摘自http://www.cnblogs.com/kuangbin/archive/2012/10/05/2712429.html?
現(xiàn)有一個(gè)由N個(gè)布爾值組成的序列A，給出一些限制關(guān)系，比如A[x] && A[y] = 0、A[x] || A[y] || A[z]=1等，要確定A[0..N-1]的值，使得其滿足所有限制關(guān)系。這個(gè)稱為SAT問題，特別的，若每種限制關(guān)系中最多只對(duì)兩個(gè)元素進(jìn)行限制，則稱為2-SAT問題。

由于在2-SAT問題中，最多只對(duì)兩個(gè)元素進(jìn)行限制，所以可能的限制關(guān)系共有11種：?
A[x]?
NOT A[x]?
A[x] AND A[y]?
A[x] AND NOT A[y]?
A[x] OR A[y]?
A[x] OR NOT A[y]?
NOT (A[x] AND A[y])?
NOT (A[x] OR A[y])?
A[x] XOR A[y]?
NOT (A[x] XOR A[y])?
A[x] XOR NOT A[y]?
進(jìn)一步，A[x] AND A[y]相當(dāng)于(A[x]) AND (A[y])（也就是可以拆分成A[x]與A[y]兩個(gè)限制關(guān)系），NOT(A[x] OR A[y])相當(dāng)于NOT A[x] AND NOT A[y]（也就是可以拆分成NOT A[x]與NOT A[y]兩個(gè)限制關(guān)系）。因此，可能的限制關(guān)系最多只有9種。

2-SAT問題在大多數(shù)時(shí)候表現(xiàn)成以下形式：有N對(duì)物品，每對(duì)物品中必須選取一個(gè)，也只能選取一個(gè)，并且它們之間存在某些限制關(guān)系（如某兩個(gè)物品不能都選，某兩個(gè)物品不能都不選，某兩個(gè)物品必須且只能選一個(gè)，某個(gè)物品必選）等，這時(shí)，可以將每對(duì)物品當(dāng)成一個(gè)布爾值（選取第一個(gè)物品相當(dāng)于0，選取第二個(gè)相當(dāng)于1），如果所有的限制關(guān)系最多只對(duì)兩個(gè)物品進(jìn)行限制，則它們都可以轉(zhuǎn)化成9種基本限制關(guān)系，從而轉(zhuǎn)化為2-SAT模型。

【建模】?
可以構(gòu)造有向圖G，G中包含2*N個(gè)頂點(diǎn)，前N個(gè)頂點(diǎn)(1~N)表示第i個(gè)元素能被選擇，后N個(gè)頂點(diǎn)(N+1~2*N)表示第i個(gè)元素不能被選擇。Ai和A(i+N)不能同時(shí)被選擇。同理Ai + Bj = ~( A(i+N) + B(j+N) )，A(i+N)和B(j+N)不能同時(shí)被選。選中~A，必須選擇B；如果選擇~B，必須選擇A。?
若圖中i到j(luò)有路徑，則若i選，則j也要選；或者說，若j不選，則i也不能選。

