论文阅读-AKS_CoRR

作者	年份	近似比
Hoogeveen	1991	53\frac{5}{3}35
An, Kleinberg, Shmoys	2012	1+52\frac{1+\sqrt{5}}{2}21+5
Sebo	2013	85\frac{8}{5}58
Rico Zenklusen	2019	1.5

Title: Improving Christofides’s Algorithm for the s-t path TSP

Alpha: 1+52\frac{1+\sqrt{5}}{2}21+5

Theorem1: Hoogeveen算法的解不超过53OPTLP\frac{5}{3}OPT_{LP}35OPTLP

定义1: Path-TSP的HK松弛

min∑e∈Ecexes.t.x(δ(v))={1,v=s,t2,v≠s,tx(δ(S))≥{1,∣S∩{s,t}∣=1,2,∣S∩{s,t}∣≠1,0≤xe≤1,∀e∈Emin \sum_{e\in E} c_ex_e\\ \begin{aligned} & s.t.\\ & x(\delta(v))=\begin{cases} 1, & v=s,t\\ 2, & v\neq s,t \end{cases}\\ & x(\delta(S)) \geq \begin{cases} 1, & |S \cap \{s,t\}|=1,\\ 2, & |S \cap \{s,t\}|\neq 1,\\ \end{cases}\\ & 0 \leq x_e \leq 1, \forall e \in E \end{aligned} mine∈E∑cexes.t.x(δ(v))={1,2,v=s,tv=s,tx(δ(S))≥{1,2,∣S∩{s,t}∣=1,∣S∩{s,t}∣=1,0≤xe≤1,∀e∈E

其中δ(S)\delta(S)δ(S)是仅有一个端点落在S中的边的边集, 同时X(E′)=∑e∈E′xeX(E')=\sum_{e \in E'} x_eX(E′)=∑e∈E′xe, 所谓的松弛就是最后一个0-1向量变成了实数.

定义2: 生成树凸集

生成树凸集由下面的不等式定义:
x(E)=∣V∣−1,x(E(S))≤∣S∣−1,∀∣S∣⊆V,∣S∣≥2,xe≥0,∀e∈E\begin{aligned} & x(E)=|V|-1,\\ & x(E(S)) \leq |S|-1, \quad \forall |S| \subseteq V, |S| \geq 2,\\ & x_e \geq 0, \quad \forall e \in E \end{aligned} x(E)=∣V∣−1,x(E(S))≤∣S∣−1,∀∣S∣⊆V,∣S∣≥2,xe≥0,∀e∈E

其中E(S)是所有两个端点都在S中的边的边集.

Lemma1: LP-relaxation的任意可行解x都在生成树凸集中.

proof: LP-relaxation的约束满足生成树凸集的定义
X(E)≡∑e∈Exe=12∑v∈Vx(δ(v))=12(∣v∣−2)⋅2+2)=∣v∣−1X(E) \equiv \sum_{e\in E} x_e = \frac{1}{2}\sum_{v\in V}x(\delta(v))\\ = \frac{1}{2}(|v|-2)\cdot 2 + 2)=|v|-1 X(E)≡e∈E∑xe=21v∈V∑x(δ(v))=21(∣v∣−2)⋅2+2)=∣v∣−1
同时,
X(E(S))=12(∑v∈Sx(δ(v))−x(δ(S)))X(E(S))=\frac{1}{2}(\sum_{v \in S}x(\delta(v))-x(\delta(S))) X(E(S))=21(v∈S∑x(δ(v))−x(δ(S)))
如果∣S∩{s,t}=1|S\cap \{s,t\}=1∣S∩{s,t}=1, 有X(E(S))≤12(1+2(∣S∣−1)−1)=∣S∣−1X(E(S))\leq \frac{1}{2}(1+2(|S|-1)-1)=|S|-1X(E(S))≤21(1+2(∣S∣−1)−1)=∣S∣−1,

