From 5609db649d3edf85c02a2b145cafc1072710384c Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Mon, 28 Apr 2025 17:08:04 +0800 Subject: [PATCH] lp for normalized mincut cont'd --- main.pdf | Bin 265196 -> 265185 bytes main.tex | 5 ++++- 2 files changed, 4 insertions(+), 1 deletion(-) diff --git a/main.pdf b/main.pdf index 565aedec0e0cfb09a16c416796b5c715ac7f52e4..febdecbf0003d5074b47a698420e44059547abda 100644 GIT binary patch delta 17047 zcmaf*Q*@?X(57SCNyqA>JGO1xwrxGJJ5Ju%>e%SmR;Odzw&(k2&CKbXU3IV8d+o!j zyK3!o5ZVq1t!|Yaf`us&9TS3u8$_B5NegTcuDPrYKj<6wjmh>!wQatYuwKfYN6j~7 zaM-gYPjh+aOG$~yDv9|A5b5MA}>>-OKp3KvE+um%Wo0B1v{ z*Ft^&@KT}BHgcgFd!)F;*hqw98cfT$mG$ycl%=2=DX-EzTGKDlxqMkA|Nf>S64{&qs4A;42I(1lD|MAVlmbe*`X-DUHeAMK9k@Zd*~ssh#WV;bue-=kpow z^I7%gdkL8Hd4Z#jI*n)wU;k!Xg^{8L_A0t@g!t7I;XBd2v2tBNet#$9QnG}!QsJv_ z7jXz!_CuQ-dWgYyXsLa+km6)EE2SnNhbqJlQUl}Pmb5k4B+$o^qs=(dHvg-xEG#7a z#r1u28vTd<*XB9leIs-E`gI@9)L&j`rlzXeKH=rd0Ym#x@Z|7a+;ms0oI(1ck6`?K z+8zCm8evG7zfYFg9uXmAnQ%DuxvUZ~0%m6j=_0R2hXIX_ib#s7OO?lD+^M=)7nJo3 z1}`b1#?B#|nZS~1_wo+q2t1mAuA8p@)_Dj#l-kMe^uP&<2eTexH81J9=>XQ&6z*o+ zcBjo(#IfMIBj-Yg%U!`9`pb88Y6-_69G3U|7_=wID3>6MW7mxl_afzjEeFjl{X*`w z$snkM3-5j#L5_ccwD9lyjFp8EQML{yu~w>-kq+^8IY5`eHNxhRd6Y$pBo0lhF`H3! zH!_mORNoonC#&HyxmOIC!`)NZwGAQnFoh*Llc@?E7%J!{`wx^_?*r&+-cK!vzE|CKGorIGCjNFp zBbq0gwg6HDlXu=oAvAjVICS#jz5Dc{^CWC?gOlM7zQYmgEIf9Db0hHOd?KbXZyKeJ zz=R$GvEO#z&fWL%QIbb@fOp^5e&f7vrZdOc32>!~ez$nsSP?YApa}uHtj42(u#QZm zwmQ@t03|fxvRHa*ZbWi-kmU$z6Enx_1AkdPLV4T3#PDP?gxW(6aUr-Y-i%qwC}v$T zUSMj&@6h^>cNGWH6d|?@bAwZ@;r!IEs$7v`=|-g+89`@Yt}^)rhRUF+?wZbvRn4_C zV1H<@_%{<~Y*~0gpY21xT<7A1q+ZN}T^Ef2$(SE`ijMjhaDS&+?{F16m+=4ujNibK;|N*rMs= ze8-8rWchsYank;nnlCcvpEwKobpO1cfr}p+wV9#{ru{?M*_c1=Q7kTZAMZl^LQSqu zFGzIMyw`RBjNab_(OLiLDUktS=UE>nP3LBj^uwt-C`Uv6(p=6 z45M|(*(PS*&nb%$F!x$uE1qPI`)1;ipl%eWg@nke)e)Yk7k+`ehlQ7RnDIeK0kGP8 zMKyM*mNatb)Rk2+SyY7_hgQ^g%PDo69u!fslW^U9IVgm%Ib^Uxepy6|YQGJbGHCa< z2Y>q=)Mku4#$8Gg>k6{O^9MLH5T$G6gj})O3H^4TF@~Vp^Oh&AU7;b0kP-<&Z`0RU zE01;q-`u{KEm^J8LYwflj6Yn;13`^EaDvIRqf%*&_1YSilY)lXMw=R8I>JFQUm3DRi>J}A`S*P<-(SpP3mjpgvrPfVd0Ac5Nh&Mrt zMP8c=rK}8{7)|gG4wsXx<{qKCY^sTJ@iwI)rv313qo#^cbbo(7nn{CwAa-=aN^Jz0 z=!$P=V>+2GCY%xicoIV2bCV+H=l3nmP2?|9oXKUjjNI+TTMbrr@1gS_CbTO|U<(8r z5KMi?*PHBBK%=Y!V;Rfrj^i1^K}v+CMokZtl1m6TU}j$ac{o9gaMxv?Lokq;+@BvM z^hAiKa7VcpBMSdV2lP-ZUn->za=gA-`;p#z-G6B0)|49q#2q`Y^jluI-#opDgonql zOdAffNFFnR|5oU(wGHdh?az~;GJXUv<_W3s-BH%Rq$xm)wnP-ALqrSzMrV742f@Y$ z@t#gf);?