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.
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
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.
Algorithms are easy Deterministic test of primality [Agrawal et al.’02] Log-space algorithm for undirected s-t- connectivity [Reingold’03] Graph isomorphism ???
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]
Our algorithmic horizon Most advanced algorithmic techniques: Semidefinite Programming[Lovász] Spectral methods … ‘‘All’’ our polynomial time algorithms have running time O(n 10 )
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).
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
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.
Hardness vs Randomness [Impagliazzo-Wigderson’98, …] Pseudorandom generators exist E requires circuits of size 2 δn.
Hardness vs Randomness [Impagliazzo et al. ‘01] NP=MA NEXP P/poly.
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.