如果∣S∩{s,t}=∅|S\cap \{s,t\}=\empty∣S∩{s,t}=∅, S-1,

如果∣S∩{s,t}=2|S\cap \{s,t\}=2∣S∩{s,t}=2,S-2

定义3: 奇数集S, 如果∣S∩T∣|S\cap T|∣S∩T∣含有奇数个, 则S是个奇数集

Lemma2: S是一个奇数集, 如果∣S∩{s,t}∣=1|S\cap \{s,t\}|=1∣S∩{s,t}∣=1, 则∣F∩δ(S)∣|F\cap \delta(S)|∣F∩δ(S)∣为偶数, 如果∣S∩{s,t}∣≠1|S\cap \{s,t\}|\neq1∣S∩{s,t}∣=1, 则∣F∩δ(S)∣|F\cap \delta(S)|∣F∩δ(S)∣为奇数.

例如,

Proof: s,t 如果在S中, 它们有偶数度, 其他点有奇数度.

定义∑v∈SdegF(v)=2∣E(S)∩F∣+∣δ(S)∩F∣\sum_{v\in S}deg_F(v)=2|E(S)\cap F|+|\delta(S)\cap F|∑v∈SdegF(v)=2∣E(S)∩F∣+∣δ(S)∩F∣

证明如下:

1.如果∣S∩{s,t}=1|S\cap \{s,t\}=1∣S∩{s,t}=1, 假设s∈Ss\in Ss∈S, s∈Ts\in Ts∈T当且仅当degF(s)deg_F(s)degF(s)even.

则Sodd→evenS_{odd}\rightarrow evenSodd→even # 个奇数度的节点在S中(∣S∩T∣odd|S\cap T| odd∣S∩T∣odd)

∑v∈SdegF(v)−2∣E(s)∩F∣=∣δ(s)∩F∣\sum_{v\in S}deg_F(v)-2|E(s)\cap F|=|\delta(s)\cap F| v∈S∑degF(v)−2∣E(s)∩F∣=∣δ(s)∩F∣
第一个子式为偶数度, 第二个子式肯定是偶数, 则右边也是偶数.

2. 如果∣S∩{s,t}≠1|S\cap \{s,t\}\neq 1∣S∩{s,t}=1,

则Sodd→oddS_{odd}\rightarrow oddSodd→odd # 个奇数度的节点在S中

∑v∈SdegF(v)−2∣E(s)∩F∣=∣δ(s)∩F∣\sum_{v\in S}deg_F(v)-2|E(s)\cap F|=|\delta(s)\cap F| v∈S∑degF(v)−2∣E(s)∩F∣=∣δ(s)∩F∣
第一个子式为奇数度, 第二个子式肯定是偶数, 则右边也是奇数.

定义4: T-join LP

以下线性规划的解是一个最小成本的T-join, 对于cost c≥0c\geq 0c≥0:
Min∑e∈Ecexes.t.x(δ(S))≥1,∀S⊆V,∣S∩T∣oddxe≥0,∀e∈EMin \sum_{e\in E}c_ex_e\\ \begin{aligned} & s.t.\\ & x(\delta(S)) \geq 1, & \forall S \subseteq V, |S\cap T| odd\\ & x_e \geq 0, & \forall e \in E \end{aligned} Mine∈E∑cexes.t.x(δ(S))≥1,xe≥0,∀S⊆V,∣S∩T∣odd∀e∈E
对于∣S∩T∣|S\cap T|∣S∩T∣为奇,
∑v∈SdegJ(v)=2∣E(S)∩J∣+∣δ(S)∩J∣\sum_{v\in S}deg_J(v)=2|E(S)\cap J|+|\delta(S)\cap J| v∈S∑degJ(v)=2∣E(S)∩J∣+∣δ(S)∩J∣
因为奇数个奇数度的节点，因此等式左边为奇，因为右边第一个子式为偶，所以第二个子式为奇数.说明S向外连接的节点一定大于等于1.

proof of Theorem1

Step1

令x∗x^*x∗为LP松弛的最优解OPT. cost of MST≤∑e∈Ecexe∗≡>OPTLP\text{cost of MST}\leq \sum_{e\in E}c_ex_e^* \equiv > OPT_{LP}cost of MST≤∑e∈Ecexe∗≡>OPTLP, 因为x∗x^*x∗总是生成树凸集的可行解.

