平面網路上防疫問題之近似演算法

୯ҥᆵ᡼εᏢႝᐒၗૻᏢଣၗૻπำᏢࣴز܌

ᅺγፕЎ

Department of Computer Science and Information Engineering College of Electrical Engineering and Computer Science

National Taiwan University Master Thesis

ѳय़ᆛၡ΢ٛࣝୢᚒϐ߈՟ᄽᆉݤ

An Approximation Algorithm for the Inoculation Problem for Planar Networks

ߋ߷ഩ Kuan-Kai Chiu

ࡰᏤ௲௤ǺֈᏢ΋ റγ Advisor: Hsueh-I Lu, Ph.D.

ύ๮҇୯ 97 ԃ 7 Д

July, 2008

誌 謝

這篇論文之所以能夠完成^, 首先要感謝我的指導教授呂學一老師 。 從一開始題目的挑 選 、 研究經驗的分享 ^, 一直到論文的寫法和架構 、 口試投影片的準備^,老師都非常費心 地指導 ^, 讓我的研究成果得以清楚完整地呈現出來 。 不管是面對面還是透過電子郵件^, 那 無數次的討論 ^,現在回想起來都還是歷歷在目 。 最重要的 ^, 是能夠通過老師嚴格的督導 完成這篇論文 ^, 相信以後的我 ^, 不管遇到任何困難的事^, 都絕對不會失去信心 。

感謝王大為老師和鍾國亮老師^, 願意在百忙之中抽空擔任口試委員^, 並對論文提供許 多寶貴的意見 ^, 使這篇論文更臻完善 。

再來^, 我也很感謝這兩年來一起陪我努力的夥伴們 。 冠伶學姐的開導跟指點 ^,讓我得 以保持信心繼續努力做研究 。 國煒學長的照顧跟搞笑 ^,讓我由衷地感激和敬佩 。 還有宗 灝 、 昱豪 、 弘偉 、 柏穎 、 世鵬 、 雅斐 、 文良 、 偉揚 、 婕妤等實驗室的好朋友們 ^,讓我在 這跌跌撞撞走來的兩年裡 ^, 充滿許多歡笑和回憶 。

最後想要感謝的^, 是我的父母還有女朋友黃韻倩小姐^, 對於他們無怨無悔地關心和支 持^, 這篇論文對我而言^, 是目前能夠回報他們最好的方式 。

能不能完成一件事情^, 全端視自己有多渴望成功^, 這是我在碩士生涯裡學會最重要的 一件事 。 僅以這篇論文 ^, 獻給所有在我拼了命努力研究的兩年裡^,一直鼓勵和支持我的 人們 。

2008年⁷月 邱冠凱

平面網路上防疫問題之近似演算法

研究生 ^: 邱冠凱 指導教授 ^: 呂學一 博士

國立臺灣大學電機資訊學院資訊工程學研究所

摘 要

給兩個數字 c 和 k ,以及 n 個點的一般網路 G ,防疫問題的定義為 ^:求出由 G 當中最 多 k 個點所組成的點集合 S ,使得 c · m + _n¹ 

in²_i 的值為最小^, 其中 m 為S 的點個 數, ni 為G \ S 的第i個連通單元 。 對於這個 ^NP-完備的問題 ^, 目前已知最好的結果是 由 Aspnes、_Chang 和 Yampolskiy 提出的O(log^1.5n) 倍比率的近似演算法 。 我們在本 篇論文中證明當G為平面網路時 ^, 這個問題仍是^NP-完備 ^, 並且提出一個O(log n)倍比 率的近似演算法 。

An Approximation Algorithm for the Inoculation Problem for Planar Networks

Student : Kuan-Kai Chiu Advisor : Hsueh-I Lu, Ph.D.

Department of Computer Science and Information Engineering College of Electrical Engineering and Computer Science

National Taiwan University

Abstract

For numbers c and k and an n-node graph G, the inoculation problem is to compute an S consisting of at most k nodes of G such that c · m + _n¹ 

in²_i is minimized, wherem is the cardinality of S and ni is the number of nodes in thei-th connected component of G \ S. The best previously known result, due to Aspnes, Chang, and Yampolskiy, for this NP-complete problem is anO(log^1.5n)-approximation algorithm. In the present article, we focus on the special case thatG is planar: We show that the problem remains NP-complete and give anO(log n)-approximation algorithm for the problem.