&op68ch(TK=s=b<>Rt#8Wt| z?B))GNcdju8^Z_Vbps9Y+YL&eio0huqG$Z#)lVf45N@04JHZdPRAFFIX-JEUPl-*MeSo4@0xS+EI=# zDnitLB&vl&H4^c!(o_J zJUd%GlJMX>O7>?_K$kF&i&dV=Ge*x=Lv&DS!RYA5ZGjV6j}_ZcK`fM)dkqa<5Ghxi z4T{J+ig20sZ}Q^uqX}?7;_!8%s$mWa1IRi6zV*;TEyCO8{LAR{fotoardokjPs~^> zt=el(T}y+*9GD_ruaG!YLsOo@XAU-Al;;bT;0NsbHoi@Xei4hU!;hbH_I1f!Xf0Q2 zY!vnbI{t+Ew;W+)pRM!iCcfRW`Sb9MDUTb9+{Xb5t;TH$9+n$?_VI5!crbhG|cFHn4dn*@ijvtwFm0U{%M1Q33-cXS7h)u3gkZ4w6JsXHL6(Pm4&1zz>f$< zu*>9l;rknjdDVm>>*C(d{U*&&3IjIxrZDF!wT5m}%^jjn{!hQF{>9!N%exmA))q?L z0b7l+NQ^DZpeiD56byjhZ(XPzo3{#(s+!S<;cy8DD)+CZ<~?5+(TzveLx|w)oOuWA z<|88b{Bn&9B1Pjv&?d>z`!N!|r=508Q~ekK=W-bt+KmiYo^9fS` zeBWOdq&oz{N|C&F21u2KUl6_J{(a&g4wpN!=`(3f7RA|0yt}F=7;~|S;SJad;f7sPM8^o%azb4R8CQ`1M3RXdj@?E3!Kz{{ zB3#VdC_wV&6OKHOL@V^dG^Gc4peambX0Nrx?#;9eH*98h%qhArgB#<|ZA0&@ds#T^ z$R)^QMvaqS z-r>L;jUEuQjJHq{VBW_KY17I5acz$Z; zr6qHj*P*`;-3A(KT#Fvig}C1s|GC@M7bcVm-zfrrffb0LIsuuFI?=-j?n^}-}SD*DdP{ctMWCyDh=R+0B(>0k3h@><|{$pewzKLq?cjPTA4A;JK zbj<`3OpR^2hUPS9U&L7$FiS1z9!;b-$_rxxt6zwBtja;jJCc~-tchq?kZfE`AhKLY z8el_j!}VJ$I`CxJH?kKSx%*~HZC@t6*JNFq%#kT=X(u@QPuqk}1wmTeJrok26g-@_^ zguJMl5OvemW3*Sfr=3j5VSvMg}N@eby(dN>uKG7M?Xp=6QBf={!DOrmr+cCq+< zwy~?v(?{T^d=n-nq#mmRtV4B~949BQoq5*KL&{ufj$>(TeYhE9FOQNIpLFOyV zyJkhmwSK0%_wQ);hQ(-$Y1N!~uya28UYX_bi|>za^S2qjWV#M(h%OLk*1@F>wgbIp z>yqp~(1A=exz~}3Pwwp;{s}7;2o4W-9|ky<5>lF~>QN%}lWL{g7q7em1su4!(c+Bg ziB@=4u6G2m@Re=k1%{$2E||+zVlqUFr!Ovz7ioNMb@~QPSx(t=fip-ie-?44|%qKRUtQ^Ed2lQs$&F|4vg*v2qa7p$CV5MADkt8qy3J zdeela(38XaP7pfN^W0MdNGh?2{iMEOhyHLrM*XzK+~nQr0mV+jH_!G2I_rB~Ex73*9zvVT|y3Am*R)UVrNY=a(GzLcZB`#+tF}PW;Ki za9XZIeZduQFVo;Uk;5UF7s0kcgk8Er-%!DqXBiH}AwG7Z_`sIyrY8#d&-w93glAzx zYl9_Sj2_VTc$$3lB#}OnWc641^M#4P@RZDRg5}P|3!!0k%=1Zle*9FAz+`9vXMgDA zLLv3ts4TPu4D$rvUbIKR!RL2J!Jy7e8npO*Z=w|#wpUtvy^w94uD0|F7A#fz4j%R+ zEWe@xY$U)R!xAK!5ij*qY8fXQ6WqWeV3G|}zPsc{3Hp7CLPOoL%qD_~_d$#aKCE(L zJ_nw~{J;qXdi2^qgAaIqa+>Jyk8VqW6j;xk%`(dY>6geC%zuw)Y{|Ze6{@R2>sKS? 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zi7Z&STMnd7)`mz1IT&DVkQAW;jW z^jHYhI-%{bgU5huCrPJPf^OIg6!6`!3)q(_^}rr}b^%*5{p7Mt1-PZ_ocDv?R|laJ zFIlKr*0X=;z;u!vyQ^ht-cdLPV=!o;(5owk*{A!Cz$l!6lW>ZEd1MTsdIXDj&VHPgIWtf92a0_NEjQv)DGF*e}Fb_B24ma0a-L_DE&Sno5|2|Y= zkeoqs#>W{Srzy_lIFsYdi!