Joint work with Noga Alon WOLA 2019 GRAPH MODIFICATION For an input - - PowerPoint PPT Presentation

Joint work with Noga Alon

Clara Shikhelman WOLA 2019

GRAPH MODIFICATION

For an input graph 𝐻 find the minimum possible number of edges/vertices we need to add/remove/edit to get a graph with given property.

INTRO

NP-HARD EDGE MODIFICATION PROBLEMS

Yannakakis ‘81 being outerplanar, transitively orientable, and line-invertible. Asano and Hirata ‘82, Asano ’87 Certain properties expressible by forbidding minors or topological minors. Natanzon, Shamir and Sharan ‘01 Hereditary properties such as being Perfect and Comparability.

INTRO

APPROXIMATION OF EDGE MOD PROBLEMS

Fernandez de la Vega ‘96, Arora, Karger and Karpinski ‘95 several NP-complete problems such as MAX-CUT and MAX-3-CNF Frieze and Kannan ‘99 Graph theo. properties Arora, Frieze and Kaplan ‘02 Quadratic assignment problems and other Alon, Vega, Kannan and Karpinski ‘02 Constraint- Satisfaction-Problem

INTRO

SOME DEFINITIONS

A graph property is called monotone if it can be defined by forbidding a family of graphs. The only relevant edge modification for monotone properties is edge deletion. For a graphs 𝐻, 𝑈 and a family of graphs ℱ let 𝑓𝑦(𝐻, 𝑈, ℱ) be the maximum possible number of copies of 𝑈 in an ℱ-free subgraph of 𝐻.

INTRODUCTION

𝑓𝑦(𝐻, 𝐿2, ℱ)

Alon, Shapira and Sudakov ’05 For any 𝜗 > 0 and ℱ there is a polynomial time algorithm that approximates 𝑓𝑦(𝐻, 𝐿2, ℱ) up to an additive error

• f 𝜗𝑜2.

A significantly (𝑜2−𝜗) better approximation is possible iff there is a bipartite graph in ℱ.

ALGORITHM FOR GENERAL 𝑈

Alon, Sh. ‘18+ For any graph 𝑈, finite family

• f graphs ℱ and 𝜗 > 0 there is a polynomial

time algorithm that approximates 𝑓𝑦(𝐻, 𝑈, ℱ) up to an additive error of 𝜗𝑜𝑤 𝑈 . Can we do better?

CAN WE DO BETTER?

𝓒 𝑼 - The family of blow ups is all the graphs obtained from 𝑈 by replacing vertices with independent sets and every edge with complete bipartite graph. Proposition (Alon, Sh. ‘18+) Let 𝑈 be a graph and ℱ a family of graphs s.t. there is a graph 𝐼 ∈ ℱ ∩ ℬ(𝑈). Then 𝑓𝑦 𝐻, 𝑈, ℱ can be calculated up to an additive error of 𝑜𝑤 𝑈 −𝑑(𝑈,ℱ) in polynomial time.

CAN WE DO BETTER?

Conjecture It is NP-hard to approximate 𝑓𝑦(𝐻, 𝑈, ℱ) up to an additive error of 𝑜𝑤 𝑈 −𝜗 iff ℱ ∩ ℬ 𝑈 = ∅. Proved for:

• 1. Both 𝑈 and ℱ are complete graphs
• 2. Both 𝑈 and ℱ are 3-connected, NP-hard up to an

additive error of 𝑜𝑤 𝑈 −2−𝜗

• 3. In progress – additive error of 𝑜𝑤 𝑈 −𝜗

INTUITION FOR THE ALGORITHM

REGULAR PAIRS

Given a graph 𝐻 and a pair of disjoint sets 𝑊

1, 𝑊 2,

we say that (𝑊

1, 𝑊 2) is an 𝝑-regular pair if for

every 𝑉𝑗 ⊆ 𝑊

𝑗 s.t. 𝑉𝑗 > 𝜗|𝑊 𝑗|

𝑒 𝑊

𝑗, 𝑊 𝑘 − 𝑒 𝑉𝑗, 𝑉 𝑘

≤ 𝜗 Where 𝑒 𝑊

𝑗, 𝑊 𝑘 = 𝑓 𝑊𝑗,𝑊𝑘 𝑊𝑗 ⋅|𝑊𝑘|

INTUITION FOR THE ALGORITHM

REGULAR PARTITIONS

For a graph 𝐻, a partition of its vertices 𝑊 = 𝑉 ∪𝑗=1

𝑙

𝑊

𝑗 is called

strongly 𝝑-regular if

• 1. For every 𝑗, 𝑘 𝑊

𝑗 = |𝑊 𝑘| and 𝑉 < 𝑙

• 2. Every pair 𝑊

𝑗, 𝑊 𝑘 is 𝜗-regular

Can be found in pol-time by changing 𝑝(𝑜2) edges.

𝑒1𝑗 𝑒1𝑙 𝑒12

𝑊

𝑙

𝑊

2

𝑊

1 𝑊

𝑗

𝑉

INTUITION FOR THE ALGORITHM

CONVENTIONAL SUBGRAPH

A conventional subgraph of 𝐻 if it is obtained by deleting 1. All of edge inside the sets 𝑊

𝑗

2. All of the edges with endpoint in 𝑉 3. All of the edges between some (𝑊

𝑗, 𝑊 𝑘)

How many conventional subgraphs are there? 2

𝑙 2

𝑒1𝑙 𝑒12

𝑊

𝑙

𝑊

2

𝑊

1 𝑊

𝑗

𝑉

𝑒1𝑗 𝑒1𝑙 𝑒12

𝑊

𝑙

𝑊

2

𝑊

1 𝑊

𝑗

𝑉

THE (SIMPLE) ALGORITHM

For a graph 𝑈, a finite family of graphs ℱ and input graph 𝐻:

• 1. Find a strong 𝜗-regular partition of 𝐻.
• 2. Find a conventional subgraph of 𝐻 which is ℱ-homorphism-free

and has max. possible 𝑈 density. Main Lemma This graph has ≥ 𝑓𝑦 𝐻, 𝑈, ℱ − 𝜗𝑜𝑤 𝑈 copies of 𝑈.

THANKS FOR YOUR ATTENTION!

𝑒1𝑙 𝑒12

𝑊

𝑙

𝑊

2

𝑊

1 𝑊

𝑗

𝑉

𝑒1𝑗 𝑒1𝑙 𝑒12

𝑊

𝑙

𝑊

2

𝑊

1 𝑊

𝑗

𝑉