令XF∈{0,1}∣E∣X_F\in \{0,1\}^{|E|}XF∈{0,1}∣E∣,并且
XF(e)={1,ife∈F0,o.w.X_F(e)= \begin{cases} 1, \quad if e\in F \\ 0, \quad o.w. \end{cases} XF(e)={1,ife∈F0,o.w.

claim: y=13XF+13x∗y=\frac{1}{3}X_F+\frac{1}{3}x^*y=31XF+31x∗是T-join LP的一个可行解.

则有c(F∪T)=c(F)+c(T)≤OPTLP+13c(F)+13OPTLP>≤53OPTLPc(F\cup T)=c(F)+c(T) \leq OPT_{LP}+\frac{1}{3}c(F)+\frac{1}{3}OPT_{LP} > \leq \frac{5}{3}OPT_{LP}c(F∪T)=c(F)+c(T)≤OPTLP+31c(F)+31OPTLP>≤35OPTLP

Step2

若要claim成立,需要证明如果∣s∩T∣|s\cap T|∣s∩T∣为奇,则y(δ(S))≥1y(\delta(S)) \geq 1y(δ(S))≥1

如果 ∣s∩{s,t}∣≠1|s\cap \{s,t\}|\neq 1∣s∩{s,t}∣=1则y(δ(S))=13∣F∪δ(S)∣+13x∗(δ(S))≥13+23=1y(\delta(S)) =\frac{1}{3}|F\cup \delta(S)|+\frac{1}{3}x^*(\delta(S)) \geq \frac{1}{3}+\frac{2}{3} = 1y(δ(S))=31∣F∪δ(S)∣+31x∗(δ(S))≥31+32=1

【第二个部分,因为HK relaxation成立】

如果∣s∩{s,t}∣=1|s\cap \{s,t\}|= 1∣s∩{s,t}∣=1, 则y(δ(S))=13∣F∪δ(S)∣+13x∗(δ(S))≥23+13=1y(\delta(S)) =\frac{1}{3}|F\cup \delta(S)|+\frac{1}{3}x^*(\delta(S)) \geq \frac{2}{3}+\frac{1}{3} = 1y(δ(S))=31∣F∪δ(S)∣+31x∗(δ(S))≥32+31=1

【第一个式子 lemma2】

claim 证毕.

定义5：凸组合

令x∗x^*x∗为LP的最优解, 令xFx^FxF表示为一个F的边集, 即
xF(e)={1e∈F0e∉Fx_F(e)=\begin{cases} 1 \quad e \in F \\ 0 \quad e \notin F \end{cases} xF(e)={1e∈F0e∈/F

因为x∗x^*x∗在生成树凸集中,因此x∗x^*x∗可以写成生成树F1,⋯,FkF_1,\cdots,F_kF1,⋯,Fk的凸组合:
x∗=∑i=1kλixFix^*=\sum_{i=1}^k \lambda_i x_{F_i} x∗=i=1∑kλixFi
其中∑i=1kλi=1,λi≥0\sum_{i=1}^k\lambda_i = 1, \lambda_i \geq 0∑i=1kλi=1,λi≥0.

对于FiF_iFi, 设TiT_iTi是其T集,JiJ_iJi是其最小成本T-join. 它们的和能够构成一个解称为best-of-many Christofide算法的解.

Theorem2: best-of-many Christofide算法的解, 同样满足上限为53OPTLP\frac{5}{3}OPT_{LP}35OPTLP.

下一步: 是否能够更优?