(3IlsFA_M#UKwXHQk%t>t4i$_Uk@pDSbXy# zz1G{OiFY%JmDtf>DtHgIU|5!-;TuS`X2kQ)o({88TuKw!dq`yvo uo&Fd){dM(MpDAPE;}-vn1gofgAN~U_NN-k`xC;ay3N$b_3MC~)PeuyuySyI& diff --git a/main.tex b/main.tex index 3f073f6..ed1f4ed 100644 --- a/main.tex +++ b/main.tex @@ -228,7 +228,7 @@ Now we focus on $L(\lambda)=\min \{w(C\setminus F)-\lambda(b-c(F)) | \forall \te Let $B$ be the minimum cost of cuts in $G$. We have $-\lambda(b-B)\leq w(C\setminus F)-\lambda(b-c(F))$ for any cut $C$ and $F\subsetneq C$. Thus the upperbound is $\e=\min \frac{w(C\setminus F)}{B-c(F)}$. -\note{It remains to show that the optimal solution at $\e$ guarantees $c(F)\leq b$? or maybe we don't need this for normalized mincut. I think normalized min-cut should not require $c(F)\leq b$. Further checks are needed. What we can guarantee is that $c(F)\leq B$.} +% \note{It remains to show that the optimal solution at $\e$ guarantees $c(F)\leq b$? or maybe we don't need this for normalized mincut. I think normalized min-cut should not require $c(F)\leq b$. Further checks are needed. What we can guarantee is that $c(F)\leq B$.} \subsection{differences} Consider $L(\lambda)$ for cut problem. One can see that the optimal $\lambda$ is clearly 0 since $L(\lambda)$ is pwl concave and the slope is negative at $\lambda=0$. What we are really interested in is the first segment on $L$. At the left end, $L(0)$ is exactly the weight of minimum cut. (the complementary slackness condition is satisfied.) At the right end, as we have shown in the previous paragraph, $\lambda$ equals to the value of the strength (which is the optimum of the linear relaxation of the cut IP). However, for cut interdiction problems $L(0)$ is not the optimum. @@ -257,6 +257,9 @@ s.t.& & \sum_{e\in T} x_e+y_e&\geq 1 & &\forall T\\ \end{aligned} \end{equation} Note that we are forcing $b+1-\sum_e c(e)y_e=1$. +\autoref{lp:normalized} is exactly the fractional optimum of \autoref{lp:conninterdict} since we have shown that to satisfy the complementary slackness condition the budget constraint has to be tight. + +How is this related to the optimal $\lambda$ in \autoref{lp:dualcutint}? The integrality gap of \autoref{ip:normalized} is also not clear. \subsection{integrality gap} I guess the 2-approximate min-cut enumeration algorithm implies an integrality gap of 2 for cut interdiction problem.