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\displaystyle B\times k^{\prime}\times\underbrace{(n_{L}+n_{R})}_{=n-(k+k^{% \prime})}=B\times k^{\prime}\times(n-(k+k^{\prime}))
P^{\prime}\subset P
0\leq i\leq k
e_{i},e_{j}
C\leq y\leq B+C
c_{i}\neq c_{j}
\alpha_{1}=B
\displaystyle\xrightarrow{\text{setting }x=A+B,y=C}=
v\in V_{s}
\mathcal{E}_{1}
|w^{*}-(c_{2}\times w^{*}+c_{j}\times w^{*}-\epsilon\times w^{*})|-|w^{*}-(c_{% 2}\times w^{*}+c_{j}\times w^{*})|\leq\epsilon\times w^{*}
e^{*}\in S
c^{\prime}_{2}=c_{2}-\epsilon
f\in\mathcal{F}=\{\text{MARK\_LEFT},\text{UNMARK\_RIGHT}\}
|w_{1}+w^{*}+w_{3}+w^{*}-w_{1}-w^{*}-w_{3}|=w^{*}
M_{L}^{*}\xleftarrow[]{}M_{L}^{*}\cup\{e_{i}\}
\mathcal{E}^{v_{i},u_{j}}_{1}=\left|\pi_{v_{i},u_{j}}-\pi^{\prime}_{v_{i},u_{j% }}\right|=\left|\pi^{\prime}_{v_{i},u_{j}}-\pi_{v_{i},u_{j}}\right|=\left|% \mathcal{W}^{\prime}(E^{(v_{i},u_{j})})-\mathcal{W}^{*}(E^{(v_{i},u_{j})})\right|
\alpha_{1}=x-A+A+B-x<B\xrightarrow{}0<0
(u,v),\;\;u\in V_{m},\;v\in\overline{V_{m}}
|\Delta E|\geq B(n_{L}+n_{R})+n_{L}n_{R}|x+y-A-B-C|
v^{\prime}_{1}
\operatorname{d}_{G^{\prime}}(u,v)\geq\varphi\left(\operatorname{d}_{G}(u,v)\right)
S\xleftarrow[]{}E_{m}
7-21\leq 3=L_{3}
{n^{2}_{R}}\times 2\big{(}{j-1\choose 2}-{j\choose 2}\big{)}={n^{2}_{R}}\times% (-2(j-1))
\mathcal{E}_{LR}=0
w^{\prime}(e_{i})=w(e_{i})+\epsilon_{i},\epsilon_{i}\in[0,w^{*}]
\displaystyle+
i+1-k+i<0\xrightarrow[]{}2i<k-1\xrightarrow[]{}i<\frac{k}{2}-\frac{1}{2}% \xrightarrow[\text{since }k\text{ is even}]{}i\leq\frac{k}{2}-1
\alpha_{1}
V_{m}=\{v_{2},v_{3}\}
\Delta({\text{MARK\_LEFT}})={n^{2}_{L}}\times(2i+{\mathcal{L}}-2i-1)+{n_{L}}{n% _{R}}(j+j-{\mathcal{R}})={n^{2}_{L}}\times({\mathcal{L}}-1)+{n_{L}}{n_{R}}(2j-% {\mathcal{R}})
E^{(v_{1},u_{5})}=\{e_{1},e^{*}_{1},e_{2},e_{3},e^{*}_{2},e_{4}\}
x=\sum_{k=i}^{n_{1}}\epsilon_{k}
v_{i},v_{j}\in V_{R}
G^{\prime}
c_{i+1},\dots,c_{k}
S_{R}-R_{i}-S_{L}\leq 0
w_{i}=w(e_{i})\;\forall i\in\{0,\dots,k+1\}
u_{1}\in T_{1}^{L},u_{2}\in T_{2}^{L}
\displaystyle\mathcal{E}_{L}^{(i+1)}=
i\geq\frac{{\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}
\displaystyle\mathcal{E}(M)=
n_{L}n_{R}|x+y-A-B-C|
V^{\prime}\subset V
R_{i}={n_{R}},\;1\leq i\leq{\mathcal{R}}