考虑yi=αXF+βx∗y_i=\alpha X_F+\beta x^*yi=αXF+βx∗, 如果是TiT_iTi-Join LP的可行解, 则best s-t 哈密顿路径的长度最多不超过(1+α+β)OPTLP(1+\alpha+\beta)OPT_{LP}(1+α+β)OPTLP.

yiy_iyi是TiT_iTi-Join LP的可行解分两种情况考虑,设S 奇数集(∣S∪Ti∣odd|S\cup T_i| odd∣S∪Ti∣odd)

如果∣s∩{s,t}∣≠1|s\cap \{s,t\}|\neq 1∣s∩{s,t}∣=1,

y(δ(S))=α∣F∪δ(S)∣+βx∗(δ(S))≥α+2βy(\delta(S)) =\alpha|F\cup \delta(S)|+\beta x^*(\delta(S)) \geq \alpha+2\beta y(δ(S))=α∣F∪δ(S)∣+βx∗(δ(S))≥α+2β

[右边第一个式大于等于1, 第二个式大于等于2, 同上]

我们希望α+2β≥1\alpha+2\beta \geq 1α+2β≥1, 则TiT_iTi-join LP约束就能够满足.

如果
∣s∩{s,t}∣=1|s\cap \{s,t\}|= 1∣s∩{s,t}∣=1, 则
yi(δ(S))=α∣F∪δ(S)∣+βx∗(δ(S))≥2α+βx∗(δ(S))y_i(\delta(S)) =\alpha |F\cup \delta(S)|+\beta x^*(\delta(S)) \geq 2\alpha+\beta x^*(\delta(S)) yi(δ(S))=α∣F∪δ(S)∣+βx∗(δ(S))≥2α+βx∗(δ(S))

注意到我们已经假设了α+2β≥1\alpha+2\beta \geq 1α+2β≥1成立, 只有2α+βx∗(δ(S))<12\alpha+\beta x^*(\delta(S)) < 12α+βx∗(δ(S))<1时, 存在问题.

注意到当α=0,β=12\alpha=0, \beta=\frac{1}{2}α=0,β=21时, 如果x∗(δ(S))≥2x^*(\delta(S)) \geq 2x∗(δ(S))≥2, 上式成立, 并且能够控制上限为32OPTLP\frac{3}{2}OPT_{LP}23OPTLP.

因此接下来只需要关注x∗(δ(S))<2x^*(\delta(S)) < 2x∗(δ(S))<2的cuts, 并且对yiy_iyi增加一个额外的修正来处理这些诶cuts.

定义6 τ\tauτ-Narrow cut

若x∗(δ(S))<1+τ,for fixed τ≤1x^*(\delta(S)) < 1+\tau, \text{for fixed }\tau \leq 1x∗(δ(S))<1+τ,for fixed τ≤1, S则是τ\tauτ-Narrow.

只有∣S∪{s,t}=1|S\cup \{s, t\} =1∣S∪{s,t}=1能够是τ\tauτ-Narrow.

定义7 τ\tauτ-Narrow cuts

CτC_{\tau}Cτ是s∈Ss \in Ss∈S的所有τ\tauτ-Narrow cuts S的全集.

CτC_{\tau}Cτ的性质:

Theorem3: 如果S1,S2∈Cτ,S1≠S2S_1, S_2 \in C_{\tau}, S_1\neq S_2S1,S2∈Cτ,S1=S2, 要么S1⊂S2S_1 \subset S_2S1⊂S2,或S2⊂S1S_2 \subset S1S2⊂S1.

