2SAT:Решение

Материал из свободной русской энциклопедии «Традиция»
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Идея[править]

Рассмотрим входную 2SAT-формулу. 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.

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>.

Обозначим через <m>x \Rightarrow y</m> существование пути из узла x в узел y. Тогда если для некоторого <m>x_i</m> будет существовать пути <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.

В противном случае, покажем, как сделать выполняющее присваивание. Для каждой переменной x, если есть путь <m>x \Rightarrow \neg x</m>, то присваиваем ей «0», в противном случае «1».

Поиск путей в графе выполняется за полиномиальное время, таким образом, задача полиномиально разрешима.

Представление 2SAT на графе[править]

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