大規模複雑システムの最適モデリング手法の構築

岩田 覚

（東京大学情報理工学系研究科）

CREST 「数理モデリング」領域

最適モデリング

現象・問題

モデル

最適モデル

モデル

モデル

モデル

数値解最適解

回路解析における最適モデリング

最小基本方程式

混合解析 Kron (1939)

甘利 (1962) 伊理 (1968)

マトロイドWhitney (1935)

修正節点解析 SPICE (1973)

マトロイド理論の工学的応用

微分代数方程式モデルによる過渡解析

最小指数混合方程式 岩田・高松 (2010)

修正節点解析に対する混合解析の優位性

高松・岩田 (2010) 岩田・高松・Tischendorf (2012)

研究計画

① 微分代数方程式

② 統計的モデリング

③ 大規模ネットワーク

H26.10 H29.3

④ 生命現象の数理モデリング

社会システムへの応用・電力システム・交通システム

生命現象の最適モデリング

H32.3

最適モデリング手法の開発

主要研究成果①微分代数方程式の最適モデリング

②統計的モデリングの最適化

③大規模ネットワークの情報圧縮

④生命現象の数理モデリング

⑤社会システムの最適モデリング

微分代数方程式の指数減少法

重み付き線形マトロイド・パリティ (STOC 2017, Best Paper Award)

離散DC計画問題の連続緩和法 (MPB 2017)

DC計画法によるスパース最適化 (MPB 2017)

低ランク行列基底問題 (MPA 2017)

蝙蝠の採餌行動時の飛行経路 (PNAS 2016)

シロイヌナズナの概日時計の階層構造 (Cell 2015)

優等列車停車駅の最適化による混雑緩和 (ATMOS 2017)

ランダム正則グラフの直径 (SODA 2018)

線形マトロイド・パリティ

最大独立パリティ集合

最大最小定理 Lovász (1980)

多項式時間解法 Lovász (1980), Gabow and Stallmann (1986)

Orlin and Vande Vate (1990)

応用

RCG 回路の構造可解性 Milic (1974)

平面トラス構造の剛性 Lovász (1980)

グラフの閉曲面への埋込み Furst, Gross, McGeoch (1988)

グラフ上の内点素路 Lovász (1980), Schrijver (2003)

𝑉:線（2要素の組）に分割

パリティ集合: 線の和𝐴 =

𝑉

𝑋

RCG回路の一意可解性抵抗 キャパシタ ジャイレータ

𝜉𝜉

𝜉1 𝜉2

ηη η1 η2𝑅𝐶 𝑔

𝜉 =𝜂

𝑅 𝜉 = 𝐶d𝜂

d𝑡

𝜉1

𝜉2=

0 −𝑔𝑔 0

𝜂1𝜂2

一意可解⟺ 各ジャイレータの両方の枝を含むか両方とも含まない全域木が存在する

→ 判定問題が線形マトロイド・パリティに帰着される．

過渡解析の微分代数方程式の指数の計算には，重み付き線形マトロイド・パリティを解く必要がある．

重み付き線形マトロイド・パリティ

A

最小重みのパリティ基

𝑤 𝐵 ≔

ℓ⊆𝐵

𝑤(ℓ)

𝑤: 𝐿 → ℝ

B

乱択擬多項式時間解法 Camerini, Galbiati, Maffioli (1992)

Cheung, Lau, Leung (2014)

応用

Steiner木問題の近似解法 Prömel and Steger (2000)

総路長最短の内点素路 Yamaguchi (2016)

主結果と方法

• 歪対称多項式行列による定式化.

• 組合せ緩和法 Murota (1990).

• 増加道アルゴリズム Gabow & Stallmann (1986).

𝑂 𝑟𝑛3 回の算術演算で重み付き

線形マトロイド・パリティ問題を解く組合せ的多項式時間アルゴリズム

S. Iwata and Y. Kobayashi, A weighted linear matroid parity algorithm,Proceedings of the 49th ACM Symposium on Theory of Computing (STOC), 2017, pp. 264-276.

labelled -regular graphs of vertices.d n‣

DIAMETER = ??