为证明上述Theorem, 首先有
x∗(δ(S1))+x∗(δ(S2))≥x∗(δ(S1−S2))+x∗(δ(S2−S1))x^*(\delta(S_1)) + x^*(\delta(S_2)) \geq x^*(\delta(S_1 - S_2)) + x^*(\delta(S_2-S_1)) x∗(δ(S1))+x∗(δ(S2))≥x∗(δ(S1−S2))+x∗(δ(S2−S1))

Theorem proof:

假设,相反的, S1−S2≠∅,S2−S1≠∅S_1-S_2\neq \empty, S_2-S_1 \neq \emptyS1−S2=∅,S2−S1=∅.
(1+τ)+(1+τ)>x∗(δ(S1))+x∗(δ(S2))≥x∗(δ(S1−S2))+x∗(δ(S2−S1))≥2+2\begin{aligned} (1 + \tau)+(1+\tau) & > x^*(\delta(S_1))+x^*(\delta(S_2)) \\ & \geq x^*(\delta(S_1-S_2)) + x^*(\delta(S_2-S_1)) \\ & \geq 2 + 2 \end{aligned} (1+τ)+(1+τ)>x∗(δ(S1))+x∗(δ(S2))≥x∗(δ(S1−S2))+x∗(δ(S2−S1))≥2+2
与定义矛盾.

根据Theorem, τ\tauτ-Narrow cuts的结构如下:

新的修正因子

令eQe_QeQ表示δ(Q)\delta(Q)δ(Q)的最小cost的边,考虑下式
yi(δ(S))=αxFi+βx∗+∑Q∈Cτ,∣Q∩Ti∣(1−2α−βx∗(δ(Q)))xeQy_i(\delta(S)) =\alpha x_{F_i}+\beta x^* + \sum_{Q \in C_{\tau},|Q\cap T_i|}(1-2\alpha - \beta x^*(\delta(Q)))x_{e_Q} yi(δ(S))=αxFi+βx∗+Q∈Cτ,∣Q∩Ti∣∑(1−2α−βx∗(δ(Q)))xeQ
对于α,β,τ≥0\alpha,\beta, \tau \geq 0α,β,τ≥0,有α+2β=1\alpha + 2\beta=1α+2β=1并且τ=1−2αβ−1\tau = \frac{1-2\alpha}{\beta} - 1τ=β1−2α−1

Theorem: yiy_iyi是一个TiT_iTi-Join LP的可行解.

proof:

对于S odd (∣S∩Ti∣odd|S\cap T_i| odd∣S∩Ti∣odd)

如果∣S∩{s,t}∣≠1|S\cap \{s,t\}|\neq 1∣S∩{s,t}∣=1
yi(δ(S))≥α+2β=1y_i(\delta(S)) \geq \alpha + 2 \beta =1 yi(δ(S))≥α+2β=1
如果∣S∩{s,t}∣=1|S\cap \{s,t\}|= 1∣S∩{s,t}∣=1

如果 S不是τ\tauτ-narrow
yi(δ(S))≥2α+β(1+τ)=1y_i(\delta(S)) \geq 2\alpha + \beta(1+\tau) =1 yi(δ(S))≥2α+β(1+τ)=1

如果 S是τ\tauτ-narrow
yi(δ(S))≥α∣Fi∩δ(S)∣+βx∗(δ(δ(S)))+(1−2α−βx∗(δ(S)))=1y_i(\delta(S)) \geq \alpha |F_i\cap \delta(S)| + \beta x^*(\delta(\delta(S))) + (1-2\alpha - \beta x^*(\delta(S))) =1 yi(δ(S))≥α∣Fi∩δ(S)∣+βx∗(δ(δ(S)))+(1−2α−βx∗(δ(S)))=1

注意到x∗=∑i=1kλixFix^*=\sum_{i=1}^k \lambda_i x_{F_i}x∗=∑i=1kλixFi,∑i=1kλi=1,λi≥0\sum_{i=1}^k\lambda_i = 1, \lambda_i \geq 0∑i=1kλi=1,λi≥0,λi\lambda_iλi可以看成FiF_iFi的概率分布, 是其概率. 紧接着, 有以下两个lemma.

