nc-rank computation, 𝑑: max degree
Special case of deg det = deg Det:
　 ～ weighted linear matroid intersection, mixed matrix

16. 𝐴(𝑡)= 𝐴 0 (𝑡)+ 𝐴 1 (𝑡) 𝑥 1 +…+ 𝐴 𝑚 (𝑡) 𝑥 𝑚
How to see
𝐴(𝑡)= 𝐴 0 (𝑡)+ 𝐴 1 (𝑡) 𝑥 1 +…+ 𝐴 𝑚 (𝑡) 𝑥 𝑚
Matrix over (skew) polynomial ring 𝕂 𝑥 [t]
Ore ring ---- ∀𝑝,𝑞,∃𝑢,𝑣:𝑝𝑢=𝑞𝑣 ≠0
Matrix over Ore quotient ring 𝕂 𝑥 (t)
𝑝 𝑞 = 𝑝 ′ 𝑞 ′ ⇔∃𝑢,𝑣, 𝑝𝑢= 𝑝 ′ 𝑣,𝑞𝑢= 𝑞 ′ 𝑣 ≠0
Degree: deg 𝑝 𝑞 ≔ deg 𝑝− deg 𝑞

17. 𝐴 = 𝐿 𝐷 𝑃 𝑈 How to define “determinant” of matrices over skew field
𝐴: nonsingular over 𝔽
(Our case: 𝔽=𝕂 𝑥 (t))
Bruhat decomposition: LU-decomposition of matrices over skew field
uni-lower-triangular
uni-upper-triangular
𝐴 = 𝐿 𝐷 𝑃 𝑈
diagonal
permutation
unique
Dieudonné determinant ∈ Abelization 𝔽 ∗ /[ 𝔽 ∗ , 𝔽 ∗ ] of 𝔽 ∗
Det 𝐴≔ sgn 𝑃 𝐷 11 𝐷 22 … 𝐷 𝑛𝑛 mod [ 𝔽 ∗ , 𝔽 ∗ ]
commutator group
Lem: Det 𝐴𝐵= Det 𝐴 Det 𝐵

18. Weighted Non-commutative Edmonds Problem
Can we compute deg Det of
𝐴(𝑡)= 𝐴 0 (𝑡)+ 𝐴 1 (𝑡) 𝑥 1 +…+ 𝐴 𝑚 (𝑡) 𝑥 𝑚
in polynomial time ?
　 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚 : variables 𝑥 𝑖 𝑥 𝑗 ≠ 𝑥 𝑗 𝑥 𝑖
　 𝐴 0 𝑡 , 𝐴 1 𝑡 ,…, 𝐴 𝑚 𝑡 : matrices over 𝕂[𝑡]　
　𝐴: matrix over 𝕂( 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑚 )(𝑡)
※ deg Det is well-defined since deg is zero on commutators
deg (𝑝𝑞 𝑝 −1 𝑞 −1 )=0

19. Min-Max Theorem 𝐴= 𝐴 0 + 𝐴 1 𝑥 1 +…+ 𝐴 𝑚 𝑥 𝑚 deg Det 𝐴=
treatable in 𝕂(𝑡)
𝐴= 𝐴 0 + 𝐴 1 𝑥 1 +…+ 𝐴 𝑚 𝑥 𝑚
deg Det 𝐴=
Min. −deg det 𝑃 − deg det Q
s.t. deg 𝑃 𝐴 𝑘 𝑄 𝑖𝑗 ≤0 (∀𝑘,∀𝑖𝑗)
𝑃,𝑄: nonsingular over 𝕂(𝑡)

20. Weak Duality deg 𝑃 𝐴 𝑘 𝑄 𝑖𝑗 ≤0 (∀𝑘,∀𝑖𝑗) ⇒deg 𝑃𝐴𝑄 𝑖𝑗 ≤0 (∀𝑖𝑗)
⇒deg Det 𝑃𝐴𝑄≤0
⇒deg (Det 𝑃 Det 𝐴 Det 𝑄)≤0
⇒deg Det 𝑃 + deg Det 𝐴+deg Det 𝑄≤0
⇒deg Det 𝐴≤− deg Det 𝑃 − deg Det 𝑄 = =
deg det 𝑃
deg det 𝑄

