From ed771de39ac3841f8bf4ad9485ba702d78137025 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Wed, 14 May 2025 10:29:02 +0800 Subject: [PATCH] typo & JP theorem --- main.pdf | Bin 102758 -> 103758 bytes main.tex | 20 ++++++++++---------- 2 files changed, 10 insertions(+), 10 deletions(-) diff --git a/main.pdf b/main.pdf index 1c1d957ea59d032dea607446c644811db555657b..3ed2f861fa23a17292f2573a22b0d7211c69cb4d 100644 GIT binary patch delta 49478 zcmV(*K;FORq6W^Y1`#DtL`E$!E;kA#(A34)3N<&e5dshaHj{xM6tgH0SP%g?kr5RE zIhWCP0VsdvTw9ad#<70iU%@;DTUn0hJ_A?zR4Iz&l4CoLckN0_&I3eZcM+4sh9Gr0 zzy5r40SJHqNLee{&Usmaz@VqQr!Qak442OCW_0%D1?_$Q`R3xY=S;F06T}!UXE(Pq z!)JyXBBhz#EN3qharv*CubMYzFXxO>ntc78yN!RBg_p);# zY{FPhMSCo_#HN_q5G7a`S`#&9W{6hIGlW-+PG@t*370ku%nt)w6cyglGeNjb!?48= zJxK(!vpFY1n<;9N7?#dj!`=>2Qy6tj?BdzY#eXgsR7PhI5i$yBS!+QkL|CjZUjB>D zmUw^o)r=C2`QG@U>zPo5DS^w??E2#GVS##jJ5Zy<)(SaWbH#|}sAg@={bh)rCOTCe zVT%#X39H+hdpE2(ru7-r=E@Sr4a9(P>BwI-Xnrr}TqR_X%FDT?$!_E=W9zqiy2f%? zhQW*|s>00w7?K4mVVxO3gPL`X3MPIS=NW(GLlj=l6-yolS%=(IrH%=s^n06PV4+1$ zV?f8Qf1+HC=u9X(ym+udrug9Mb!X!ST-)|^c4RI^8~M8=oYHKqHD{rCmQsuf&$t}6 ze!+>PvcqC^7zbqut2~D!HLz-Em#1AxU?vB0peFdNHR@`38Cz!7nFhg0kImB77fF8w z9Ehg6J4;_BOr)clJu4Ieu4~$m!~}8i!m+3J=mba<~${oIV=EV#^jl)?Eg(BV`s!8=D^!; zJ(0=ZUm`=u9!+GbNn{kRnf_i9nN^eS`#%%zOTD~|rm)JHgvkkcNJ_j(N^^fwVVNt{e`mJ+4pCk^Bp%PPd_Kt9K^QsyJMU{TtKQ*p_;n1NH! z;9+g=I+!zq@V-bo7_^ri3&yppI#2<~BSUX{U2vyn)YZ@|2kk)F+y&SHy8W8WxPU3t zWCjtcYl6d(#}I1xNTjp_RycnVND^gq|jdk0Q!JJ z5Q_@0HE!4p!MyaC=L76;&c-d#8=Mh+JJg@l4twbWaa<53#PB<)Q$}#F zB^=rX9JADXWb;3^Iy(%qnIgP5+b}FaMIlptriJGroGVJ0g+!2(kkfSn71Wj}C#=?v=iutDL;IPiQn zsA^83Ht7sXXggU*G2TVwT~4}GQ^G}C#mhnTj_jwr7Ke=hGYr?WV}OL&wBe^e6IiX6 z>bPDrt~m?gEHR%n$4h@CG0K0PV(pxEr65As4`0V>Flb03b9@pJ0RBclQXsk16gY-+pu>yv>CCPjs*OqyijW1a7|#Ufmn zo3YxB%H+r4?v23o$YQW*nes!xlC>8cHZJB;Kyn8fbTk@9IwOBRdoHx_T69s!dsE7j z7vIG}K>I2Fpa!8xoU@;2N4NfZ;fVHV9@RU7*;Xv>PboBudn@Np$@^3GbVT5MaU;U- z%b>2%f@m1a%^I(h%egcr`LcMOZ!WL8trkkb^qepkN z?2L7%o~k=rOiM&C3iR3`o^6(s97zG(#?tOfr@BjeD)O{Bk`Tl8h*!mWzuIM+yxdo- zXc%vb-F+xjWF{S&lrFN>CBS4cW`=lEB*cK zZXL({4mG%=Ff6&b&x7W?wJzgzc2td!=Z(l#||woLnz-w_U`gsWNFuM zpRMCn`Rd>KVuv6TzP`>@`<(Pbbjpy5U3G=j&SKrm&xz6>ELT6(I~Knf)1!Uw`NRo5 z)~o*NUC@8da-Xe=-MdRBWYSX-jp~W6AwGtBrhkQ5on$Vb-UxImJ-BNsh^ zip_Gr$fFr#LjpCi*QdQh&WxE)Aa=QrCfCpH5yx zFy~UWhf6esZg;PI*cIz+<-XAO!+zGG2mYz#6I2kTy0c}gstl|)3+v(DF|4K0Qlmni zmYaX7SmxW#`1aC|0y9P(^szI6iYNufJ_{37rOh(?$f2q`&5x&;69Rdc_L$ovkl%$V^W|OS zbd_)J0=yruLI6RG(PtX_8MfYHRJ#36KDfF*Upp1^Hw$7H>GD+JD;Y@)e&B4VvmJqBaK&QY+9oNE24S%&VjqAeHo}Q-nreD>u{BY zk8O?u3n=qgd z285&OdZ^J?|ErFojvJLsDg1TB#gBi=WFA`}9?drvrJY5M;7Nsb=8^0+Uze|46F^WE z%5B#Y4M+`lTBWu7`YK