The Diameter of Dense Random Regular Graphs

‣Bollobás and de la Vega (1982) :diam(Gn,d) ⇡

log n

log(d� 1)

for constant d � 3 . NOT KNOWN FOR

‣THIS WORK :

limn!1

diam(Gn,d) = b↵�1c+ 1 for d / n↵

Gn,d : uniform random element from the set of

diam(Gn,d) :=

.

‣Application to designing of network topologies.

Nobutaka Shimizu(to appear in SODA ’18)The University of Tokyo

A 3-regular graph

as n ! 1

d = d(n) = !(1)

1/ 2

Combinatorial secretary problemsCombinatorial secretary problems [Kleinberg’05]We want to hire the best combination of candidates

Interviewing from left in random order

At each arrival, we must decide whether to hire,and this decision cannot be revokedThe number of candidates is known in advance

Algorithms for k-submodular secretary problemsKaito Fujii (UTokyo)

2/ 2

We want to hiremultiple positions at the same timee.g.) not only secretaries, but CTO

Interviewing from left in random order

Secretary

CTOWe formulate this problem ask-submodular secretary problemsWe propose algorithms for several constraints

Algorithms for k-submodular secretary problemsKaito Fujii (UTokyo)

大城 泰平（東京大学）joint work with 岩田覚 教授（東京大学），高松瑞代 准教授（中央大学）

Index Reduction for Differential-Algebraic Equations

with Mixed Matrices

微分代数方程式 Differential-Algebraic Equation

fi(t, x(t), x(t), x(t), . . .) = 0(i = 1, . . . , n)

指数が高い DAE は数値的に解きにくい！数値計算の前に，低指数の DAE に変換したい

グラフアルゴリズムを用いた既存手法 [Mattsson−Söderlind ’93] では，数値キャンセルに起因する失敗が起こることがある．

問題点

本研究で与えたアルゴリズム

入力定数係数線形 DAE

ここで各 は混合行列Ak

出力 入力と同一の解を持つ低指数の DAE

NX

k=0

Akx(k) = f(t)

混合行列🌸数値的要素（正確な定数）と 🌿組合せ的要素（非精確なパラメータ；物理量）を持つ行列

キーワード：組合せ緩和，マトロイド，組合せ行列理論

1

2

( 1)

N

iji j iL d

N N

12

( 1)

N

ij jm mij m j

ii i

a a aC

N k k

,i i j

i j G

k a

,

1 1

1i

i j G i j

EN d

Study the Network Development Patterns of Brain by Graph TheoryDUAN Fang* and Kazuyuki Aihara

Institute of Industrial Science, the University of Tokyo

(M. Kikuchi et al, 2011)

Correlation maps between the nodal level indices and the achievement scale.

Correlation coefficient between the small-world scalars and the raw scores of K-ABC.

The well-functioning brain was better able to drive the functional network structure towardsincreased efficiency in local.

Network Development Patterns of beta-band neural rhythm.

➢ PPG is widely used in clinical settings and sport equipment

➢ PPG can be recorded with green (gPPG) and NIR light, but gPPG is

movement artifacts resistant

➢ PPG is chaotic has short-term and doesn’t have long term prediction

noise doesn’t affect its trajectory equally

➢ Improvement of PPG prediction can benefit health monitoring area

Nina Sviridova

Institute of Industrial Science, The University of Tokyo, Tokyo, Japan

Introduction

➢ Data were collected from six 20-40 year old healthy human

subjects.

➢ Finger reflectance-type recorder based on Arduino UNO

board, with sensor equipped with green-light LED.

➢ Data recording was conducted in a sitting-relaxed position,

in quiet air-conditioned room with moderate light.