21. Strong Duality + Algorithm (SDA)
𝑃𝐴𝑄= 𝑃𝐴𝑄 0 +𝑂( 𝑡 −1 ) =
𝑃 𝐴 0 𝑄 𝑃 𝐴 1 𝑄 0 𝑥 1 +…+ 𝑃 𝐴 𝑚 𝑄 0 𝑥 𝑚
over 𝕂( 𝑥 )
over 𝕂
𝑃𝐴𝑄 0 : nonsingular on 𝕂( 𝑥 )⇒
deg Det 𝑃𝐴𝑄=0
⇒ deg Det 𝐴=−deg det 𝑃−deg det 𝑄 ⇒ 𝑃,𝑄: optimal
𝑃𝐴𝑄 0 : singular on 𝕂( 𝑥 ) ⇒
We can augment 𝑃,𝑄→ 𝑃 ′ ,𝑄′ s.t.
deg det 𝑃 ′ +deg det 𝑄 ′ >deg det 𝑃+deg det 𝑄

22. 𝐼 𝑂 𝑂 𝑡 𝐼 𝑟 𝑆𝑃𝐴𝑄𝑇 𝐼 𝑂 𝑂 𝑡 −1 𝐼 𝑛−𝑠 ⇒ deg 𝑃 ′ 𝐴 𝑄 ′ 𝑖𝑗 ≤0
𝑃𝐴𝑄 0 : singular on 𝕂( 𝑥 ) ⇒
nc-rank of 𝑃𝐴𝑄 0 <𝑛
𝑡 −1 ∗ 𝑟 𝑠
∃𝑆,𝑇: nonsingular
over 𝕂
𝑆𝑃𝐴𝑄𝑇= 𝑂( 𝑡 −1 ) 𝑡
𝑟+𝑠>𝑛
𝐼 𝑂 𝑂 𝑡 𝐼 𝑟 𝑆𝑃𝐴𝑄𝑇 𝐼 𝑂 𝑂 𝑡 −1 𝐼 𝑛−𝑠 ⇒ deg 𝑃 ′ 𝐴 𝑄 ′ 𝑖𝑗 ≤0 𝑃′ 𝑄′
deg det 𝑃 ′ +deg det 𝑄 ′ =(𝑟+𝑠−𝑛)+deg det 𝑃+deg det 𝑄
>0
Long-step: use 𝐼 𝑂 𝑂 𝑡 𝛼 𝐼 𝑟 , 𝐼 𝑂 𝑂 𝑡 −𝛼 𝐼 𝑛−𝑠 , 𝛼>0

23. Remarks Essence of this argument / algorithm is
in combinatorial relaxation method (Murota 1990,1995)
developed for computing deg det.
short-step
Naive iteration bound for initial 𝑃,𝑄 =( 𝑡 −𝑑 𝐼,𝐼)
𝑛𝑑 (𝑑: maximum degree)
𝑑−(deg Det 𝐴−max deg (𝑛−1-minor))
We can improve this bound to
= max deg – min deg of Smith-McMillan form of 𝐴
from view of discrete convex analysis beyond ℤ 𝑛

