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2SAT:Решение — различия между версиями
Материал из CustisWiki
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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 <m>x_i</m>, то для выполнимости формулы необходимо <m>x_i=1</m>, соответственно мы фиксируем <m>x_i \equiv 1</m> 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 <m>(\neg x_i)</m>. 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 <m>x \lor y</m> эквивалентна формуле <m>(\neg x \rightarrow y) \land (\neg y \rightarrow x)</m>. Последней формуле, легко придать интерпретацию на графе: для [[2SAT]]-формулы, содержащей ''n'' переменных <m>x_i</m>, сопоставим ориентированный граф из ''2n'' узлов: <m>\forall i\ x_i,\ \neg x_i</m>, а для каждой дизъюнкции <m>(x \lor y)</m> он будет содержать два ребра <m>(\neg x \rightarrow y)</m> и <m>(\neg y \rightarrow x)</m>. В разрешимой формуле, истинность терма <m>x^{\sigma_i}_i</m> означает истинность всех термов <m>x^{\sigma_j}_j</m> достижимых (в смысле путей в ориентированном графе) в графе из узла, соответствующему терму <m>x^{\sigma_i}_i</m>.
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| − | Обозначим через <m>x \Rightarrow y</m> существование пути из узла ''x'' в узел ''y''. Тогда
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| − | если для некоторого <m>x_i</m> будет существовать пути
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| − | <m>x_i \Rightarrow \neg x_i</m> и <m>\neg x_i \Rightarrow x_i</m>, то формула будет неразрешима. Действительно, при <m>x_i=1</m>, "нарушается" первый путь, а при <m>x_i=0</m>, «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 <m>x \Rightarrow \neg x</m>, 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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