too long: finish in polynomial time Transformation must not expand input size too much: polynomial in original input size There is a polynomial-time reduction from every problem A Î NP to B. B NP Invalid transformation: “do an exponential search to find the answer, and output that”

Y. We already know B does not have property Y. Proof by Contradiction: Assume ! has some property ". Since ! has ", there is a solution $ to ! that satisfies ". Show how $ could be used to solve %.

Y. We already know B does not have property Y. Proof by Contradiction: Assume ! has some property ". Since ! has ", there is a solution $ to ! that satisfies ". Show how $ could be used to solve %. Since we know B does not satisfy Y, but having S would imply B satisfies Y, S cannot exist. Thus, S cannot exist, and A does not have Y.

property Y. We already know B does not have property Y. Proof by Contradiction: Assume ! has some property ". Since ! has ", there is a solution $ to ! that satisfies ". Show how $ could be used to solve %. NP-Hardness: " = “is NP-Hard”, % = a known NP-Hard problem, $ = “a TM that solves ! in polynomial-time”

problem A Î NP to B. B NP Show there is a polynomial- time reduction from one problem X Î NP-Hard to B. X NP B This assumes we already know some problem X that is in NP-Hard. To get the first one, we need to prove it the hard way!

Prove A does not have some property Y. We already know B does not have property Y. Proof by Contradiction: Assume ! has some property ". Since ! has ", there is a solution $ to ! that satisfies ". Show how $ could be used to solve %. NP-Hardness: " = “is NP-Hard”, % = a known NP-Hard problem, $ = “a TM that solves ! in polynomial-time”

Prove Multi-Unit Auction with Single-Minded Bids is NP-Hard. We already know KNAPSACK is NP-Hard. Proof by Contradiction: Assume !"# is not NP-Hard. Since !"# is not NP-Hard, there is a polynomial-time solution % to !"#. Show how % could be used to solve &'#(%#)&. NP-Hardness: * = “is NP-Hard”, + = KNAPSACK, % = “a TM that solves !"# in polynomial-time”

profoundly different place than we usually assume it to be. There would be no special value in ‘creative leaps’, no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss...” Scott Aaronson Charge Project 4: Final auction next Thursday Project 4 reports Due next Friday