24. Special case of deg det = deg Det
𝐴= 𝐴 0 + 𝐴 1 𝑥 1 + 𝐴 2 𝑥 2 +…+ 𝐴 𝑚 𝑥 𝑚
Thm [Ivanyos et al 2010]
rank 𝐴 𝑖 =1 (∀𝑖 ≥1)⟹ rank 𝐴=nc-rank A.
bipartite matching 𝑖𝑗∈𝐸 𝐸 𝑖𝑗 𝑥 𝑖𝑗
linear matroid intersection 𝑘 𝑎 𝑘 𝑏 𝑘 𝑇 𝑥 𝑘
mixed matrix (Murota-Iri 1985) 𝐴 0 + 𝑖𝑗∈𝐸 𝐸 𝑖𝑗 𝑥 𝑖𝑗
rank 𝐴 𝑖 (𝑡)=1 (∀𝑖 ≥1)⟹ deg det 𝐴=deg Det A.
𝐴(𝑡)= 𝐴 0 (𝑡)+ 𝐴 1 (𝑡) 𝑥 1 +…+ 𝐴 𝑚 (𝑡) 𝑥 𝑚
Thm [H. 18]
mixed polynomial matrix
weighted ver.

25. bipartite matching 𝑖𝑗∈𝐸 𝑡 𝑐 𝑖𝑗 𝐸 𝑖𝑗 𝑥 𝑖𝑗
SDA + long-step ≈ Hungarian
Interpretation
via
Euclidean building
Linear matroid 𝑘 𝑡 𝑐 𝑘 𝑎 𝑘 𝑎 𝑘 𝑇 𝑥 𝑘
SDA + long-step ≈ Greedy
Linear matroid intersection 𝑘 𝑡 𝑐 𝑘 𝑎 𝑘 𝑏 𝑘 𝑇 𝑥 𝑘 ?
SDA + long-step ≈ Lawler’s primal dual
Lawler 1975
Mixed polynomial matrix 𝐴 0 𝑡 + 𝑖𝑗∈𝐸 𝑡 𝑐 𝑖𝑗 𝐸 𝑖𝑗 𝑥 𝑖𝑗
SDA + optimization over ℤ 𝑛 -sublattice (apartment)
≈ combinatorial relaxation algo.
Iwata-Takamatsu 2013
Iwata-Oki-Takamatsu 2017
dual of bipartite matching

26. = View from Discrete Convex Analysis beyond ℤ 𝑛 Dual of nc-rank
Dual of deg Det
Max. 𝑟+𝑠
Max. deg det 𝑃 + deg det Q ∗ ∗
s.t.
𝑃 𝐴 𝑘 𝑄= ∗
(∀𝑘)
s.t. deg 𝑃 𝐴 𝑘 𝑄 𝑖𝑗 ≤0 (∀𝑘,∀𝑖𝑗) 𝑟 ∗ 𝑠
𝑃,𝑄: nonsingular over 𝕂(𝑡)
𝑃,𝑄: nonsingular over 𝕂 =
L-convex optimization
on the modular lattice
of free modules over PID 𝕂 𝑡 −
Max. dim 𝑋+ dim 𝑌
s. t. 𝐴 𝑘 𝑋,𝑌 =0 ∀𝑘
Submodular optimization
on the modular lattice
of vector subspaces
𝑋,𝑌⊆ 𝕂 𝑛
vector subspaces
where 𝐴 𝑘 𝑥,𝑦 ≔ 𝑥 𝑇 𝐴 𝑘 𝑦

27. 𝑃,𝑄: nonsingular over 𝕂(𝑡)
Max. deg det 𝑃 + deg det Q
s.t. deg 𝑃 𝐴 𝑘 𝑄 𝑖𝑗 ≤0 (∀𝑘,∀𝑖𝑗)
𝑃,𝑄: nonsingular over 𝕂(𝑡)
𝕂 𝑡 − ≔ 𝑝 𝑞 ∈𝕂 𝑡 deg 𝑝 𝑞 ≤0}
𝑃 𝕂 𝑡 − ⟷ full-rank free 𝕂 𝑡 − -submodule of 𝕂 𝑡 𝑛
Max. deg 𝐿 + deg 𝑀
s.t. deg 𝐴 𝑘 𝐿,𝑀 ⊆[−∞,0] (∀𝑘)
𝐿,𝑀: full-rank free 𝕂 𝑡 − -submodules of 𝕂 𝑡 𝑛 =
where 𝐴 𝑘 𝑥,𝑦 ≔ 𝑥 𝑇 𝐴 𝑘 𝑦

