From 55913ffe2e8fd4adbab90af28cc002c92ff054e3 Mon Sep 17 00:00:00 2001 From: ArrogantGao Date: Sun, 11 May 2025 16:40:50 +0800 Subject: [PATCH 1/2] add hw1 --- hw1/2025-05-11-14-55-56.png | Bin 0 -> 146791 bytes hw1/2025-05-11-14-56-48.png | Bin 0 -> 159677 bytes hw1/2025-05-11-14-57-11.png | Bin 0 -> 182506 bytes hw1/2025-05-11-15-02-41.png | Bin 0 -> 56933 bytes hw1/hw.md | 33 +++++++++++++++++++++++++++++++++ 5 files changed, 33 insertions(+) create mode 100644 hw1/2025-05-11-14-55-56.png create mode 100644 hw1/2025-05-11-14-56-48.png create mode 100644 hw1/2025-05-11-14-57-11.png create mode 100644 hw1/2025-05-11-15-02-41.png create mode 100644 hw1/hw.md diff --git a/hw1/2025-05-11-14-55-56.png b/hw1/2025-05-11-14-55-56.png new file mode 100644 index 0000000000000000000000000000000000000000..d9fa288a0d7b6257a9e29ceab566a5c5f1556d80 GIT binary patch literal 146791 zcmbTdby!L6Kifbvw-QC?SxD(u6-tfKe zN3P%d*K?oiI@!(HojsY^*_qGGZh}9k$YQ@Fdx?O6fGsa4^$`I9-5LP_Q3D+n{)CU` z*A)T+mI_Ex@`JpjB+UnVTMLl2IRb)QaGWNZmU@|_pcQRMoFE%gBC@*Ix+V4-s6HD;i^f>c$ z*lu}Bx!F!}usm!+M#z?=iP_B3LD-N#+bp9++{fr+w*E-m>cf?R$ib;V^QpIXYz!YE zzU}g8bI9rYALzSqmLl)Vr;|sk(h{;)4@}>Q#Texfq8)xgZ(}A@*h3G%YO9g4u$%F-t zF0ce6gj+>l_uhGafO)HJz4D4G!=_}Ze2NnFWapj0fH1fz_xGwNqb?J;8V zWILuqlH(yNC1FV~k)L$IL)(=vdMmQeckI~G-@f@Yd>hSCyie12-#Mw=_aBId?N4z>R#!)H1g zO(KQ$CV;><*twSWFvMdW|9BU-0CTua1a1BLtE#Kg7bXCW9=#);jdIkhzCbrYDo3%0 z6`%VY?_}x)vaj{4h_$$%0{aCp`BB1a-=zeG+_n8$q!}PT){T38$$8KseLO@+R+=rK z{-GcJr)Zb+wVbktat-b4&&4f65zv|QOvqV)d%(!Z9pfF{YyfUeXS`8hX~nGkK43bO z7lF^2!m3lJESm*TxkhzSQrk9Xa`({ZBh)c7IKx!XpVMA9T{mzOQ_o=tpf{`{byGLz z*^C}6gzyo2adlcZf7*Hf^I&a%4fXr(UA9+K(@)q_F-_rg$wK=w;<4b;%*@qEFZEGZw? zNGwLlqly)w1YejDpBBPTab_)yX`h#+FRam~j5q|)$;5aBFuYKSTN%D#F9a0i5dx4! zR6JPWxTFj5&^lkN(lpEA5%_e{^~sS%2g~6zaFc5X#nKf-y)+~L=u06M zPQw&+CMRr7flR&K}M{DSPq>6B)B2%A4{>6!iTyPO1TSP7IY-7I5huoGFqJVND1Px z<2yvx{Ps5OJLU=KS`R{M;G?;IaSghE=|z3i)`GW#LjMD~Eu@u6iN=iJH{nkVbH7x- z4nLl698LsFc)?OcV*xveS0>zC$pRs@p|!!a5w1UGRUmTv3{i5ZZ_uJ~nSZ{a*^K!q zPcG*n*CU@GpPJ)hhF#7)=TD)eLaW#(y}^BIbINsUeoA;M`;(~a=bChIR?u4>=67#} z-zvw3#H#em{^gqr*p&>+vrs4gFp%#w$ulWDsYymgwndgk#zjUDc0;yFcE>vmT9|QF zoyeSLHa;eL_B_gn2wAPdi_` z#7+B5(6_zR7NY@;ozg)~#8Ae5gZo%@i__}6u^S#lw-pvAa(|N9HxxWr;!5t~~y-H0Lc?ec-S zyg3mI8H-E{g$l3**n)ZpJV-bgGI;fCxhg~F8!I>Kv`(egxmJ=+cufje+v+Cyb=ObR zIP^yEgE8<@%}*^YJ16_E%;L8kL3>I}3MSr6!^;z_b!*e6Bsf2B<*Jp`~af(f5hKl(Y z6K&QPi5jYEJ)N(+kL_E-H^LOT}; zHQ6rVS2jm^>(<@DJqILLqdg0fkh@2X!niMxo!_TXy@2vvXyZ>xp*iYtB7Q!MH(&_};>8eB!@GqkH9)lndn{ z$*&SCh4M|?cQ70n_2l@J;@|N9j{HcV?)>{)1uLo6#>Dck?pM3r3wgd63nd}Zxo`G| z8`

