how to make barcode in c#.net 3S AT 3D M ATCHING in Software

Encoding Quick Response Code in Software 3S AT 3D M ATCHING

3S AT 3D M ATCHING
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b0 p0 g1
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Suppose that the two boys b0 and b1 and the two girls g0 and g1 are not involved in any other triples (The four pets p0 , , p3 will of course belong to other triples as well; for otherwise the instance would trivially have no solution) Then any matching must contain either the two triples (b0 , g1 , p0 ), (b1 , g0 , p2 ) or the two triples (b0 , g0 , p1 ), (b1 , g1 , p3 ), because these are the only ways in which these two boys and girls can nd any match Therefore, this gadget has two possible states: it behaves like a Boolean variable! To then transform an instance of 3 SAT to one of 3D MATCHING, we start by creating a copy of the preceding gadget for each variable x Call the resulting nodes p x1 , bx0 , gx1 , and so on The intended interpretation is that boy b x0 is matched with girl gx1 if x = true, and with girl gx0 if x = false Next we must create triples that somehow mimic clauses For each clause, say c = (x y z), introduce a new boy bc and a new girl gc They will be involved in three triples, one for each literal in the clause And the pets in these triples must re ect the three ways whereby the clause can be satis ed: (1) x = true, (2) y = false, (3) z = true For (1), we have the triple (bc , gc , px1 ), where px1 is the pet p1 in the gadget for x Here is why we chose p 1 : if x = true, then bx0 is matched with gx1 and bx1 with gx0 , and so pets px0 and px2 are taken In which case bc and gc can be matched with px1 But if x = false, then px1 and px3 are taken, and so gc and bc cannot be accommodated this way We do the same thing for the other two literals of the clause, which yield triples involving b c and gc with either py0 or py2 (for the negated variable y) and with either pz1 or pz3 (for variable z) We have to make sure that for every occurrence of a literal in a clause c there is a different pet to match with bc and gc But this is easy: by an earlier reduction we can assume that no literal appears more than twice, and so each variable gadget has enough pets, two for negated occurrences and two for unnegated The reduction now seems complete: from any matching we can recover a satisfying truth assignment by simply looking at each variable gadget and seeing with which girl b x0 was matched And from any satisfying truth assignment we can match the gadget corresponding to each variable x so that triples (b x0 , gx1 , px0 ) and (bx1 , gx0 , px2 ) are chosen if x = true and triples (bx0 , gx0 , px1 ) and (bx1 , gx1 , px3 ) are chosen if x = false; and for each clause c match b c and gc with the pet that corresponds to one of its satisfying literals But one last problem remains: in the matching de ned at the end of the last paragraph, some pets may be left unmatched In fact, if there are n variables and m clauses, then exactly 248
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2n m pets will be left unmatched (you can check that this number is sure to be positive, because we have at most three occurrences of every variable, and at least two literals in every clause) But this is easy to x: Add 2n m new boy-girl couples that are generic animallovers, and match them by triples with all the pets!
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Recall that in ZOE we are given an m n matrix A with 0 1 entries, and we must nd a 0 1 vector x = (x1 , , xn ) such that the m equations Ax = 1 are satis ed, where by 1 we denote the column vector of all 1 s How can we express the 3D MATCHING problem in this framework Z OE and ILP are very useful problems precisely because they provide a format in which many combinatorial problems can be expressed In such a formulation we think of the 0 1 variables as describing a solution, and we write equations expressing the constraints of the problem For example, here is how we express an instance of 3D MATCHING (m boys, m girls, m pets, and n boy-girl-pet triples) in the language of ZOE We have 0 1 variables x 1 , , xn , one per triple, where xi = 1 means that the ith triple is chosen for the matching, and x i = 0 means that it is not chosen Now all we have to do is write equations stating that the solution described by the x i s is a legitimate matching For each boy (or girl, or pet), suppose that the triples containing him (or her, or it) are those numbered j1 , j2 , , jk ; the appropriate equation is then xj1 + xj2 + + xjk = 1, which states that exactly one of these triples must be included in the matching For example, here is the A matrix for an instance of 3D M ATCHING we saw earlier
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