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| | '''Numerical 3-dimensional matching''' is an [[NP-complete]] decision problem. It is given by three [[multisets]] of [[integer]]s <math>X</math>, <math>Y</math> and <math>Z</math>, each containing <math>k</math> elements, and a bound <math>b</math>. The goal is to select a subset <math>M</math> of <math>X\times Y\times Z</math> such that every integer in <math>X</math>, <math>Y</math> and <math>Z</math> occurs exactly once and that for every triple <math>(x,y,z)</math> in the subset <math>x+y+z=b</math> holds. |
| | This problem is labeled as [SP16] in.<ref name=garey>Garey, Michael R. and David S. Johnson (1979), Computers and Intractability; A Guide to the Theory of NP-Completeness. ISBN 0-7167-1045-5</ref> |
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| Opulence. Maybe there's something in the air of Paris that inspires it.<br><br>There's the proliferation of palais (petit or grand) for designers to show their collections in, the galleries filled with florid, lurid art, and the haute couture, of course, opulence cubed and then wrapped in duchesse satin for good measure. They all do something to designers, no matter how minimally minded.<br><br>It's certainly done something to Marc Jacobs. Compare and contrast the first Louis Vuitton collection he presented oh-so-quietly 16 years ago with the ball-busting extravaganza unveiled last Wednesday, part �Turn Back Time� video, part Paul Verhoeven's Showgirls<br><br>
| | == Example == |
| It was an indulgent farewell to a house that has helped redefine the meaning of luxury goods in the 21st century. Indulgent for Jacobs, but also for the audience, a funereal feast for the eyes, Jacobs' all-black swansong. The Stephen Jones headdresses were inspired by Ziegfeld Follies and Cher's 1986 Oscars outfit<br>
| | Take <math>X=\{3,4,4\}</math>, <math>Y=\{1,4,6\}</math> and <math>Z=\{1,2,5\}</math>, and <math>b=10</math>. This instance has a solution, namely <math>\{(3,6,1), (4,4,2), (4,1,5)\}</math>. Note that each triple sums to <math>b=10</math>. The set <math>\{(3,6,1), (3,4,2), (4,1,5)\}</math> is not a solution for several reasons: not every number is used (a <math>4\in X</math> is missing), a number is used too often (the <math>3\in X</math>) and not every triple sums to <math>b</math> (since <math>3+4+2=9\neq b=10</math>). However, there is at least one solution to this problem, which is the property we are interested in with decision problems. |
| Says it all<br>
| | If we would take <math>b=11</math> for the same <math>X</math>, <math>Y</math> and <math>Z</math>, this problem would have no solution (all numbers sum to <math>30</math>, which is not equal to <math>k\cdot b=33</math> in this case). |
| Jacobs declared that the collection was obsessed with �pure adornment�, reasoning that �connecting with something on a superficial level is as honest as connecting with it on an intellectual level�. That's an interesting conceit. Contemporary fashion, like contemporary art, tends to be overthoug<br><br>
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| If not by designers, then certainly by critics. After all, we have to justify our presence there. And the Air Miles we rack <br>.
| | == Related problems == |
| Marco Zanini also showed his final collection for the house of Rochas. He's moving to Schiaparelli. Like Jacobs, he was obsessed with surface, with dumb, straightforward beauty. His was sugary sweet, in pastels that made your teeth ache, dedicated to Tennessee Williams' The Glass Menager<br>.
| | Every instance of the Numerical 3-dimensional matching problem is an instance of both the [[3-partition problem]], and the 3-dimensional matching problem. |
| Banish thoughts of tormented southern belles. Zanini focused on the crystalline beauty of said ornaments, bonding devor� velvet to organza and freckling iridescent fabrics with Swarovski g<br><br>
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| Maria Grazia Chiuri and Pier Paolo Piccioli took a precious approach, too. Their Valentino collection was inspired by the Rome Opera, apparently. In actual fact, it was all about the heavily embellished surfaces of lace and tulle. Lawn shirts seemed included purely as foil for all that decoration, a palette clean<br><br>
| | == Proof of NP-completeness == |
| | NP-completeness of the 3-partition problem is stated by Garey and Johnson in "Computers and Intractability; A Guide to the Theory of NP-Completeness".<ref name=garey/> It is done by a reduction from 3-dimensional matching via 4-partition. |
| | To prove NP-completeness of the numerical 3-dimensional matching, the proof is similar, but a reduction from 3-dimensional matching via the numerical 4-dimensional matching problem should be used. |
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| It would have made it easier to enjoy Hedi Slimane's latest Saint Laurent collection if you could take it at face value. Look! Sequinned lips! Flames! Lurex pop-socks! Removed from the [http://Www.bing.com/search?q=heritage&form=MSNNWS&mkt=en-us&pq=heritage heritage] of Saint Laurent, it had a pop, pap appeal. If you stopped looking for a hidden depth, the soul-searching that Yves Saint Laurent made an intrinsic part of his fashion, Slimane ticked bo<br><br>
| | == References == |
| | {{reflist}} |
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| Girls who want to look like that will love to dress in t<br>s.
