平面グラフとその周辺の話題kfujiwara/sendai/...until graph minors appeared. • perhaps,...
TRANSCRIPT
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日常生活には,平面構造のネットワークが多い.例えば, 1.鉄道網 2.道路網 etc….
マイナー操作
Introduction
ネットワークの解析のためには「平面グラフ」の研究が必要丌可欠!
平面判定
これらの「平面構造」という特徴を利用することで,ネットワークの理論的な解析が可能 → VLSI等のネットワーク構築 → 迅速なカーナビゲーションシステムの 情報アップデート etc…
私の研究
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平面グラフの一例
平面グラフとは,「辺をそれぞれ交差しないように 平面上に描くことができるグラフ」を指す
Introduction マイナー操作 平面判定 私の研究
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The Four Color Theorem
Straight line embedding
3-polytope
Polyhedral etc
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Every planar graph can be drawn
in the plane with straight edges
Fáry-Wagner
planar graph
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Steinitz
Every 3-connected planar graph
is the skeleton of a polytope.
3-connected planar graph
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Polyhedral version
Andre’ev
Every 3-connected planar graph
is the skeleton of a convex polytope
such that every edge
touches the unit sphere
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The Four Color Theorem
Straight line embedding
3-polytope
Polyhedral etc
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Planar network(平面ネットワーク)
Subway, Road etc (鉄道網、道路網等).
How can computer recognize? (コンピューターにどうやって認識さえるか?)
This needs some deep THEORY! (理論研究が必要!)
Need to convert into graphs! (グラフの言葉に変える必要がある!)
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How can computer recognize planar graph?
Remember, computer can only accept 0 or 1!
How can computer recognize a planar graph??
From the “0/1” input, everybody should share the same “picture” (without seeing the actual map).
We need some deep results in graph theory.
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Every planar graph is 4-colorable
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Unique embedding theorem
Every planar graph (essentially) has a unique embedding (Whitney, 1935).
So your embedding and my embedding are SAME! Point: Save the following information: Face, vertices and edges. Give “orientation” to each face. The above information can give an embedding
(uniquely “extendable”)
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Whitney Equivalence
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Introduction
問題: 不えられたネットワークが「平面グラフ」かどうかを,どのように判定するか?
問題: 不えられたネットワークが「平面グラフ」ならどのように「よい」平面グラフを得るか?→この問題の解答を得るためには
「マイナー操作」を理解する必要がある.
マイナー操作
ネットワークを理論的に解析するためには,実際のネットワークが平面(またそれに近い構造の)グラフかどうかを判定する必要がある!!
また、平面に高速に埋め込む必要がある!!
平面判定 私の研究
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Introduction マイナー操作
どんな平面グラフでも,次の3つの操作を加えた後も平面グラフの性質は変わらない
1.辺を取り除く.
2.頂点を取り除く.
3.辺を縮約する.
マイナー操作とは???
この3つの操作をグラフ理論の分野では「マイナー操作」という
平面判定 私の研究
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マイナー操作
1.辺を取り除く.
2.頂点を取り除く.
3.辺を縮約する.
Introduction マイナー操作
平面判定 私の研究
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マイナー操作
マイナー操作
1.頂点を取り除く.
2.辺を取り除く.
3.辺を縮約する.
Introduction 平面判定 私の研究
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この3つの操作(1.頂点を取り除く 2.辺を取り除く 3.辺を縮約する)をしても平面グラフがその性質を保存することを
「マイナー操作に関して閉じている」
と呼ぶ
マイナー操作
Introduction 平面判定
重要:「全ての平面グラフはマイナー操作に関して閉じている」が,「マイナー操作に関して閉じているグラフが全て平面グラフ」ではない!
私の研究
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平面判定
不えられたグラフが平面グラフかどうかを判定するためには,「非平面グラフ」から,「平面グラフ」になる「最小」のグラフは何かを調べる必要がある.
マイナー操作
Introduction
非平面グラフ
大きいグラフ平面グラフ マイナー操作
1.頂点を取り除く.
2.辺を取り除く.
3.辺を縮約する.
小さいグラフ
平面グラフ
非平面グラフとは限らない.
私の研究
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クラトスキーの定理 その1
非平面の「最小」なグラフは,K3,3とK5である. 与えられたグラフが平面グラフである必要十分条件は,「どのようなマイナー操作を
しても,K3,3とK5にならない」というこ
とである
非平面から平面になる「最小」のグラフは何か?
