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2SAT:Решение — различия между версиями
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| (не показаны 3 промежуточные версии 3 участников) | | Строка 1: |
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| − | ==Идея==
| + | #REDIRECT [[discopal:{{PAGENAME}}]] |
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| − | Рассмотрим входную [[2SAT]]-формулу.
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| − | Во-первых, ясно, что можно быстро исключить все дизъюнкции, состоящие из одного терма — если это дизъюнкция типа <math>x_i</math>, то для выполнимости формулы необходимо <math>x_i=1</math>, соответственно мы фиксируем <math>x_i \equiv 1</math> и автоматически исключаются все дизъюнкты, куда эта переменная входит в положительной степени, т.к. их выполнимость гарантирована. Если есть дизъюнкт, куда такая переменная входит в отрицательной степени - формула неразрешима. Аналогично (с точностью до наоборот) избавляемся от переменных, "засветившихся" в дизъюнкции <math>(\neg x_i)</math>. Если после редукции, неразрешимость формулы еще не проявилась, у нас остается формула, состоящая из дизъюнктов включающих ровно два терма.
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| − | Теперь заметим, что формула <math>x \lor y</math> эквивалентна формуле <math>(\neg x \rightarrow y) \land (\neg y \rightarrow x)</math>. Последней формуле, легко придать интерпретацию на графе: для [[2SAT]]-формулы, содержащей ''n'' переменных <math>x_i</math>, сопоставим ориентированный граф из ''2n'' узлов: <math>\forall i\ x_i,\ \neg x_i</math>, а для каждой дизъюнкции <math>(x \lor y)</math> он будет содержать два ребра <math>(\neg x \rightarrow y)</math> и <math>(\neg y \rightarrow x)</math>. В разрешимой формуле, истинность терма <math>x^{\sigma_i}_i</math> означает истинность всех термов <math>x^{\sigma_j}_j</math> достижимых (в смысле путей в ориентированном графе) в графе из узла, соответствующему терму <math>x^{\sigma_i}_i</math>.
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| − | Обозначим через <math>x \Rightarrow y</math> существование пути из узла ''x'' в узел ''y''. Тогда
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| − | если для некоторого <math>x_i</math> будет существовать пути
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| − | <math>x_i \Rightarrow \neg x_i</math> и <math>\neg x_i \Rightarrow x_i</math>, то формула будет неразрешима. Действительно, при <math>x_i=1</math>, "нарушается" первый путь, а при <math>x_i=0</math>, «нарушается» второй путь.
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| − | В противном случае, покажем, как сделать выполняющее присваивание.
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| − | Для каждой переменной ''x'', если есть путь <math>x \Rightarrow \neg x</math>, то присваиваем ей «0», в противном случае «1».
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| − | Поиск путей в графе выполняется за полиномиальное время, таким образом, задача полиномиально разрешима.
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| − | ==Представление 2SAT на графе==
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Текущая версия на 20:46, 13 июня 2011
- REDIRECT discopal:2SAT:Решение
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