<code class="hljs cpp has-numbering" style="display: block; padding: 0px; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include<iostream></span> <span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include<algorithm></span> <span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include<cstdio></span> <span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include<cstring></span> <span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include<cmath></span> <span class="hljs-preprocessor" style="color: rgb(68, 68, 68); box-sizing: border-box;">#include<queue></span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">using</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">namespace</span> <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">std</span>; <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">const</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> MAXN = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2200</span>; <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">const</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> MAXM = MAXN*MAXN;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">struct</span> EdgeNode {<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> to;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> next; }Edges[MAXM];<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> Head[MAXN]; <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> dfn[MAXN],low[MAXN],belong[MAXN],Stack[MAXN],vis[MAXN]; <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> m,id,lay,scc,N,M; <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//belong[]來(lái)判斷i和i+N是否在一個(gè)強(qiáng)連通分量里</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">void</span> AddEdges(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> u,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> v) {Edges[id].to = v;Edges[id].next = Head[u];Head[u] = id++; }<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">void</span> TarBFS(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> pos) {dfn[pos] = low[pos] = ++lay;Stack[m++] = pos;vis[pos] = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = Head[pos]; i != -<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; i = Edges[i].next){<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> v = Edges[i].to;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>( !dfn[v] ){TarBFS(v);low[pos] = min(low[pos],low[v]);}<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">else</span> <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(vis[v])low[pos] = min(low[pos],low[v]);}<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> v;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(dfn[pos] == low[pos]){++scc;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">do</span>{v = Stack[--m];belong[v] = scc;vis[v] = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;}<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">while</span>(v != pos);} }<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> main() {<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> u,v;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">while</span>(~<span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">scanf</span>(<span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"%d%d"</span>,&N,&M)){id = m = scc = lay = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;<span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">memset</span>(Head,-<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">sizeof</span>(Head));<span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">memset</span>(vis,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">sizeof</span>(vis));<span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">memset</span>(low,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">sizeof</span>(low));<span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">memset</span>(dfn,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">sizeof</span>(dfn));<span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">memset</span>(belong,<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>,<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">sizeof</span>(belong));<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; i < M; ++i){<span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">scanf</span>(<span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"%d%d"</span>,&u,&v);<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> a = <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">abs</span>(u);<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> b = <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">abs</span>(v);<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(u > <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> && v > <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>){ <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//a、b至少一個(gè)被選中</span>AddEdges(a+N,b); <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果a不被選，b就必須被選</span>AddEdges(b+N,a); <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果b不被選，a就必須被選</span>}<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(u < <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> && v < <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>){ <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//a、b至少有一個(gè)不被選中</span>AddEdges(a,b+N); <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果a被選，b就必須不被選</span>AddEdges(b,a+N); <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果b被選，a就必須不被選</span>}<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(u > <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> && v < <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>){ <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//a被選中和b不被選中兩件事至少發(fā)生一件</span>AddEdges(a+N,b+N); <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果a不被選中，b必須不被選中</span>AddEdges(b,a); <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果b被選中，那么a必須被選中</span>}<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(u < <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span> && v > <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>){ <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//a不被選中和b被選中至少發(fā)生一件</span>AddEdges(a,b); <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果a被選中，b必須被選中</span>AddEdges(b+N,a+N); <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果b不被選中，a必須不被選中</span>}}<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; i <= <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>*N; ++i)<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>( !dfn[i] )TarBFS(i);<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> ans = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">1</span>; i <= N; ++i){<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(belong[i] == belong[i+N]) <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//如果i和i+N同在一個(gè)連通分量里，則2-SAT不滿足</span>{ans = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">break</span>;}}<span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">printf</span>(<span class="hljs-string" style="color: rgb(0, 136, 0); box-sizing: border-box;">"%d\n"</span>,ans); <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">//不存在就滿足2-SAT</span>}<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span> <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>; } </code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; 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padding: 0px 5px;">56</li><li style="box-sizing: border-box; padding: 0px 5px;">57</li><li style="box-sizing: border-box; padding: 0px 5px;">58</li><li style="box-sizing: border-box; padding: 0px 5px;">59</li><li style="box-sizing: border-box; padding: 0px 5px;">60</li><li style="box-sizing: border-box; padding: 0px 5px;">61</li><li style="box-sizing: border-box; padding: 0px 5px;">62</li><li style="box-sizing: border-box; padding: 0px 5px;">63</li><li style="box-sizing: border-box; padding: 0px 5px;">64</li><li style="box-sizing: border-box; padding: 0px 5px;">65</li><li style="box-sizing: border-box; padding: 0px 5px;">66</li><li style="box-sizing: border-box; padding: 0px 5px;">67</li><li style="box-sizing: border-box; padding: 0px 5px;">68</li><li style="box-sizing: border-box; padding: 0px 5px;">69</li><li style="box-sizing: border-box; padding: 0px 5px;">70</li><li style="box-sizing: border-box; padding: 0px 5px;">71</li><li style="box-sizing: border-box; padding: 0px 5px;">72</li><li style="box-sizing: border-box; padding: 0px 5px;">73</li><li style="box-sizing: border-box; padding: 0px 5px;">74</li><li style="box-sizing: border-box; padding: 0px 5px;">75</li><li style="box-sizing: border-box; padding: 0px 5px;">76</li><li style="box-sizing: border-box; padding: 0px 5px;">77</li><li style="box-sizing: border-box; padding: 0px 5px;">78</li><li style="box-sizing: border-box; padding: 0px 5px;">79</li><li style="box-sizing: border-box; padding: 0px 5px;">80</li><li style="box-sizing: border-box; padding: 0px 5px;">81</li><li style="box-sizing: border-box; padding: 0px 5px;">82</li><li style="box-sizing: border-box; padding: 0px 5px;">83</li><li style="box-sizing: border-box; padding: 0px 5px;">84</li><li style="box-sizing: border-box; padding: 0px 5px;">85</li><li style="box-sizing: border-box; padding: 0px 5px;">86</li><li style="box-sizing: border-box; padding: 0px 5px;">87</li><li style="box-sizing: border-box; padding: 0px 5px;">88</li><li style="box-sizing: border-box; padding: 0px 5px;">89</li><li style="box-sizing: border-box; padding: 0px 5px;">90</li><li style="box-sizing: border-box; padding: 0px 5px;">91</li><li style="box-sizing: border-box; padding: 0px 5px;">92</li><li style="box-sizing: border-box; padding: 0px 5px;">93</li><li style="box-sizing: border-box; padding: 0px 5px;">94</li><li style="box-sizing: border-box; padding: 0px 5px;">95</li><li style="box-sizing: border-box; padding: 0px 5px;">96</li><li style="box-sizing: border-box; padding: 0px 5px;">97</li><li style="box-sizing: border-box; padding: 0px 5px;">98</li><li style="box-sizing: border-box; padding: 0px 5px;">99</li><li style="box-sizing: border-box; padding: 0px 5px;">100</li><li style="box-sizing: border-box; padding: 0px 5px;">101</li><li style="box-sizing: border-box; padding: 0px 5px;">102</li><li style="box-sizing: border-box; padding: 0px 5px;">103</li><li style="box-sizing: border-box; padding: 0px 5px;">104</li><li style="box-sizing: border-box; padding: 0px 5px;">105</li><li style="box-sizing: border-box; padding: 0px 5px;">106</li><li style="box-sizing: border-box; padding: 0px 5px;">107</li><li style="box-sizing: border-box; padding: 0px 5px;">108</li><li style="box-sizing: border-box; padding: 0px 5px;">109</li><li style="box-sizing: border-box; padding: 0px 5px;">110</li><li style="box-sizing: border-box; padding: 0px 5px;">111</li><li style="box-sizing: border-box; padding: 0px 5px;">112</li><li style="box-sizing: border-box; padding: 0px 5px;">113</li><li style="box-sizing: border-box; padding: 0px 5px;">114</li><li style="box-sizing: border-box; padding: 0px 5px;">115</li><li style="box-sizing: border-box; padding: 0px 5px;">116</li><li style="box-sizing: border-box; padding: 0px 5px;">117</li><li style="box-sizing: border-box; padding: 0px 5px;">118</li></ul>

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