28. ℒ 𝕂 𝑡 𝑛 ≔{ full-rank free 𝕂 𝑡 − -submodules of 𝕂 𝑡 𝑛 }
≈ Euclidean building of SL(𝕂 𝑡 𝑛 )
Lem: ℒ 𝕂 𝑡 𝑛 is a uniform modular lattice, i.e.,
𝑥⟼ 𝑥 + ≔ 𝑦 𝑦 covers 𝑥}
is order-preserving bijection
Rem [H. 17] : uniform modular lattice
≈ Euclidean building of type A

29. Uniform modular lattice [H.17]
𝑥⟼ 𝑥 + ≔ 𝑦 𝑦 covers 𝑥} is order-preserving bijection
Ex: ℤ 𝑛 , ∧ = min, ∨ = max, 𝑥 + =𝑥+𝟏 𝑥
𝑥 +
Rem: ℒ 𝕂 𝑡 2 ≈ ℤ⊠ infinite regular tree
Ex: ℤ⊠ Tree

30. L-convexity on uniform modular lattice
Def [Murota 96] 𝑓: ℤ 𝑛 →ℝ∪{∞} is L-convex:
𝑓 𝑝 +𝑓 𝑞 ≥𝑓 𝑝∧𝑞 +𝑓 𝑝∨𝑞
∃𝛼,𝑓 𝑝+𝟏 =𝑓 𝑝 +𝛼
ℒ: uniform modular lattice
Def [H. 17] 𝑓:ℒ→ℝ∪{∞} is L-convex:
𝑓 𝑝 +𝑓 𝑞 ≥𝑓 𝑝∧𝑞 +𝑓 𝑝∨𝑞
∃𝛼,𝑓 𝑝 + =𝑓 𝑝 +𝛼
Several DCA concepts/properties are naturally extended

31. deg Det 𝐴=
Min. − deg 𝐿 − deg 𝑀
s.t. deg 𝐴 𝑘 𝐿,𝑀 ⊆[−∞,0] (∀𝑘)
𝐿,𝑀∈ ℒ 𝕂 𝑡 𝑛
is viewed as L-convex function minimization
on uniform modular lattice
SDA = steepest descent algorithm for this L-convex function:
𝑝⟶ 𝑝 ′ : minimizer over [𝑝, 𝑝 + ]
submodular optimization
# iterations = 𝑑 ∞ (initial,opt)
adapting Murota-Shioura 2014

32. Summary Edmonds problem, rank v.s. nc-rank
Weighted Edmonds problem, deg det v.s. deg Det
formulation, algorithm, special case of deg det = deg Det
Submodularity / discrete convexity aspect
L-convexity on uniform modular lattice (= Euclidean building)

33. Problems If 𝐴= 𝐴 0 𝑡 𝑐 0 + 𝐴 1 𝑡 𝑐 1 𝑥 1 +…+ 𝐴 𝑚 𝑡 𝑐 𝑚 𝑥 𝑚
where 𝐴 𝑖 :𝑛×𝑛 over 𝕂,
can we compute deg Det A in time polynomial
in 𝑛,𝑚, and log max 𝑖 𝑐 𝑖 ?
Representable by deg det but deg det < deg Det :
Non-bipartite matching: Edmonds 1965
Matching forest: Giles 1982
Path matching: Cunningham-Geelen 1997
Linear matroid matching: Lovász 1980, Iwata-Kobayashi 2017
Can we develop a unified theory ?

34. References
H. Hirai: Uniform modular lattice and Euclidean building, 2017
H. Hirai: Uniform semimodular lattice and valuated matroid, 2018
H. Hirai: Computing degree of determinant via discrete convex optimization over Euclidean building, 2018.
Thank you for your attention