From 30dd262f6d3d146eab737af49a82a9267c1f1bca Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Sat, 19 Apr 2025 12:01:23 +0800 Subject: [PATCH] fix typo. different cut? --- main.pdf | Bin 253598 -> 253600 bytes main.tex | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/main.pdf b/main.pdf index 1a46645c1dcdd7270cb012bc78b6799c8a9701e4..accd4f06e2179112da475cc28bd234591a12aa57 100644 GIT binary patch delta 3991 zcmV;I4`}e7y$_(h53p4*0y8(4Ax#4)f9+dKZzH)8zWZ17v4tHme3Ok}FIntN5)6V( z#y|qh?m@PCY-KE2E6HO|etoJ)cB@H|Em?L33uI4fck@v%R((}P)9R+8)w>HCe*biN z@rSpJ+ln#LD8;MGPZgt-G*nlHT0*6&F6Y(9`tr7&Ou4c3YQmcOVR_YV{NMKze{Ske z{`qRt&L+&(pV#*jChNs&x9tWo0W*KO{44Cn4DKe4!OcESg;w?Z@QL&2lcM#{RMBpp z9)4u-iL2~9oW!~>5o;eEHH$#iZ0CQM$O4ea>+P;UBXHA^&uR3DSq=Nqo=BT~HFbfM zn&A6h6EMK&;eaXQL`hpsxg?S{e_jngb4Jr?@w%WpE8!DR37TYtYgK70h!(oKT*4CX zfoV~GBBH0k|-N!q*P>Cjz|-VSS{lnuF%G;sAvA? z(472x_gB`MI{G=R-ZTe{KikO8v9TeSTyw&xiRoCZ=4fya6K3k`b{=^Bf9ikR`Hd5T ztheoMj76ah5ls^>{q`tD!hkiYgd#GT3cVyOMLKr5S*jkMs@}$)hT@4<(7gaD(*i zcjE@E3RY(Iavj158mBWtW4iD&CR;8X-_iO-E%sz_K?zT&B4a+2e+zJgTE{cSd?u?R zqboSgh$b)-&66#BG}B()r&PtAnlL!>0)cD@Lg> zl$)&|7poipX|}m}e^`P5w&*Lltlw_Dc#;*?oAyh0iWs;=Rd2&y&P4`+@BHA$3DSa4yrYmNk9Hr09ukoKr5aHcw%p3@qTdlf9vn|Y!5$Kl`xu+Qd*Jf zDE`yTqIp2!KjwXcRfGyzatcEVBoVqvV%CU2H+e3M9+@Vay&e+vnK9nwsE+ixw7f0Y zL=)M>D{B|I`7V#^VQDsk3*8C*zK}~$`|^BeXg5p(cwo(mhGSB@DWllnNhAZLi7t2O zNze@f82n)}e>EYy+b-rVClsAKga>rS5`Aa89nCisw}b(%iMDuo>`dZ@LAZ>Y10r`j zJl-x4(O%znX$PdX-bH)wKWV^yXH2X2u>h$oNHkjX9Fr5m_U4k9I2WS+<1y|aD+Dxe zV?qLWa}d5utX3g|TJIJC0{9mZ{Db5wWCR~PsJ|W4f1NOpNhp~L;fU_+>8cZe_$}j! z>L$EFS}6q8dCa7@K8q=v3&Jt3P>9F)i5)N0q^^|@wV%pI*|Sc~ zP+8AAHTql=muH$d0~@D;|8C>wS)(jlqdvtNOD-QXVI!K@f%{;^k~COZ2sli4X>gZt z8oN&};rVukiR~BvCTk_Up=C$N09%ysR$NM>e}F-fQku+VkpTCa1{eiMrVet&B-tEj zDN0%$B0eEWO;#M9rHdagFMhq~3pf;Tv;mO<28Ba$y}bDN3$5nx4MBpoP4(zTE&;BR z21e5AuKIBC_s~R%8HBYE$SMP>q=Cmxo}vRw9Da=BUWt)ZWgn9>k%;CTLSnanNC?d< zf9b_KIyfjw5Th#rkdS$?lD?$a6}iE!Daa<2vab3bP?(~C3^%Q>Zgo|&SFjT^nbjca{`f4G8{D^QvGAyfkQ&O_I8zuYeJKZaLI0`BkO z;FKSUzdp0@>_L{KDxrHFF=OQDllT6+WBb?JLxU<9&XSlgmA$ie~x3H z0pm>t_=iYLw($^g!lUu-6jGYCwuXUve|_*J;a&?3+t-@mxwprz`@8k;_bkF6*UsSJ z0kW}nGxyo}aMH2n2HGKnLN2+t!@uDQj*Lgl`V?BSa+VCwur<(l-33W_6FL@0k|_*Q z(JrssuAZ{4XatDj`#hq+Py6C+e~f|h?GJ?GzH}Mg|r~|zz1==6f7`xRWA;a1 zT11cm{SF%=|jg#!}S-7k?x?w1H{2~~+ykKQji zsorb)s-Ym9#XDRDbOTyzXTC% zP-$S0(MAnMmJ4)}7?9dv)SE&l$?nslbRQ1DI=4|$&oE$qQNL5Me`AhJfG^;v^?@k% z7dx`j6M_;uEhdK3)GKH#lGmHaW-s_U%}qiRtiv55_|MJfZD@r38He3t&&j98UogdJHxmZ0-}sGp z-R-%96g2EOyfr6}LIT-X(ij^v{X;Rth+eYPHEW}3%f-#@e?k5(v2BdyTpY2&KHq)F zlCRRF8f8tMnNEsqI*EB+s6^Z?FjUF4OXqg)t|q3sowZ(bnAr6 z_qyPHXBXC9Z|Ro1O6XsO;~Vojp%NY7P4kv_K5H3WfcekjVakF z5PYZz_5Hm1{sG~xzxM=6`>=a(#jB&Hp@{Bok_*DJYj4_c4;y6irK>-|9#{8Ip;!>m z%5F1rH_LODGNnX$iQ68tNhsbQ=S0(>!hhw; 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(there is a $\pm1$ difference between principal partition and graph strength... but we dont care those $c\lambda$ terms since the difficult part is minimize $L(\lambda)$ for fixed $\lambda$) \subsection{principal sequence of partitions for cut interdiction} -Now we focus One $L(\lambda)=\min \{w(C\setminus F)-\lambda(b-c(F)) | \forall \text{cut } C\;\forall F\subset C\}$. We can still assume that $G$ is connected and see that $L(\lambda)$ is pwl concave. Let $\lambda^*$ be a breakpoint on $L$. Suppose that there are two optimal solutions $(C_1,F_1)$ and $(C_2,F_2)$ at $\lambda^*$. For fixed $C$ ($C_1=C_2$), the same argument for principal partition still works. However, the difficult part is that $C$ might not be the same. +Now we focus on $L(\lambda)=\min \{w(C\setminus F)-\lambda(b-c(F)) | \forall \text{cut } C\;\forall F\subset C\}$. We can still assume that $G$ is connected and see that $L(\lambda)$ is pwl concave. Let $\lambda^*$ be a breakpoint on $L$. Suppose that there are two optimal solutions $(C_1,F_1)$ and $(C_2,F_2)$ at $\lambda^*$. For fixed $C$ ($C_1=C_2$), the same argument for principal partition still works. However, the difficult part is that $C$ might not be the same. \subsection{integrality gap} I guess the 2 approximation cut enumeration algorithm implies a integrality gap of 2 for cut interdiction problem.