M&M: gPPG

ቐ

ሶx = −σ x − y ,ሶy = ρx − y − xz,ሶz = xy − βz,

σ = 10, ρ = 28, β = 8/3

M&M: Lorenz model and local noise sensitivity

The Local Translation Error (LTE) => determinism of the signal :

𝑒𝑙𝑜𝑐=1

𝑘+1σ𝑗=0𝑘 𝑣𝑗− 𝑣

2

𝑣 2 ,

where x0 is a fixed and arbitrary chosen vector on reconstructed trajectory, xj

(j=1,2,…,k) are its k nearest neighbors, yi are images of xi (i=0,1,…,k), vi = yi–xi

is translation vectors, 𝑣 is the average of 𝑣𝑗.

Acknowledgement. The

author would like to

acknowledge Dr. Zhao Tiejun

from National Agriculture and

Food Research Organization

(Tsukuba, Japan) for providing

PPG data for this study.

DNP: y 𝑡 + 𝑝 = 𝑥 𝑡 +1

𝑛σ𝑖=1𝑛 𝜗𝑖 ,

WDNP: 𝑦∗ 𝑡 + 𝑝 + 𝑗 = ቀ

ቁ

𝑥 𝑡 +

+1

𝑛σ𝑖=1𝑛 𝑦𝑖

∗ 𝑡 + 𝑝 + 𝑗 − 𝑥∗(𝑡 − 𝑗) ,

𝑀𝑡 =1

𝑘 + 1

𝑖=0

𝑘

𝐿𝑇𝐸(𝑥𝑖 (𝑡))

𝑀𝑡 is compared with averaged

LTEs of preceding points and

its neighbors – Mt-j (j=1,..,l).

The point 𝑥0∗ with the smallest

value M is chosen as a new

predictee; if j>0 then the

prediction is conducted for

(p+j) steps.

Results

Results

The correlation coefficient between original and predicted gPPG time series for direct prediction (blue dashed) and modified prediction (red solid line).

Local noise sensitivity7 in gPPG. Distribution of local determinism along a time-delay

reconstructed gPPG trajectory for typical time-series from a subject in 20s, the

translation error is 0.16

CONCLUSION

In this study modification of

deterministic nonlinear

prediction based on the

information regarding the

effects produced by noise

on the dynamics of the

chaotic signal was

proposed. The obtained

results demonstrated that

approach to account

presence of additive noise

in the data and its effect on

the data dynamics can

provide an improvement in

the prediction of chaotic

Lorenz model. For the

gPPG case, only a slight

prediction improvement

was achieved for the

number of prediction steps

close to the heart cycle

time. Based on the results

for the Lorenz system it can

be noticed that for the

greater amount of noise in

data, suggested prediction is

more beneficial. It is

important to notice that

while the Lorenz model

data were artificially noise-

induced, the raw signal was

used for calculations

corresponding to the gPPG

data. Therefore the further

investigation required for

improvement of the gPPG

prediction.

x(t)y(t+p)

x1(t)

y1(t+p)

x(t-j)

Envy-Free Matchings with Lower Quotas横井優 (国立情報学研究所, CREST 岩田プロジェクト)

選好 h2 ≻ h1 ≻ h3

h1

h2

h3

d2

d3

d4

d5

選好 d5 ≻ d1 ≻ d4 ≻ d3

上限制約 𝑞h1: 2𝐷 → 𝐙+

下限制約 𝑝h1: 2𝐷 → 𝐙+

𝐷: doctors 𝐻: hospitals

上下限制約を守りながら各人の選好に考慮しながら マッチングしたい

d1

2

目標：Envy-freeマッチングの存在判定 & 発見

系：上下限制約が層族上に定義されている場合は効率的に解ける

ℎ ≻𝑑(現割当)ℎ • 𝑑 ≻ℎ 𝑑′

• (現割当)+𝑑 − 𝑑′が上下限制約を満たす

𝑑′

𝑑

𝑑 は 𝑑′ に対して justified-envy をもつ

結果： Envy-freeマッチングの存在判定は一般にはNP-hard,上下限がparamodularなら (i.e., 実行可能域がM♮凸なら) P.