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Are lower bounds hard to prove? Michal Koucký Institute of Mathematics, Prague

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Computational complexity P = NP ? (for a record) I don’t believe P=NP. I don’t believe P NP. → Too early to tell.

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Common wisdom “P differs from NP because we are so good at proving upper bounds but so bad in proving lower bounds.” “Lower bounds are so much harder to prove than designing an algorithm.”

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STOC 2011 Lower bounds 9 Tight bounds15 Upper bounds50 Unclassified 7

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You get what you pay for ACM 100 000members* SIGACT 1 600members + lower bound people 100 → We should see 100 - 1000x more upper bounds than lower bounds. Source: * Barbara Ryder, + Lance Fortnow.

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Complexity landscape AC 0 ACC 0 TC 0 L NL NC P NP PH PSPACE EXP NEXP P L or P PSPACE → There are lower bounds we know are true but we cannot prove. Efficiently parallelizable

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Polynomial identity testing can be done deterministically in polynomial time, or E has sub-exponential size circuits. → There are upper bounds we know are true but we cannot prove.

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Algorithms are easy Deterministic test of primality [Agrawal et al.’02] Log-space algorithm for undirected s-t- connectivity [Reingold’03] Graph isomorphism ???

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Our intuition never fails Nondeterministic space is closed under complement [Immerman-Szelepsenyi’88] Evaluating arithmetic formula using 3 registers [Barrington’86, Ben-Or-Cleve’88] Linear Programming [Khachiyan’79]

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Our algorithmic horizon Most advanced algorithmic techniques: Semidefinite Programming[Lovász] Spectral methods … ‘‘All’’ our polynomial time algorithms have running time O(n 10 )

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Time hierarchy n 10 n 100 n 1000 n 10000

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Lower bounds via upper bounds SAT TimeSpace( n 1.58, n δ ) [Fortnow’00, …] Idea: 1. Assume that SAT is efficiently solvable then some harder problem is efficiently solvable as well. 2. Iterate, until you get a contradiction with a known lower bound (time hierarchy).

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Lower bounds via upper bounds Lower bound amplification [K.-Allender’08] Idea: For certain problems (downwards self- reducible) if they are solvable by circuits of size n k then they are solvable by circuits of size n 1+ε

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Lower bounds via upper bounds NEXP ACC 0 [Williams’11] Idea [Williams’10] : If Circuit-SAT can be solved in time 2 n /n ω(1) then NEXP P/poly.

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Hardness vs Randomness [Impagliazzo-Wigderson’98, …] Pseudorandom generators exist E requires circuits of size 2 δn.

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Hardness vs Randomness [Impagliazzo et al. ‘01] NP=MA NEXP P/poly.

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Conclusions Difficult problems are hard to resolve be it upper bound or lower bound. Until we have an optimal bound it is hard to predict which way it will go.

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