Lemma:

令F为随机采样的生成树FiF_iFi, T为对应的点集TiT_iTi, Q∈CτQ\in C_{\tau}Q∈Cτ是一个τ\tauτ-narrow cut.
Pr[∣δ(Q)∩F∣=1]≥2−x∗(δ(Q))Pr[∣Q∩T∣odd]≤x∗(δ(Q))−1Pr[|\delta(Q)\cap F|=1] \geq 2 - x^*(\delta(Q)) \\ Pr[|Q\cap T| odd] \leq x^*(\delta(Q)) - 1 Pr[∣δ(Q)∩F∣=1]≥2−x∗(δ(Q))Pr[∣Q∩T∣odd]≤x∗(δ(Q))−1

proof:

x∗(δ(Q))=E[∣F∩δ(Q)∣]≥Pr[∣F∩δ(Q)∣=1]+2Pr[∣F∩δ(Q)∣≥2]x^*(\delta(Q)) = E[|F \cap \delta(Q)|] \geq Pr[|F \cap \delta(Q)|=1] + 2Pr[|F \cap \delta(Q)| \geq 2] x∗(δ(Q))=E[∣F∩δ(Q)∣]≥Pr[∣F∩δ(Q)∣=1]+2Pr[∣F∩δ(Q)∣≥2]
并且Pr[∣F∩δ(Q)∣=1]+Pr[∣F∩δ(Q)∣≥2]=1Pr[|F \cap \delta(Q)|=1] + Pr[|F \cap \delta(Q)| \geq 2]=1Pr[∣F∩δ(Q)∣=1]+Pr[∣F∩δ(Q)∣≥2]=1, 因此,
Pr[∣F∩δ(Q)∣=1]≥2−x∗(δ(Q))Pr[∣F∩δ(Q)∣≥2]≤x∗(δ(Q))−1Pr[|F\cap \delta(Q)|=1] \geq 2 - x^*(\delta(Q))\\ Pr[|F\cap \delta(Q)| \geq 2] \leq x^*(\delta(Q)) -1 Pr[∣F∩δ(Q)∣=1]≥2−x∗(δ(Q))Pr[∣F∩δ(Q)∣≥2]≤x∗(δ(Q))−1
因为, ∣Q∩Ti∣odd|Q\cap T_i| odd∣Q∩Ti∣odd有∣Fi∩δ(Q)∣≥2|F_i \cap \delta(Q)| \geq 2∣Fi∩δ(Q)∣≥2,

所以Pr[∣Q∩Ti∣odd]≤Pr[∣F∩δ(Q)∣≥2]≤x∗(δ(Q))−1Pr[|Q\cap T_i| odd]\leq Pr[|F \cap \delta(Q)| \geq 2] \leq x^*(\delta(Q))-1Pr[∣Q∩Ti∣odd]≤Pr[∣F∩δ(Q)∣≥2]≤x∗(δ(Q))−1

Lemma:

∑Q∈CτCeQ≤∑e∈Ecexe∗\sum_{Q \in C_{\tau}}C_{e_Q} \leq \sum_{e \in E} c_ex_e^* Q∈Cτ∑CeQ≤e∈E∑cexe∗

Proof:
∑Q∈CτCeQ≤cost MST≤∑e∈Ecexe∗\sum_{Q \in C_{\tau}}C_{e_Q} \leq \text{cost MST} \leq \sum_{e \in E} c_ex_e^* Q∈Cτ∑CeQ≤cost MST≤e∈E∑cexe∗
构造过程, 对于Q∈CτQ\in C_{\tau}Q∈Cτ, 将一条MST的边e映射到Q, 每次移除一个e, 然后构造s和v.

Theorem: Best-of-Many Christofides算法是1+52\frac{1+\sqrt{5}}{2}21+5算法.