Contents

Acknowledgements i

Chinese Abstract ii

English Abstract iii

1 Introduction 1

2 Preliminaries 4

2.1 Hardness . . . 4

2.2 Minimum quotient cut . . . 5

3 Our algorithm 7 3.1 A reduction . . . 7

3.2 Finding a bicriterion approximate solution . . . 8

3.3 Finding a node set with good cost effectiveness . . . 12

3.4 Proving Theorem 1.1 . . . 16

4 Concluding remarks 17

Bibliography 18

List of Figures

1.1 IfG is the graph as shown in the left and S = {a, b, c}, then φ(S) = 23. . 2

Chapter 1

Introduction

For any setS, let |S| denote the cardinality of S. For any node subset S of G, let G \ S denote the graph obtained fromG by deleting the nodes in S and the edges incident to the nodes inS. For any node subset S of G, let

φ(S) =

i

n²_i,

whereniis the number of nodes in thei-th connected component of G\S. See Figure 1.1 for an example.

Given ann-node graph G and two numbers c and k, the inoculation problem is to find a node subsetS of G with |S| ≤ k that minimizes

c · |S| + 1

n · φ(S).

To address a game-theoretical model of network security (see, e.g., [2, 3, 6–8, 10–18, 20]), Aspnes, Chang, and Yampolskiy [4] formulated the problem with k = n to describe the following scenario of virus attack. Suppose that each infected node incurs 1 unit of penalty and it takes c units of cost to secure a node by, say, installing an anti-virus

c b

a

Figure 1.1: IfG is the graph as shown in the left and S = {a, b, c}, then φ(S) = 23.

software. The virus spreads by infecting some initial node chosen uniformly at random.

An insecure node gets infected if any of its neighbors in the graph is infected. IfS consists of the secured nodes, thenc·|S| is the inoculation cost and_n¹·φ(S) is the expected penalty incurred by all infected nodes.

Aspnes et al. [4] ensured the NP-completeness of the problem. They also showed that it takes in ˜O(n⁴) time¹to compute an O(log²n)-approximation solution for the problem with k = n. In the journal version [5], they further reduced the approximation ratio to O(log^1.5n) while raised the time complexity ˜O(n^6.5). Moscibroda, Schimid, and Watten- hofer [19] studied the problem for highly regular and low-dimensionalG.

The present article focuses on planar G, for which case we show that the problem remains NP-hard and obtain an approximation algorithm as summarized in the following theorem.

Theorem 1.1. For anyn-node planar graph G, any number c, and any positive constant

, it takes ˜O(n^6+) time to compute an O(log n)-approximate solution for the inoculation problem withk = n.

Similar to the method of Aspnes, Chang, and Yampolskiy [4, 5], our approach repeat-

1The ˜O(·) notation suppresses the polylog(n) factors.

edly removes a near-optimal sparse vertex cut from some connected component of the current graph. The near-optimal sparse vertex cut causes a sufficient decrease of risk per removed node, quantified as its cost effectiveness. We compute a node set with sufficient cost effectiveness by resorting to the approximation algorithm of Amir, Krauthgamer, and Rao [1] that finds a near optimal solution for the minimum quotient vertex cut problem for planar graphs.

The rest of the paper is organized as follows. Chapter 2 gives the preliminaries. Chap- ter 3 gives our algorithm. Chapter 4 concludes the paper.

Chapter 2

Preliminaries

For the rest of the paper, letG be the input n-node planar graph.

2.1 Hardness

The following lemma ensures that the inoculation problem withk = n remains NP-hard even ifG is planar.

Lemma 2.1. The inoculation problem for an n-node planar graph G with k = n and

n1 < c < _n³ is NP-hard.

