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Eleven. The E-approximation scheme HE for the knapsack challenge. Theorem 2. 6. 2 set of rules he's an E-approximation scheme. evidence we'll continue in a totally analogous option to the evidence of Theorem 2. five. five. If the optimum answer z* contains below l goods we are going to enumerate this resolution sooner or later throughout the execution of the 1st for-loop. another way allow us to give some thought to the set L* containing the l goods which give a contribution the top revenue within the optimum resolution. set of rules HE generates this set throughout the execution of the second one for-loop and applies Ext-Greedy to the knapsack challenge ultimate after packing all goods in L *. Denote back by means of Zs the optimum answer of the corresponding subproblem S with merchandise set N:= {j I Pj ~ min{Pi liE L*}} \L* and potential c - LjEL* Wj and the approximate answer computed by way of Ext-Greedy by means of respectively. As sooner than we have now for HE that I' ~ L jEL* Pj + and from ~ back there are instances to be thought of. Theorem 2. five. four that rp, zsG ! zs. Case I: LjEL* Pj ~ z. hz* From above we instantly get A ~ '~Pj+zs " eG ~ '~Pj+ " 1 * z'2zs· jEL* jEL* The situation of Case I yields tiP 2. 6 Approximation Schemes 39 Case II: LiEL* Pi < e:hz* at the very least one of many £ goods in L * should have a revenue smaller than l~2 z* . by way of definition Zs is composed merely of things with revenue smaller than l~2Z*. For the LP-relaxation of subproblem S with price we get from Theorem 2. 2. 1 by way of including the full break up merchandise s' *_E_=I_£ 1-1 __ £+2 1E and therefore have proven that during either circumstances zA ~ (1 - £ )z* . o additionally the "bad" instance from above could be generalized to teach that HE can achieve a relative functionality ratio rather as regards to 1 - £. r1 r1 enable n = ~ + 1 and c = ~ M. merchandise 1 is given by means of WI = 1 and PI = 2 and goods 2 to n are all exact with W2 = ... = Wn = M and P2 = ... = Pn = M. these things are back referred to as "big goods" as within the facts of Theorem 2. five. five. Any subset of £ = n - three goods both includes simply monstrous goods during which case Ext-Greedy returns merchandise 1 and one other huge merchandise as answer of the subproblem yielding a complete resolution worth of (£ + I)M + 2, or it comprises merchandise 1 and £ - 1 colossal goods. within the latter case ExtGreedy packs one other massive goods which results in an identical answer as ahead of. even if, the optimum resolution contains all £ + 2 immense goods with overall revenue (£ + 2)M. The ratio among approximate and optimum resolution price converges to for expanding M. m contemplating the operating time of HE there are quite often the (l) (trivially bounded via O(nl)) executions of Ext-Greedy for all attainable subsets of cardinality £ to accomplish. given that each one of them calls for linear time this yields an O(nl+I) operating time sure. In Theorem 6. 1. 1 of part 6. 1 this time certain could be decreased to O(nl) through a greater implementation of HE . For the sake of completeness we even have to say that Ext-Greedy calls for an extra O(nlogn) time for sorting the goods (of path just once) whether it is no longer changed by way of the simplified model with operating time O(n) pointed out within the earlier part.