Introduction マイナー操作
平面判定 私の研究
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Introduction マイナー操作
平面判定
K3,3とK5グラフの例
左: K5グラフ 右: K3,3グラフ
私の研究
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平面グラフに対しては,
「禁止」すべき「最小グラフ」は,K3,3とK5のみである.
Introduction マイナー操作
平面判定
クラトスキーの定理 その2
平面グラフかどうかを判断するための指標になる.平面グラフかどうかの判定に利用されている.
私の研究
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Kuratowski定理の数学的応用
と計算機(アルゴリズム的)応用
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Planar graphs are closed under taking minors(minor-closed).
There are only two “forbidden” minors, namely K5 and K3,3
There are many other “minor-closed” graphs, for example, graphs on surfaces, linklessly embeddable graphs, knotless embeddable graphs in 3-space etc.
What about other minor-closed graph property (preserved when taking minors)?
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3.曲面上に埋め込めるグラフ(ドーナツ状のトーラス)
Introduction マイナー操作
私の研究 平面判定
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Given a “link” in 3-space. Is it knotted??
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Seminal series of ≥ 23 papers(> 500 pages) Perhaps the deepest theory in graph theory Powerful results on excluded minors:
• Every minor-closed graph property (preserved when taking minors) has a finite set of excluded minors [Wagner’s Conjecture]
• Every minor-closed graph property can be decided in polynomial time [Graph Minor Algorithm]
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1. A far-generalization of Kuratowski’s theorem (Wagner’s conjecture).
2. Many new “concepts” in the proof. Tree-width, Tangle, Representativty, clique-sum etc.
3. A real “tour de force”. A seminal decomposition theorem and structure theorem.
4. A solution to one of the original Gary-Johnson’s problems (the disjoint paths problem) .
5. Many algorithmic applications.
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Graph Minor Decomposition Thm Structure of H-minor-free Graphs [GM16—Robertson & Seymour 2003]
• Every H-minor-free graph can be written
as O(1)-clique sums of graphs (gluing).
(Tree-structure (Tree decomposition))
• Each summand is a (basic) graph that can be O(1)-almost-embedded into a bounded-genus surface (almost 2-dimentional)
• O(1) constants depend only on |V(H)|
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Tree-Decomposition
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Almost-Embeddable Graphs
Apex Surface
Vortex
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A shorter proof of graph minor algorithm (Wollan and KK, STOC’10)
(< 50 pages, it does not depend on graph minor decomposition theorem) Much shorter proof of the graph minor
structure theorem (Wollan and KK, 2010) (< 50 pages) Faster graph minor testing.
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Seminal series of ≥ 23 papers(> 500 pages) Perhaps the deepest theory in graph theory Powerful results on excluded minors:
• Every minor-closed graph property (preserved when taking minors) has a finite set of excluded minors [Wagner’s Conjecture]
• Every minor-closed graph property can be decided in polynomial time [Graph Minor Algorithm]
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平面グラフの判定の計算量 (Hopcroft, Tarjan)
平面グラフは,線形時間で判定できる (ここでいう『線形時間』とは,入力の頂点数に関して線形関数のステップという意味である).
この判定アルゴリズムを導き出すために,
クラトスキーの定理が利用されている.
Introduction マイナー操作
平面判定 私の研究
実用の場面では,平面グラフの判定を可能な限り迅速に行う必要がある!!
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クラトスキーの定理 その1
非平面の「最小」なグラフは,K3,3とK5である. 与えられたグラフが平面グラフである必要十分条件は,「どのようなマイナー操作を
しても,K3,3とK5にならない」というこ
とである
非平面から平面になる「最小」のグラフは何か?
Introduction マイナー操作
平面判定 私の研究
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私の研究の一例
実際のネットワーク網は,「完全な平面」とは限らない.(例えば,交通網における立体交差や鉄道網での地下トンネル等)
→ 不えられたグラフが,k交差グラフであるかの判定を線形時間で行うことは可能か?
→ 不えられたグラフが,k-平面グラフであるかの判定を線形時間で行うことは可能か?