Proof. The proof is modified from that of Aspnes et al. [5, Theorems 3 and 10]. Since computing the minimum cardinality of vertex cover for planar G is NP-complete [9], it suffices to show that the cardinality of any optimal solution for the inoculation problem withk = n and ¹_n < c < _n³ forG is the same as the cardinality of any minimum vertex cover ofG. S is a minimum vertex cover for G if and optimal solution of the inoculation problem withc = _n².

Let S1 be a minimum vertex cover for G. Let S2 be an optimal solution for the inoculation problem for G with k = n and _n¹ < c < _n³. Aspnes et al. [5, Theorem 10]

guarantees thatS2is a vertex cover ofG. Therefore,

|S1| ≤ |S2|. (2.1)

SinceS1 andS2 are both vertex covers ofG, we have

φ(S1) = n − |S1| (2.2)

φ(S2) = n − |S2|. (2.3)

SinceS1 is a feasible solution for the inoculation problem, we have

c · |S2| +φ(S2)

n ≤ c · |S1| +φ(S1) n . It follows fromc > _n¹ and Equations (2.2) and (2.3) that

|S2| ≤ |S1|. (2.4)

Combining Inequalities (2.1) and (2.4), we have|S1| = |S2|. The lemma is proved.

2.2 Minimum quotient cut

A, B, R is a cut of a graph H if A, B, and R with |A| ≤ |B| form a partition of the nodes of H such that H does not have any edge with one endpoint in A and the other endpoint inB. The quotient of A, B, R is

θH(A, B, R) = |R|

|A| + |R|.

The minimum quotient cut problem forH is to find a cut of H with minimum quotient.

Lemma 2.2 (Amir et al. [1]). For anyn-node planar graph H and any positive constant , it takesO(n^4+) time to compute a ⁴₃(1 +¹_+ o(1))-approximate solution for the minimum quotient cut problem forH.

Chapter 3

Our algorithm

3.1 A reduction

For numbers α, β ≥ 1, a node subset S of G is a bicriterion (α, β)-approximate solution for the inoculation problem forG if the following conditions hold for S, α, and β:

• Condition B1: |S| ≤ α · k.

• Condition B2: φ(S) ≤ β · φ( ˆS) holds for any node subset ˆS of G with | ˆS| ≤ k.

Therefore, aβ-approximate solution for the problem is a (1, β)-approximate solution for the problem. Our result is based on the following reduction of Aspnes et al. [5], which ensures that it suffices to focus on finding an (α, β)-approximate solution for the problem withc = 0.

Lemma 3.1 (Aspnes et al. [5, Corollary 13]). If an (α, β)-approximate solution for the inoculation problem withc = 0 can be computed in O(f(n)) time, then it takes O(n·f(n))

time to compute a max(α, β)-approximate solution for the inoculation problem with k = n.

3.2 Finding a bicriterion approximate solution

Given a number γ ≥ 1, we show how to compute an (O(γ · log n), O(γ))-approximate solution for the inoculation problem with c = 0. Observe that if k ≥ n, then the node set ofG is a trivial (O(γ · log n), O(γ))-approximate solution. Therefore, the rest of the section assumesk < n. For any node subsets S and R of G, the cost effectiveness of R with respect toS is

μ(R, S) = φ(S) − φ(S ∪ R)

|R| .

With the following conditions, Algorithm 1 gives the main procedure.

• Condition C1: μ(R, S) < _(γ+1)k¹ · φ(S).

• Condition C2: |S ∪ R| ≥ min{n, (1 + (γ + 1) log2(n − k)) · k}.

• Condition C3: μ(R, S) ≥ ¹_γ·μ( ˆS, S) holds for any node subset ˆS of G with | ˆS| ≤ k.

Observe that Algorithm 1 can only abnormally abort at the step of finding a node subset R of the current G \ S such that Condition C3 holds. If Algorithm 1 does not abnormally abort, then the following lemma ensures the correctness of Algorithm 1.

Lemma 3.2. If Algorithm 1 does not abnormally abort, then it takes Algorithm 1 at most n iterations to output an (O(γ · log n), O(γ))-approximate solution for the inoculation problem withc = 0.

