$c_{j}=\epsilon_{2}$

$\{e_{1},e_{2}\}$

$\Delta_{2}({\text{MARK\_LEFT}})=(c_{1})\times X+(1-c_{1})\times X$

$A$

$S$

$\alpha\geq 1$

$|w_{3}+w^{*}+w_{4}-w_{3}-w_{4}|=w^{*}$

$\displaystyle\pi_{v_{i},u_{j}}$

$\{v^{\prime}_{4},v^{\prime}_{5},v^{\prime}_{6}\}$

$x=w_{0}$

$v_{n_{1}+1}$

$\displaystyle=$

$e^{*},$

${n_{L}}{n_{R}}\times(({\mathcal{L}}-i-1)({\mathcal{R}}-j)-({\mathcal{L}}-i)({% \mathcal{R}}-j))={n_{L}}{n_{R}}\times({\mathcal{L}}{\mathcal{R}}-{\mathcal{L}}% j-i{\mathcal{R}}+ij-{\mathcal{R}}+j-{\mathcal{L}}{\mathcal{R}}+{\mathcal{L}}j+% i{\mathcal{R}}-ij)={n_{L}}{n_{R}}\times(j-{\mathcal{R}})$

${R}_{j}$

$-2\times R_{i}\times(S_{RM}-R_{i})$

$V_{m}=\{v|v,u\in V,\exists e=(u,v)\in E_{m}\}$

$v_{2}$

$\Delta_{v_{i},u_{j}}=\mathcal{E}^{v_{i},u_{j}}_{2}-\mathcal{E}^{v_{i},u_{j}}_{% 1}=\left|w^{*}_{k}\right|-\left|\mathcal{W}^{\prime}(E^{(v_{i},u_{j})})-% \mathcal{W}^{*}(E^{(v_{i},u_{j})})\right|\leq\left|w^{*}_{k}-\mathcal{W}^{% \prime}(E^{(v_{i},u_{j})})+\mathcal{W}^{*}(E^{(v_{i},u_{j})})\right|$

$k$

$w^{\prime}(e_{i})=w(e_{i})+\epsilon_{i},\epsilon_{i}\in\{0,w^{*}\}$

$c_{1}$

$n_{R}=3$

$\displaystyle=n_{L}\times(w_{0}-x+w_{0}+w_{1}-x+\dots+w_{0}+w_{1}+\dots+w_{k}-% x)=n_{L}\times\big{(}\big{(}\sum_{j=0}^{k}(k+1-j)w_{j}\big{)}-(k+1)\times x% \big{)}$

$\mathcal{C}(v)=k$

$\{v_{1},v_{2}\}$

$\mathcal{E}_{L}^{(\frac{k}{2})}$

$k^{\prime}$

$c_{i}+c_{j}\leq 1$

$S_{L}-L_{i}-S_{R}\leq 0$

$\{v^{\prime}_{1},v^{\prime}_{2},v^{\prime}_{3}\}$

$E^{\prime}\subset E$

$v_{1},v_{3}\in\overline{V_{m}}$

$w^{*}_{1},w^{*}_{2},\dots,w^{*}_{k}$

$\mathcal{F}$

$x<0$

$C\leq x\leq B+C$

$T_{j}^{L}$

$\mathcal{E}(M_{L}^{*})\leq\mathcal{E}(M_{R}^{*})\xrightarrow[]{}\sum_{e_{i}\in E% _{L}}L_{i}\times w^{*}\times(S_{L}-L_{i}-S_{R})\leq\sum_{e_{i}\in E_{R}}R_{i}% \times w^{*}\times(S_{R}-R_{i}-S_{L})$

$x\geq A+B$

$\mathcal{E}_{LR}$

$\displaystyle\geq\mathcal{E}(M_{0})+\sum_{e_{i}\in E_{L}}L_{i}\times w^{*}% \times(S_{L}-L_{i}-S_{R})=\mathcal{E}(M_{L}^{*})$

$x

$S_{RM}>0$

$\mathcal{E}(M)<\mathcal{E}(M^{\prime})$

${L}_{i}$

${n_{R}}$

$e_{1}=(v_{1},v_{3})$

${\mathcal{L}}\xleftarrow[]{}|E_{L}|$

$\mathcal{E}_{L}$

$\mathcal{E}(M^{\prime\prime})\leq\mathcal{E}(M)$

$w^{*}$

$0<\epsilon

$|\sum_{k=i}^{n_{1}}\epsilon_{k}|$

$|\Delta E|$

$-R_{i}\times S_{LM}$

$w^{\prime}(e)=w(e)+w(e^{*})$

$u_{1}\in T_{i}^{L},u_{2}\in T_{j}^{R}$

$S_{RM}=0$

$\Delta({\text{MARK\_LEFT}})\leq 0$

$C_{R}$

$-L_{i}\times(S_{LM})+L_{i}\times(S_{LU}-L_{i})$

$c_{0},c_{1},\dots,c_{i}$

$k+1-j$

$|\Delta E|\geq(n-2)B=|\overline{V_{m}}|B$

$B\times k^{\prime}\times(n-(k+k^{\prime}))=B\times(n-2)$

$P^{\prime}$

$|\Delta E|^{\prime}\geq w^{*}$

$C$

$|y-C|+|y-B-C|$

$E_{m}\subset E$

$j

$\overline{V_{m}}=V-V_{m}$

$v_{i}\in V_{L},v_{i}\neq v_{n_{1}+1}$

${\mathcal{L}}$

$e^{*}=(v_{1},v_{2})$

$y=C$

$\mathcal{E}(M_{L}^{*})\leq\mathcal{E}(M_{R}^{*})$

$u\in T_{2}^{L}$

$\Delta({\text{UNMARK\_RIGHT}})$

$v_{i},v_{j}\in V_{L}\;(i

$c_{i}=1$

${n^{2}_{R}}\times((j-1)({\mathcal{R}}-j+1)-j({\mathcal{R}}-j))={n^{2}_{R}}% \times(j{\mathcal{R}}-j^{2}+j-{\mathcal{R}}+j-1-j{\mathcal{R}}+j^{2})={n^{2}_{% R}}\times(2j-{\mathcal{R}}-1)$

$\mathcal{C}(u)=k^{\prime}$

$E^{\prime}$

$0.4$

$\Delta({\text{UNMARK\_RIGHT}})=0$

${n_{L}}{n_{R}}((i+1)j-ij)={n_{L}}{n_{R}}\times j$

$-\left|w^{*}-\sum_{k=i}^{n_{1}}\epsilon_{k}\right|$

$c_{i}

$e\in E^{\prime}$

$k-k^{\prime}\geq 0$

$-R_{i}\times(S_{RU})+R_{i}\times(S_{RM}-R_{i})$

$w^{*}_{k}$

$\operatorname{CONTRACTION}(\pi)$

$T-\{v_{1},v_{2}\}$

$V_{R}$

$\begin{array}[]{cc}\Delta&=R_{i}\times\bigg{(}\underbrace{-(S_{RM}-R_{i})}_{<0% }-S_{RU}\underbrace{-S_{LM}}_{<0}+S_{LU}\bigg{)}

$G=P_{n}$

$\triangleright$