Arjun Suresh (talk | contribs) |
Arjun Suresh (talk | contribs) |
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$P \subseteq NP \subseteq NPC \subseteq NPH$ | $P \subseteq NP \subseteq NPC \subseteq NPH$ | ||
| − | Consider problems $A$ and $ | + | Consider problems $A$, $B$ and $C$ |
*If <math>A</math> is reduced to <math>B</math> and <math>B</math> $\in$ P, then <math>A</math> $\in$ <math>P</math> | *If <math>A</math> is reduced to <math>B</math> and <math>B</math> $\in$ P, then <math>A</math> $\in$ <math>P</math> | ||
| − | *If <math>A</math> is reduced to <math>B</math> and <math>B</math> $\in$ NP, then <math>A</math> $\in$ <math>NP</math> (<math> | + | *If <math>A</math> is reduced to <math>B</math> and <math>B</math> $\in$ NP, then <math>A</math> $\in$ <math>NP</math> (<math>A</math> may also be in <math>P</math>, but that cannot be inferred from the given statement) |
| − | *If <math>A</math> is reduced to <math>B</math> and <math>B</math> $\in$ NPC, then <math>A</math> $\in$ <math> | + | *If <math>A</math> is reduced to <math>B</math> and <math>B</math> $\in$ NPC, then <math>A</math> $\in$ <math>NP</math> (<math>A</math> may also be in <math>NPC</math>, but that cannot be inferred from the given statement) |
| − | *If <math>A</math> is reduced to <math>B</math> and <math>B</math> $\in$ NPH, then <math>A</math> $\in$ <math> | + | *If <math>A</math> is reduced to <math>B</math> and <math>C</math> is reduced to <math>A</math> and <math>B, C</math> $\in$ NPC, then <math>A</math> $\in$ <math>NPC</math> |
| + | *If <math>A</math> is reduced to <math>B</math> and <math>B</math> $\in$ NPH, then <math>A</math> $\in$ <math>?</math> | ||
| + | Here we can't say anything about A. It can be as hard as NPH, or as simple as P | ||
*If problem <math>A</math> is reduced to a problem <math>B</math> and <math>B</math> $\in$ P, then <math>A</math> $\in$ <math>P</math> | *If problem <math>A</math> is reduced to a problem <math>B</math> and <math>B</math> $\in$ P, then <math>A</math> $\in$ <math>P</math> | ||
*If problem <math>A</math> is reduced to a problem <math>B</math> and <math>B</math> $\in$ P, then <math>A</math> $\in$ <math>P</math> | *If problem <math>A</math> is reduced to a problem <math>B</math> and <math>B</math> $\in$ P, then <math>A</math> $\in$ <math>P</math> | ||
*If problem <math>A</math> is reduced to a problem <math>B</math> and <math>B</math> $\in$ P, then <math>A</math> $\in$ <math>P</math> | *If problem <math>A</math> is reduced to a problem <math>B</math> and <math>B</math> $\in$ P, then <math>A</math> $\in$ <math>P</math> | ||
*If problem <math>A</math> is reduced to a problem <math>B</math> and <math>B</math> $\in$ P, then <math>A</math> $\in$ <math>P</math> | *If problem <math>A</math> is reduced to a problem <math>B</math> and <math>B</math> $\in$ P, then <math>A</math> $\in$ <math>P</math> | ||
Assume all reductions are done in polynomial time
$P \subseteq NP \subseteq NPC \subseteq NPH$
Consider problems $A$, $B$ and $C$
Here we can't say anything about A. It can be as hard as NPH, or as simple as P
Assume all reductions are done in polynomial time
$P \subseteq NP \subseteq NPC \subseteq NPH$
Consider problems $A$ and $B$
GATECSE