Algorithm 1 main procedure

LetS initially be an arbitrary node subset of G with |S| = k. The algorithm proceeds in iterations, each of which computes a nonempty node subsetR of the current G \ S such that Condition C3 holds. If neither of Conditions C1 and C2 hold for the current S and R, then the algorithm lets S = S ∪ R and proceeds to the next iteration. Otherwise, the algorithm halts. The output depends on Condition C1 for the finalS and R:

• If Condition C1 holds, then the algorithm outputs the final S.

• If Condition C1 does not hold, then the algorithm outputs the final S ∪ R.

Proof. Observe that each iteration of Algorithm 1 increases the size ofS by at least one.

By definition of Condition C2, Algorithm 1 halts in at most n iterations. The rest of the proof argues that Algorithm 1 computes an (O(γ · log n), O(γ))-approximate solution.

Consider the finalS and R at the last iteration of Algorithm 1. We first show that

|S| = O(γ · log n) · k, (3.1)

which holds trivially if Algorithm 1 runs for exactly one iteration. If Algorithm 1 runs for at least two iterations, letS^(respectively,R^) be the setS (respectively, R) at the second- to-last iteration. Since Condition C2 does not hold forS^andR^, we obtain Equation (3.1) as follows.

|S| = |S^∪ R^| < min{n, (1 + (γ + 1) log2(n − k)) · k} = O(γ · log n) · k.

The following case analysis is according to whether Condition C1 holds forS and R.

Case 1: Condition C1 holds for the finalS and R. We prove the lemma by showing that Conditions B1 and B2 hold with S = S, α = O(γ · log n), and β = γ + 1. Note that

Condition B1 is immediate from Equation (3.1). To see thatφ(S) ≤ (γ + 1) · φ( ˆS) holds for any node subset ˆS of G with | ˆS| ≤ k, observe that Conditions C1 and C3 hold for S andR. Thus,

φ(S)

(γ + 1)k > μ(R, S) ≥ μ( ˆS, S)

γ = φ(S) − φ(S ∪ ˆS)

γ · | ˆS| ≥ φ(S) − φ( ˆS) γ · k . Therefore, we have Condition B2.

Case 2: Condition C1 does not hold for the finalS and R. By definition of Algorithm 1, Condition C2 holds for S and R. We prove the lemma by showing that Conditions B1 and B2 hold withS = S ∪ R, α = O(γ · log n), and β = 1. We first prove Condition B1, i.e.,|S ∪ R| = O(γ · log n) · k. By Equation (3.1), it remains to ensure |R| ≤ (γ + 1) · k as follows

φ(S)

|R| ≥ φ(S) − φ(S ∪ R)

|R| = μ(R, S) ≥ φ(S)

(γ + 1)k,

where the last inequality is by the assumption that Condition C1 does not hold for S and R.

We next prove Condition B2, i.e.,φ(S ∪ R) ≤ φ( ˆS) holds for any node subset ˆS of G with| ˆS| ≤ k. Let S_i(respectively,R_i) be the setS (respectively, R) at the i-th iteration of Algorithm 1. Letδ be the number of iterations executed by Algorithm 1. For notational brevity, we defineSδ+1 to be the finalS ∪ R. For each i = 1, . . . , δ, one can verify that S_i+1= S_i∪ R_i, we have

μ(Ri, Si) = φ(S_i) − φ(S_i∪ R_i)

|Ri| = φ(S_i) − φ(S_i+1)

|Si+1| − |Si| . Since Condition C1 never holds throughout the execution, we have

φ(Si) − φ(Si+1)

|S_i+1| − |S_i| ≥ φ(Si) (γ + 1)k.

Thus,

φ(S_i+1) ≤



1 − |Si+1| − |Si| (γ + 1)k



φ(S_i). (3.2)

Note that Condition B2 holds trivially, if Condition C2 holds with |S ∪ R| = n. For the rest of the proof, we have Condition C2 holds with

|S ∪ R| ≥ (1 + (γ + 1) log2(n − k)) · k.

Since it may not disconnectG by removing S1 fromG with |S1| = k , we have φ(S1) ≤ (n − k)². Since it may divideG into n − | ˆS| nodes by removing ˆS from G with | ˆS| ≤ k, we haveφ( ˆS) ≥ n − k. Thus, we have the following result.