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+Since we know that $U \leq \ell v_{max}$, and in each step we need to compute $\ell$ times, the time complexity is $O(\ell^2 v_{max})$. However, $v_{max}$ can be exponential to $\ell$, so the time complexity can be exponential and the problem is still in NP. + +![](2025-05-11-14-56-48.png) + +With the method proposed in the question, the cost of the approximation algorithm is $O(\ell^2 \lfloor v_{max} / K \rfloor)$. In the worst case the error is $\ell v_{max} (1 - K^{-1})$, so that the approximation ratio is at most $\ell(1 - K^{-1})$. Then to reach a $1 - \epsilon$ approximation ratio, we need to have $\ell(1 - K^{-1}) \sim \epsilon$, clearly $K$ is polynomial in ${\epsilon}^{-1}$ and the algorithm is a FPTAS for the knapsack problem. + + +![](2025-05-11-14-57-11.png) + +At the begining, just assume $x_1 = x_2 = ... = x_n = 0$, which is not a feasible solution. Then define a vector $s$ with length $m$ with $s_i = b_i$. Then $(x, s)$ is a feasible solution of the following constraint: +```math + \left(\begin{matrix} + A & 0\\ + 0 & I + \end{matrix}\right) + \left(\begin{matrix} + x\\ + s + \end{matrix}\right) = b +``` +Under this constraint, we minimize the target function $\sum_{i=1}^m s_i$ until it equals to $0$, which gives a feasible solution for the original problem. From abcfc265be003d29981ae78fbe913d1d8be0564b Mon Sep 17 00:00:00 2001 From: ArrogantGao Date: Sun, 11 May 2025 16:42:36 +0800 Subject: [PATCH 2/2] clean up --- hw1/{ => xzgao}/2025-05-11-14-55-56.png | Bin hw1/{ => xzgao}/2025-05-11-14-56-48.png | Bin hw1/{ => xzgao}/2025-05-11-14-57-11.png | Bin hw1/{ => xzgao}/2025-05-11-15-02-41.png | Bin hw1/{ => xzgao}/hw.md | 0 5 files changed, 0 insertions(+), 0 deletions(-) rename hw1/{ => xzgao}/2025-05-11-14-55-56.png (100%) rename hw1/{ => xzgao}/2025-05-11-14-56-48.png (100%) rename hw1/{ => xzgao}/2025-05-11-14-57-11.png (100%) rename hw1/{ => xzgao}/2025-05-11-15-02-41.png (100%) rename hw1/{ => xzgao}/hw.md (100%) diff --git a/hw1/2025-05-11-14-55-56.png b/hw1/xzgao/2025-05-11-14-55-56.png similarity index 100% rename from hw1/2025-05-11-14-55-56.png rename to hw1/xzgao/2025-05-11-14-55-56.png diff --git a/hw1/2025-05-11-14-56-48.png b/hw1/xzgao/2025-05-11-14-56-48.png similarity index 100% rename from hw1/2025-05-11-14-56-48.png rename to hw1/xzgao/2025-05-11-14-56-48.png diff --git a/hw1/2025-05-11-14-57-11.png b/hw1/xzgao/2025-05-11-14-57-11.png similarity index 100% rename from hw1/2025-05-11-14-57-11.png rename to hw1/xzgao/2025-05-11-14-57-11.png diff --git a/hw1/2025-05-11-15-02-41.png b/hw1/xzgao/2025-05-11-15-02-41.png similarity index 100% rename from hw1/2025-05-11-15-02-41.png rename to hw1/xzgao/2025-05-11-15-02-41.png diff --git a/hw1/hw.md b/hw1/xzgao/hw.md similarity index 100% rename from hw1/hw.md rename to hw1/xzgao/hw.md