Proof:

Best s-t Path≤∑iλic(Fi∪Ji)=∑iλi[c(Fi)+αc(Fi)+β∑e∈Ecexe∗+∑Q∈Cτ,∣Q∩Ti∣(1−2α−βx∗(δ(Q)))ceQ]≤(1+α+β)∑e∈Ecexe∗+∑Q∈Cτ(x∗(δ(Q))−1)(1−2α−βx∗(δ(Q)))ceQ≤(1+α+β)∑e∈Ecexe∗+max0≤z<τE(1−2α+β(1+z))∑Q∈CτceQ≤(1+α+β+max0≤z<τE(1−2α+β(1+z)))∑e∈Ecexe∗=(1+α+β+max0≤z<τE(βτ−βz))∑e∈Ecexe∗[Maximizedatz=τ/2]≤(1+α+β+β(τ2)2)OPTLP≤(2−β+(3β−1)24β)OPTLP\begin{aligned} & \text{Best s-t Path} \leq \sum_i \lambda_i c(F_i \cup J_i)\\ & = \sum_i \lambda_i[c(F_i) + \alpha c(F_i) + \beta\sum_{e\in E} c_e x_e^* + \sum_{Q \in C_{\tau},|Q\cap T_i|}(1-2\alpha - \beta x^*(\delta(Q)))c_{e_Q}]\\ & \leq (1+\alpha + \beta) \sum_{e\in E} c_e x_e^* + \sum_{Q\in C_{\tau}}(x^*(\delta(Q))-1)(1-2\alpha -\beta x^*(\delta(Q)))c_{e_Q}\\ & \leq (1+\alpha + \beta) \sum_{e\in E} c_e x_e^* + max_{0 \leq z < \tau}E(1-2\alpha + \beta(1+z))\sum_{Q\in C_{\tau}}c_{e_Q}\\ & \leq (1+\alpha + \beta + max_{0 \leq z < \tau}E(1-2\alpha + \beta(1+z)))\sum_{e\in E} c_e x_e^*\\ & = (1+\alpha + \beta + max_{0 \leq z < \tau}E(\beta \tau - \beta z))\sum_{e\in E} c_e x_e^* [Maximized at z=\tau/2]\\ & \leq (1+\alpha + \beta + \beta (\frac{\tau}{2})^2)OPT_{LP}\\ & \leq (2 - \beta + \frac{(3\beta -1)^2}{4\beta})OPT_{LP} \end{aligned} Best s-t Path≤i∑λic(Fi∪Ji)=i∑λi[c(Fi)+αc(Fi)+βe∈E∑cexe∗+Q∈Cτ,∣Q∩Ti∣∑(1−2α−βx∗(δ(Q)))ceQ]≤(1+α+β)e∈E∑cexe∗+Q∈Cτ∑(x∗(δ(Q))−1)(1−2α−βx∗(δ(Q)))ceQ≤(1+α+β)e∈E∑cexe∗+max0≤z<τE(1−2α+β(1+z))Q∈Cτ∑ceQ≤(1+α+β+max0≤z<τE(1−2α+β(1+z)))e∈E∑cexe∗=(1+α+β+max0≤z<τE(βτ−βz))e∈E∑cexe∗[Maximizedatz=τ/2]≤(1+α+β+β(2τ)2)OPTLP≤(2−β+4β(3β−1)2)OPTLP

证毕.

前后四篇工作的算法分析总体思路是一致的,都和wolsey分析的过程是相似, 以最后1.5的为例,

找到一个生成树F和一个满足HK relaxation的点z, 证明:

l(F)≤OPTl(F) \leq OPTl(F)≤OPT,
l(z)≤OPTl(z) \leq OPTl(z)≤OPT,
z/2∈PQT−joinz/2 \in P_{Q_{T-join}}z/2∈PQT−join, 其中Q_T := odd(T)△\bigtriangleup△{s,t}

T和一个T-join构成解,并且l(F)+l(J)≤l(T)+l(z)/2≤3/2OPTl(F)+l(J) \leq l(T) + l(z)/2 \leq 3/2 OPTl(F)+l(J)≤l(T)+l(z)/2≤3/2OPT

而之前的工作主要是弱化了第二条的要求,使得l(z)≤(1+c)OPTl(z) \leq (1+c) OPTl(z)≤(1+c)OPT.

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