φ(S ∪ R) = φ(Sδ+1)

≤ φ(S1) ·

δ i=1



1 −|Si+1| − |Si| (γ + 1)k



≤ (n − k)²·

δ i=1



1 −|Si+1| − |Si| (γ + 1)k



≤ (n − k)²·

δ i=1



1 − 1

(γ + 1)k

_{|Si+1|−|Si|}

= (n − k)²·



1 − 1

(γ + 1)k

_{|Sδ+1|−|S1|}

≤ (n − k)²·



1 − 1

(γ + 1)k

_{(γ+1)k log2(n−k)}

< (n − k)²·

1 2

log₂(n−k)

= (n − k)

≤ φ( ˆS),

where the first inequality is by Inequality (3.2). Therefore, we have Condition B2.

3.3 Finding a node set with good cost effectiveness

The following lemma ensures the feasiblity of each iteration of Algorithm 1.

Lemma 3.3. For any positive constant , if γ = 8(2 + ¹_), then each iteration of Algo- rithm 1 requiresO(n^4+) time and does not abnormally abort.

Proof. Clearly, it suffices to focus on the step to compute a node subsetR of the current G \ S such that Condition C3 holds. At first, for each connected component H of G \ S, we compute a node subsetR of H such that the following condition holds.

• Condition D1: μ(R, S) ≥ _γ¹ · μ( ˆR, S) holds for any node subset ˆR of H.

The detail is left to the second part. Then, by choosing the one with maximumμ(R, S) over allR, we can derive the node subset R such that Condition D1 holds with R = R for anyH of G \ S. Let VH be the node subset ofH. By definition of φ, we have

φ(S) − φ( ˆS) = 

H∈G\S

φ(S) − φ(S ∪ { ˆS ∩ VH}) ,

then

μ( ˆS, S) = φ(S) − φ(S ∪ ˆS)

| ˆS|

=



H∈G\S

φ(S) − φ(S ∪ { ˆS ∩ VH})

| ˆS|

=



H∈G\S

μ( ˆS ∩ V_H, S) · | ˆS ∩ V_H|

| ˆS| .

Since ˆS ∩ V_H is a node subset of H, and Condition D1 holds with R = R for any H of

G \ S, Condition C3 holds by the follows

μ( ˆS, S) =



H∈G\S

μ( ˆS ∩ VH, S) · | ˆS ∩ VH|

| ˆS|

≤



H∈G\S

γ · μ(R, S) · | ˆS ∩ VH|

| ˆS|

≤ γ · μ(R, S).

The rest of the proof proves that, for any connected component H of G \ S, it takes O(|V_H|^4+) time to compute a node subset R of H such that Condition D1 holds. There- fore, we can deriveR in O(n^4+) time, which proves Lemma 3.3.

For any node subset ˆR of H, let  ˆA, ˆB, ˆR be a cut of H, and | ˆA| is maximized with respect to ˆR. We give the proof in three parts.

1. μ( ˆR, S) ≤ _θ ^3|VH|

H( ˆA, ˆB, ˆR).

2. A cut A, B, R with θ_H(A, B, R) ≤ ⁴(2+¹_)

3 · θ_H( ˆA, ˆB, ˆR) can be computed in O(|VH|^4+) time.

3. θH(A, B, R) ≥ _2·μ(R,S)^|VH| .

Combining the three statements, we can compute a node subset R of H in O(|VH|^4+) time such that

μ( ˆR, S) ≤ 3|VH|

θH( ˆA, ˆB, ˆR) ≤ 4 2 + ¹_

|VH| θ_H(A, B, R) ≤ 8

 2 + 1





· μ(R, S) = γ · μ(R, S).

Therefore, we have Condition D1.

LetV_Hⁱ be the node set of the i-th connected component of the remaining graph in- duced by removing ˆR from H. By definition of φ, we have

φ(S) − φ(S ∪ ˆR) = |V_H|²−

i

|V_Hⁱ|². (3.3)

SinceR is a node subset of H, Equation (3.3) also holds with ˆR = R for the rest of the proof.

The first statement can be proved by two cases of| ˆB|.