| | [[Category:Strongly NP-complete problems]] |
| Karl Lagerfeld has always been about surface. His toying with the hallmarks of Chanel - tweeds, camellias, pearls, chains, those two-tone shoes - has always been about ironic appropriation, post-modern reinterpretation. It's the fashion equivalent of Jeff Ko<br><br>
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| He showed his latest Chanel collection in an art gallery. At least, it was on the surface. It was all fake, only the clothes were real. And they were pure Chanel, the art-house backdrop just that. I kept thinking of something Dinos Chapman once said to me: �I think that the art world and the high-end fashion [http://www.pcs-systems.co.uk/Images/celinebag.aspx Celine Bag Online] world� are the same peop<br>.�
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| People wanted to buy the Chanel works of art as surely as they wanted to buy the Chanel clothes. They both became post-modern commod<br>ies.
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| That's looking below the surface, though, behind the opulence of Chanel's specially woven, artfully unravelling tweeds that resembled rag-rugs, the canvas bags with a 2.55 fa�ade painted on the front, the graffitied art-student backpacks. They were just great, covetable fashion, brilliant pro<br><br>s.
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| That brings us, inevitably, to Phoebe Philo. She's known for great products - photography is banned in the C�line showroom for fear of rampant copying. And rightfully so. Still, that oft-imitated C�line hallmark is an ascetic aesthetic. Philo is the last designer one imagines inclined to opu<br><br>e.
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| But her riotously messy spring C�line collection felt fresh and energetic, pleated skirts bouncing below an elongated torso, fringe swaying on latticed leather bags, mashed-up metal formed into enamelled bracelets and <br>els.
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| Couture, of course, breeds opulence. Riccardo Tisci halted his made-to-measure line this year, but the handicraft of couture infected his spring collection, from the crystal-encrusted masks, to feather-embroidered bodices, to a series of sequinned, sinuous multi-pleat evening <br><br>s.
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| Those were old-school opulent. For modern couture, and a contemporary opulence, fashion turns to Raf Simons. His last Dior haute-couture collection met mixed reviews, but has inspired many a designer. The throbbing mood of Africa that beat through the collections, albeit slightly hackneyed, can be traced to S<br>ons.
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| This time, he pushed his aesthetic further still. It felt a collection in flux, sitting halfway between Dior past and Simons future, a cross-pollination. The inspiration was flowers, a theme at the very root of Dior. But the best summary was the least ostentatious: Simons' shirtdresses, twisted takes on the white cotton coats of workers corkscrewing around the body in a fascinating surface of complex <br><br>s.
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| So, French fashion is all about surface. Where does that leave Rei Kawakubo and Junya Watanabe, or Hussein Chalayan? The visual is just a fragment of what they offer, thinking clothes for the thinking woman. Jacobs may be in love with �beauty for beauty's sake�, but this trio of talents begs for somethin<br>more.
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| They all had a stellar, cerebral season: Watanabe creating an idiosyncratic ode to the spaghetti Western, Chalayan a paean to the windswept beach, while Kawakubo presented 23 non-outfits that challenged our perceptions of what fashion actually represents. She stated that she had no new ideas, so didn't create c<br><br>es.
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| If only other designers could follow her lead and thin out th<br>herd.
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| The surface of this deep-thinking threesome's clothing was universally impressive, but it's what lies beneath that really interests. Opulent intelligence? Their clothes beg dissection and discussion. To intellectualise a fashion show isn't automatically to over- intellectual<br><br>it.
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| Ultimately, that's what fashion is about. Surface is all well and good. But you have to get someone inside the damn clothes for it to all make<br><br>se.
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| It's ideological, as well as physical. At least, it is when it's really great.
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Numerical 3-dimensional matching is an NP-complete decision problem. It is given by three multisets of integers , and , each containing elements, and a bound . The goal is to select a subset of such that every integer in , and occurs exactly once and that for every triple in the subset holds.
This problem is labeled as [SP16] in.[1]
Example
Take , and , and . This instance has a solution, namely . Note that each triple sums to . The set is not a solution for several reasons: not every number is used (a is missing), a number is used too often (the ) and not every triple sums to (since ). However, there is at least one solution to this problem, which is the property we are interested in with decision problems.
If we would take for the same , and , this problem would have no solution (all numbers sum to , which is not equal to in this case).
Related problems
Every instance of the Numerical 3-dimensional matching problem is an instance of both the 3-partition problem, and the 3-dimensional matching problem.
Proof of NP-completeness
NP-completeness of the 3-partition problem is stated by Garey and Johnson in "Computers and Intractability; A Guide to the Theory of NP-Completeness".[1] It is done by a reduction from 3-dimensional matching via 4-partition.
To prove NP-completeness of the numerical 3-dimensional matching, the proof is similar, but a reduction from 3-dimensional matching via the numerical 4-dimensional matching problem should be used.
References
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- ↑ 1.0 1.1 Garey, Michael R. and David S. Johnson (1979), Computers and Intractability; A Guide to the Theory of NP-Completeness. ISBN 0-7167-1045-5