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zUd2|(mTdqjrjJLVmqMYBV?qSi(gcal2{{Uj53c(snjKri&ZsH*Yqa}SnwS4k>XY-VgsOt7^yfnHHnzd?S$^Y-qK%?u zCdkFe#ZdfQ@96L9m8nFFINJVQ_T~&2f zX%?+sNmUxPdQ^=PUfuD{C@HNr!*3``a+J&*wE;#qltB!Obo}3fj2=a`D*V1Xu$d9o zT%o*+6#zU~_fG^ob678R?#!p?Znr7VTW{XTEGH6w%&RC3)8;datH>-*eu+LgHl$V*j7W(bNXc d)y>4k&CA8Y5{{LHm4}xX8jg}uQdtV_{{UJ0rK11< diff --git a/main.tex b/main.tex index 908de55..29257dc 100644 --- a/main.tex +++ b/main.tex @@ -113,22 +113,22 @@ s.t.& & \sum_i D_i d(s_i,t_i)&=1 & &\\ \end{equation} \end{minipage} -\newcommand{\lp}{LP\ref{LP}} -\newcommand{\ip}{IP\ref{IP}} -\newcommand{\dual}{LP\ref{dual}} -\newcommand{\metric}{LP\ref{metric}} +\newcommand{\lp}{\texttt{LP\ref{LP}}} +\newcommand{\ip}{\texttt{IP\ref{IP}}} +\newcommand{\dual}{\texttt{LP\ref{dual}}} +\newcommand{\metric}{\texttt{LP\ref{metric}}} \begin{enumerate} \item \ip{} $\geq$ \lp{}. Given any feasible solution to \ip{}, we can scale all $x_e$ and $y_i$ simultaneously with factor $1/\sum_i D_i y_i$. The scaled solution is feasible for \lp{} and gets the same objective value. \item \lp{} $=$ \dual{}. by duality. \item \metric{} $=$ \lp{}. It is easy to see \metric{} $\geq$ \lp{} since any feasible metric to \metric{} induces a feasible solution to \lp{}. In fact, the optimal solution to \lp{} also induces a feasible metric. Consider a solution $x_e,y_i$ to \lp{}. Let $d_x$ be the shortest path metric on $V$ using edge length $x_e$. It suffices to show that $y_i=d_x(s_i,t_i)$. This can be seen from a reformulation of \lp{}. The constraint $\sum_i D_i y_i=1$ can be removed and the objective becomes $\sum_e c_e x_e / \sum_i D_i y_i$. This reformulation does not change the optimal solution. Now suppose in the optimal solution to \lp{} there is some $y_i$ which is strictly smaller than $d_x(s_i,t_i)$. Then the denominator $\sum_i D_i y_i$ in the objective of our reformulation can be larger, contradicting to the optimality of solution $x_e,y_i$. -\begin{theorem}[Japanese Theorem] -$D$ is a demand matrix. $D$ is routable in $G$ iff $\forall l:E\to \R^+$, $\sum_e c_e l(e)\geq \sum_{uv} D(u,v) d_l(u,v)$, where $d_l(s,t)$ is the short path distance based on $l$. -\end{theorem} -\begin{proof} -$D$ is routable iff $\opt(\dual{})\geq 1$ -\end{proof} \end{enumerate} +\begin{theorem}[Japanese Theorem] +$D$ is a demand matrix. $D$ is routable in $G$ iff $\forall l:E\to \R^+$, $\sum_e c_e l(e)\geq \sum_{uv} D(u,v) d_l(u,v)$, where $d_l(s,t)$ is the short path distance induced by $l(e)$. +\end{theorem} +Note that $D$ is routable iff the optimum of the LPs is at least 1. Then the theorem follows directly from \metric{}. + +\subsection{Flow-cut gap} \bibliographystyle{plainnat} \bibliography{ref}