• | ˆB| ≤ ^2|V₃^H|. From Equation (3.3), we have μ( ˆR, S) = φ(S) − φ(S ∪ ˆR)

| ˆR|

= |V^H|²−

i|V_Hⁱ|²

| ˆR|

≤ |VH|²−

i|V_Hⁱ|

| ˆR|

= |V^H|²− (| ˆA| + | ˆB|)

| ˆR|

= |V^H|²− (|VH| − | ˆR|)

| ˆR|

= |V^H|(|VH| − 1)

| ˆR| + 1

= |V^H|(| ˆA| + | ˆB| + | ˆR| − 1)

| ˆR| + 1

= |V^H|(| ˆA| + | ˆR|)

| ˆR| + |V^H|(| ˆB| − 1)

| ˆR| + 1

≤ |VH|

θH( ˆA, ˆB, ˆR) +2|VH|(|VH| − | ˆB|)

| ˆR|

= |V_H|

θ_H( ˆA, ˆB, ˆR) +2|V_H|(| ˆA| + | ˆR|)

| ˆR|

= |V_H|

θH( ˆA, ˆB, ˆR) + 2|V_H| θH( ˆA, ˆB, ˆR)

= 3|V_H| θH( ˆA, ˆB, ˆR).

• | ˆB| > ^2|V₃^H|. We prove that there is only one connected component in ˆB. As- sume for contradiction that ˆB consists of more than one connected components.

The following two cases show that there exists a cut A^, B^, ˆR with |A^| > | ˆA|, contradicting to that| ˆA| is maximized with respect to ˆR.

– If there exists a connected componentHb of ˆB with |Hb| ≤ ^|V₃^H|, we letA^ = min{ ˆA ∪ H_b, ˆB \ H_b}, then |A^| > | ˆA|.

– Otherwise, we letA^ be any connected component of ˆB, then |A^| > ^|V₃^H| >

| ˆA|.

Thus, we have

μ( ˆR, S) = φ(S) − φ(S ∪ ˆR)

| ˆR|

= |V^H|²−

i|V_Hⁱ|²

| ˆR|

= |VH|²− | ˆB|²−

V_Hⁱ∈ ˆA|V_Hⁱ|²

| ˆR|

≤ |V_H|²− | ˆB|²− | ˆA|

| ˆR|

= |V^H|²− (|VH| − | ˆA| − | ˆR|)²− | ˆA|

| ˆR|

= 2|VH|(| ˆA| + | ˆR|) − (| ˆA| + | ˆR|)²− | ˆA|

| ˆR|

< 2|V_H|(| ˆA| + | ˆR|)

| ˆR|

= 2|V_H| θ_H( ˆA, ˆB, ˆR), where the second equality is by Equation (3.3).

The second statement follows immediately from Lemma 2.2. We then prove the third statement. Since|A| ≤ |B|, we have

|B| ≥ |A| + |B|

2 = |V^H| − |R|

2 . (3.4)

From Equation (3.3) and Inequality (3.4), we have

μ(R, S) = φ(S) − φ(S ∪ R)

|R|

= |V^H|²−

i|V_Hⁱ|²

|R|

≥ |V_H|²− |A|² − |B|²

|R|

= (|A| + |B| + |R|)²− |A|²− |B|²

|R|

= 2|A||B| + 2|VH||R| − |R|²

|R|

≥ 2|A||B| + |VH||R|

|R|

≥ |A|(|VH| − |R|) + |VH||R|

|R|

= |V^H|(|A| + |R|)

|R| − |A||R|

|R|

≥ |V_H|

θH(A, B, R) −|V_H|(|A| + |R|) 2|R|

= |VH|

2 · θ_H(A, B, R).

3.4 Proving Theorem 1.1

Proof. Lemmas 3.2 and 3.3 together ensure that an (O(log n), O(1))-approximate solu- tion for the inoculation problem withc = 0 can be found in O(n^5+) time. The theorem follows immediately from Lemma 3.1.

Chapter 4

Concluding remarks

We leave open the approximability of the inoculation problem for general parameter k.

It would be interesting to see if our techniques can be extended to work for this general version for planar graphs.

Bibliography

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