2-SAT Group Aha 01 Party problem CONTENTS 02 Model 03 Algorithm 04 - - PowerPoint PPT Presentation

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Dec 06, 2023 411 likes •840 views

2-SAT Group Aha 01 Party problem CONTENTS 02 Model 03 Algorithm 04 Existence of possible solution 05 Find a possible solution Part I Party Problem Party Problem There is a party to be held in ACM class. There are three types of

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2-SAT

Group Aha

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01 Party problem 02 Model 03 Algorithm 04 Existence of possible solution 05 Find a possible solution

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Part I Party Problem

There is a party to be held in ACM class.
There are three types of relationship between them.
Friend 𝐵 ⟶ 𝐶
A think B is his/her friend. So if B go to the party, A will go with B.
Pair 𝐵 ⟶ 𝐶
A and B are in a same pair, so at least one of them will go to the party.
Contradiction 𝐵 ⟶ 𝐶
A and B dislike each other, so at most one of the will go the party.

Party Problem

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Part I Party Problem

7 students:
Car(Yang Jiacheng)， Fang Bohui，QiuWenda，Su Yufeng，Tan Bowen，

Xu Zhenjia，Yang Zhuolin

Car⟶ Xu，Qiu ⟶Tan，Tam ⟶Qiu，Qiu ⟶Fang，Su ⟶Tan
Xu ⟶Yang，Su ⟶Qiu
Yang ⟶Car，Yang ⟶Tan

Do we have an approach to satisfy all the relationships?

Example:

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Part II Model

Every variable has a value 0 or 1
Car(Yang Jiacheng) ⟶ C
Fang Bohui ⟶ F
Qiu Wenda ⟶ Q
Su Yufeng ⟶ S
Tan Bowen ⟶ T
Xu Zhenjia ⟶ X
Yang Zhuolin⟶ Y

Variable

C = 1 means Car will go to the party
C = 0 means Car won’t go to the party

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Part II Model

relationship ->disjunctive normal form

Xu ⟶ Yang 𝑌 ∨ Y ¬𝑌 ⟹ 𝑍 ¬𝑍 ⟹ 𝑌 Yang ⟶ Car ¬𝑍 ∨ ¬𝐷 𝑍 ⟹ ¬𝐷 𝐷 ⟹ ¬𝑍 Car ⟶ Xu 𝐷 ∨ ¬𝑌 𝐷 ⟹ 𝑌 ¬𝑌 ⟹ ¬𝐷

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Part II Model

So our aim is give each variable a value(0 or 1) to make the expression

equals 1

Relationship

(C∨¬X)∧(Q∨¬T)∧(T∨¬Q)∧(Q∨¬F)∧(S∨¬T)∧(X∨Y)∧(S∨Q)∧(¬Y∨¬C)∧(¬Y∨¬T)

Car⟶ Xu，Qiu ⟶Tan，Tam ⟶Qiu，Qiu ⟶Fang，Su ⟶Tan
Xu ⟶Yang，Su ⟶Qiu
Yang ⟶Car，Yang ⟶Tan

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Part II Model

C = 1
F = 0
Q = 1
S = 1
T = 1
X = 1
Y = 0
R = True

Example

C = 0
F = 1
Q = 1
S = 1
T = 1
X = 0
Y = 0
R = False

R = (C∨¬X)∧(Q∨¬T)∧(T∨¬Q)∧(Q∨¬F)∧(S∨¬T)∧(X∨Y)∧(S∨Q)∧(¬Y∨¬C)∧(¬Y∨¬T)

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Part III 2-Sat

2-satisfability

In computer science, 2-satisfiability is a computational problem of

assigning values to variables, each of which has two possible values, in order to satisfy a system of constraints on pairs of variables.

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Part II Model

𝐷𝑏𝑠

𝐷 ¬𝐷

>1 𝑢𝑝 𝑕𝑝 0 𝑜𝑝𝑢 𝑢𝑝 𝑕𝑝 >1 𝑜𝑝𝑢 𝑢𝑝 𝑕𝑝 0 𝑢𝑝 𝑕𝑝

𝑄 𝑅

1 𝑄 = 1 ⟹ 𝑅 = 1

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Part II Model

Friend

Car ⟶ Xu 𝐷 ∨ ¬𝑌 𝐷 ⟹ 𝑌 ¬𝑌 ⟹ ¬𝐷

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Part II Model

Friend

Qiu ⟶ Tan

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Part II Model

Friend

Tan ⟶ Qiu

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Part II Model

Friend

Qiu ⟶ Fang

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Part II Model

Friend

Su ⟶ Tan

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Part II Model

Pair

Xu ⟶ Yang 𝑌 ∨ Y ¬𝑌 ⟹ 𝑍 ¬𝑍 ⟹ 𝑌

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Part II Model

Pair

Su ⟶ Qiu

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Part II Model

Contradiction

Yang ⟶ Car ¬𝑍 ∨ ¬𝐷 𝑍 ⟹ ¬𝐷 𝐷 ⟹ ¬𝑍

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Part II Model

Contradiction

Yang ⟶ Tan

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Part II Model

Obs1 :

𝑣

⋯⋯

𝑥

⋯⋯

𝑣 𝑤L 𝑤M 𝑥

⋯⋯

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Part II Model

Obs2 :

In the graph theory, a graph is

said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected.

SCC(strongly connected component)

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Part II Model

Obs3 : 𝑣 ⋯ ⋯

𝑣 = 0

𝑣 = 1 ⟹ 𝑣 N = 1

¬𝑣

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Part II Model

Obs4 : 𝑣 ⋯ ⋯ ¬𝑣 ⋯ ⋯

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Part III Algorithm

1.Construct graph. Find Strongly Connected Components(SCC) of it.
Time complexity: O(m)
2. Construct the condensation of the implication graph.
3.Check whether any SCC contains a variable and its negation.
3.1 if true: Sad. We can’t find a perfect plan. Halt.
3.2 else: Happy! We can find a perfect plan.
4. Topologically order the vertices.
5. Based on the lemma proved before, we can select them from bottom and

make a perfect plan.

Algorithm

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Part III Algorithm

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Part III Algorithm

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Part III Algorithm

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Part III Algorithm

1.Construct graph. Find Strongly Connected Components(SCC) of it.
Time complexity: O(m)
2. Construct the condensation of the implication graph.
3.Check whether any SCC contains a variable and its negation.
3.1 if true: Sad. We can’t find a perfect plan. Halt.
3.2 else: Happy! We can find a perfect plan.
4. Topologically order the vertices.
5. Based on the lemma proved before, we can select them from bottom and

make a perfect plan.

Algorithm

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Part III Algorithm

1 1 1 1 1 1

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Part III Algorithm

1.Construct graph. Find Strongly Connected Components(SCC) of it.
Time complexity: O(m)
2. Construct the condensation of the implication graph.
3.Check whether any SCC contains a variable and its negation.
3.1 if true: Sad. We can’t find a perfect plan. Halt.
3.2 else: Happy! We can find a perfect plan.
4. Topologically order the vertices.
5. Based on the lemma proved before, we can select them from bottom and

make a perfect plan.