…等を明らかにする必要がある
Introduction マイナー操作
私の研究 平面判定
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最近の研究で,以下の性質のグラフを
線形時間で判定することが可能であることを明らかにした(STOC’07,STOC’08,FOCS’08,
STOC’09、FOCS‘09等).
1.k-平面グラフ.
2.k交差グラフ.
3.曲面上に埋め込めるグラフ
(例えばドーナツ状のトーラス).
Introduction マイナー操作
私の研究 平面判定
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3.曲面上に埋め込めるグラフ(ドーナツ状のトーラス)
Introduction マイナー操作
私の研究 平面判定
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• Mainly joint work with B. Reed (and his students), partially with B. Mohar.
• (half (1/3?) of) Algorithmic Graph Minor Program (10+ STOC FOCS papers).
• Problem: Either give an embedding or a certificate(NOT embeddable).
• All of these embedding problems should be done in LINEAR time !!
• We are “almost” there !
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New Algorithmic Program • After Ripton-Tarjan’s algorithm, not so many known
until Graph Minors appeared.
• Perhaps, this is because we did NOT know “certificates” (Not having an embedding), until Graph Minors appeared.
• Right now, there is a “significant” generalization of Graph Minors’ algorithm.
• This gives rise to many new “LINEAR TIME ALGORITHMS” for graph embedding.
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• Embedding graphs on a fixed surface, FOCS’08 .
• Crossing Number (at most k) STOC’07.
• “Densely” embeddable on a fixed surface (face-width at least k) STOC’08
• Planar graph with k apices, FOCS’09.
• Linkless embedding& Knotless embedding.
• “Almost” embeddable graphs (Graph Minors)
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Surface
• By a surface, we mean a compact 2-dimensional manifold.
• We consider plane with crosscaps or handles.
• Plane + crosscap: Projective planar.
• Plane + handle : Torus
• Plane + two crosscaps : Klein bottle
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Torus
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Klein Bottle
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Embedding into a surface
• NP-complete to determine Euler genus of a given graph. (Thomassen, 1988)
• For fixed Euler genus k, O(nk) algorithm. (Filotti, Miller and Reif, 1990)
• O(n3) algorithm. (Robertson and Seymour, 1995)
• Linear time algorithm. (Mohar, 1997)
• Mohar’s algorithm gives either an embedding in the surface of Euler genus k or an obstruction (One of minimal forbidden minors.)
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Mohar’s algorithm
• First LINEAR time algorithm. • One of the most important graph
algorithms. But,,, 1. Proof > 100 pages. 2. Proof consists of 6 other papers ! 3. Hidden constant is huge…
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New Algorithm(FOCS’08)
• With B. Reed and B. Mohar
• We get another linear time algorithm.
• Better in the following sense:
1. Proof < 15 pages
2. Hidden constant is better..
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• Planarity Testing (Hopcroft-Tarjan etc)
• Embedding graphs on a fixed surface (Mohar + FOCS’08) .
• Crossing Number (at most k) STOC’07.
• “Densely” embeddable on a fixed surface (face-width at least k) STOC’08
• Planar graph with k apices FOCS’09.
• Linkless embedding& Knotless embedding.
• “Almost” embeddable graphs(Graph Minors)
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What is crossing number ?
• Roughly, planar graph + k edge crossings. • So it is close to planar graphs. • Formally, a graph has crossing number at
most k if it can be embedded in the plane with at most k edge-crossings.
• NP-complete to compute the exact crossing number.
• For fixed k, however, it is more feasible.
• O(nk+1) is easy for crossing number
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Crossing Number
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Property of Crossing Number
• Planar graphs and Graphs on a fixed surface are closed under taking minors, but a graph with k crossings is not.
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Not closed under minors
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Some hard crossing number problems
• What is crossing number of the complete graph of order k ?
• Only known up to k=14! • What is crossing number of the balance
complete bipartite of order 2k ? • This is a well-known conjecture by
Turan. (Called Turan’s “Brick Factory” Problem.)
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Computing Crossing Number
• NP-complete to decide the crossing number (Gary and Johnson, 1982)
• For fixed k, however, it is easy to decide whether or not a give graph has crossing number at most k in O(nk+1).
• For fixed k, Grohe (STOC’01 and JCSS 2004) gave an O(n2) algorithm (i.e, f(k) n2)
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Result
• Theorem [KK and REED]
For fixed k, there is a linear time algorithm for deciding whether or not a given graph can be embedded in a plane with at most k crossings.