\pi_{v_{i},u_{j+1}}=\pi_{v_{i},u_{j}}+w^{*}_{k}\text{, }\pi^{\prime}_{v_{i},u_% {j}}=\pi^{\prime}_{v_{i},u_{j+1}}\text{, and }\pi^{\prime\prime}_{v_{i},u_{j}}% =\pi^{\prime\prime}_{v_{i},u_{j+1}}
u_{1}\in T_{i}^{L},u_{2}\in T_{j}^{L}
\epsilon=0.4
e\in E_{m}
e_{i},w_{i}=w(e_{i})
\displaystyle\mathcal{E}^{(x<w_{0})}_{L}
V_{m}=\{u_{1},\dots,u_{2k}\}
\displaystyle\underbrace{n_{L}\times n_{R}\times|x+y-A-B-C|}_{\text{between % the subpath of }w_{1}\text{ and }w_{2}}
\mathcal{E}^{v_{i},u_{j}}_{2}=\left|\pi^{\prime\prime}_{v_{i},u_{j}}-\pi_{v_{i% },u_{j}}\right|=\left|w^{*}_{k}\right|
\frac{\mathcal{E}^{(\frac{k}{2})}_{L}}{n_{L}}-\frac{\mathcal{E}_{L}}{n_{L}}\leq 0
\mathcal{E}^{v_{i},u_{j+1}}_{2}=0
\displaystyle\xrightarrow[]{}\mathcal{E}(M)\geq\mathcal{E}(M_{L}^{*})
|x+y-A-B-C|
E^{(u,v)}
x=w_{0}+w_{1}+\dots+w_{\frac{k}{2}}
w_{1}+w_{2}+w_{3}
j\xleftarrow[]{}0
S_{R}=S_{RU}+S_{RM}
c_{1}+c_{2}=0.6+0.8=1.4>1
S_{L}=\sum_{i=1}^{{\mathcal{L}}}L_{i}
c_{1}+c_{2}=1+\epsilon
w^{\prime}(e_{i})=w(e_{i})+w^{*}
\{e_{1},e_{3}\}
G^{{}^{\prime}}
i\xleftarrow{}i+1
\Delta(f)
\displaystyle n_{L}\times k^{\prime}\times\underbrace{\big{(}|x-A|+|x-A-B|\big% {)}}_{\geq B\text{ (Lemma \ref{l12})}}+n_{R}\times k^{\prime}\times\underbrace% {\big{(}|y-C|+|y-B-C|\big{)}}_{\geq B\text{ (Lemma \ref{l12})}}
i<\frac{{\mathcal{L}}}{2}+\frac{{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}
\epsilon_{2}\in\{0,w^{*}\}
\displaystyle\mathcal{E}(M)
V_{L}=\{u|(u,v_{1})\in E^{\prime}\},{\mathcal{L}}=|V_{L}|
n_{L}+n_{R}=n-2=\left|\overline{V_{m}}\right|
T_{i}^{L}
i={\mathcal{L}}
\epsilon>0
E^{\prime}\subseteq E
S_{R}^{\prime}\geq S_{L}^{\prime}
u_{1}\in T_{i}^{R},u_{2}\in T_{j}^{R}
\displaystyle|\Delta E|=
V_{L},V_{R}\subset V
\displaystyle=\mathcal{E}(M_{0})+(\epsilon_{1}+\epsilon_{2})\times\sum_{e_{i}% \in E_{L}}L_{i}\times w^{*}\times(S_{L}-L_{i}-S_{R})\xrightarrow[S_{L}-L_{i}-S% _{R}\leq 0]{\epsilon_{1}+\epsilon_{2}\leq 1}
\{u_{1},v_{1}\}
\Delta({\text{UNMARK\_RIGHT}})=R_{i}\times\bigg{(}-(S_{RM}-R_{i})-S_{RU}-S_{LM% }+S_{LU}\bigg{)}
\alpha_{1}=|x-A|+|x-A-B|
|z|\leq|x|+|z-x|\xrightarrow[]{}|z|-|x|\leq|z-x|
n_{R}\geq 0
21-7>1=R_{2}
v^{*}=\{v_{2},v_{3},\dots,v_{k+2}\}
w:E\rightarrow\mathbb{R}_{\geq 0}
i=\{1,\dots,{\mathcal{L}}\}
\Delta_{1}({\text{MARK\_LEFT}})<0
0-\left|\sum_{k=i}^{j-1}\epsilon_{k}\right|
u,v\in G
V_{m}=\{v_{1},v_{2}\}