• Actually the algorithm does give a desired embedding if one exists.
• It also implies the planarity with at most K edges problem.
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• Embedding graphs on a fixed surface, FOCS’08 .
• Crossing Number (at most k) STOC’07.
• “Densely” embeddable on a fixed surface (face-width at least k) STOC’08
• Planar graph with k apices, FOCS’09.
• Linkless embedding& Knotless embedding. SOCG’10
• “Almost” embeddable graphs (Graph Minors)
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How can computer recognize in 3-space
Remember, computer can only accept 0 or 1!
How can computer recognize a “knot” in 3-space?
This is actually a big open question in Math!
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The Unknotting Problem
Given a “link” in 3-space.
Is it knotted??
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The Unknotting Problem
Given a “link” in 3-space.
Is it knotted??
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History
Haken gave an “algorithm” to decide in
1950’s. Not polynomial time. In P?? One of the most challenging problems
in Topology, Graph Theory and Theoretical Computer Science.
Open for >50 years…
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Linklessly embeddable graphs
homological, homotopical,…
equivalent
embeddable in 3-space without linked cycles
Apex graph
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Linkless Embedding Linkless embedding in 3-space: Embedding a
graph in 3-space so that there is no “link”.
Given a graph G, can you decide if there is a linkless embedding? (Yes, by Robertson-Seymour-Thomas, 1995)
Theorem[RST]
A graph G has a linkless embedding if and only if G does not contain Petersen family as a minor.
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The Petersen family
(graphs arising from K6 by Δ-Y and Y- Δ)
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K6 and Petersen Graph
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Introduction マイナー操作
平面判定
Planar graphs = No K5 nor K3,3 as a minor
左: K5グラフ 右: K3,3グラフ
私の研究
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Linkless Embedding Linkless embedding in 3-space: Embedding a graph
in 3-space so that there is no “link”. Given a graph G, can you decide if there is a
linkless embedding? (Yes, by Robertson-Seymour-Thomas, 1995)
Theorem[RST] A graph G has a linkless embedding if and only if G
does not contain Petersen family as a minor. Theorem[RST] Essentially unique embedding (for 4-connected
graphs)
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Unique embedding theorem
Every planar graph (essentially) has a unique embedding (Whitney, 1935).
So your embedding and my embedding is SAME! Point: Save the following information: Face, vertices and edges. Give “orientation” to each face. The above information can give an embedding
(uniquely “extendable”)
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Recent Progress
Flat embedding: Embedding in 3-space so that each cycle bounds a disk. So a generalization of planar embedding to 3-space! Flat embedding -> Linkless embedding But Linkless embedding does not imply flat embedding.
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Flat embedding VS unknotting problem
Conjecture 1 (Lovasz, Schrijver, RST, 1995) Poly time algorithm to test if a GIVEN
embedding is FLAT. Conjecture => the unknotting problem! Converse?? Theorem(SOCG’10): Two big conjectures are equivalent!
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Linkless Embedding Linkless embedding in 3-space: Embedding a graph
in 3-space so that there is no “link”. Given a graph G, can you decide if there is a
linkless embedding? (Yes, by Robertson-Seymour-Thomas, 1995)
If there is, can you construct?? (conjectured yes, by Lovasz, Schrijver, RST)
Yes! O(n2) algorithm (first P time algorithm! SOCG’10))
Theorem: O(n) time algorithm to find a flat
embedding in 3-space!
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平面グラフの判定の計算量 (Hopcroft, Tarjan)
平面グラフは,線形時間で判定できる (ここでいう『線形時間』とは,入力の頂点数に関して線形関数のステップという意味である).
この判定アルゴリズムを導き出すために,
クラトスキーの定理が利用されている.
Introduction マイナー操作
平面判定 私の研究
実用の場面では,平面グラフの判定を可能な限り迅速に行う必要がある!!
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Open Problem The Unknotting Problem
Given a “link” in 3-space.
Is it knotted??
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まとめ
平面グラフのKuratowski定理は、グラフ理論、理論計算機分野における中心的課題。
数学においてはグラフマイナー理論を導き出す(Robertson&Seymour)。
平面性の線形時間判定(Hopcroft&Tarjan)
トポロジーの中心的課題にも関連
研究課題は多数残っている!