AM with Multiple Merlins

We introduce and study a new model of interactive proofs: AM(k), or Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known MIP, here the assumption is that each Merlin receives an independent random challenge from Arthur. One motivation for this model (which we explore in detail) comes from the close analogies between it and the quantum complexity class QMA(k), but the AM(k) model is also natural in its own right. We illustrate the power of multiple Merlins by giving an AM(2) protocol for 3SAT, in which the Merlins' challenges and responses consist of only n^{1/2+o(1)} bits each. Our protocol has the consequence that, assuming the Exponential Time Hypothesis (ETH), any algorithm for approximating a dense CSP with a polynomial-size alphabet must take n^{(log n)^{1-o(1)}} time. Algorithms nearly matching this lower bound are known, but their running times had never been previously explained. Brandao and Harrow have also recently used our 3SAT protocol to show quasipolynomial hardness for approximating the values of certain entangled games. In the other direction, we give a simple quasipolynomial-time approximation algorithm for free games, and use it to prove that, assuming the ETH, our 3SAT protocol is essentially optimal. More generally, we show that multiple Merlins never provide more than a polynomial advantage over one: that is, AM(k)=AM for all k=poly(n). The key to this result is a subsampling theorem for free games, which follows from powerful results by Alon et al. and Barak et al. on subsampling dense CSPs, and which says that the value of any free game can be closely approximated by the value of a logarithmic-sized random subgame.Comment: 48 page

Varying the Abundance of O Antigen in \u3cem\u3eRhizobium etli\u3c/em\u3e and Its Effect on Symbiosis with \u3cem\u3ePhaseolus vulgaris\u3c/em\u3e

Judged by migration of its lipopolysaccharide (LPS) in gel electrophoresis, the O antigen of Rhizobium etli mutant strain CE166 was apparently of normal size. However, its LPS sugar composition and staining of the LPS bands after electrophoresis indicated that the proportion of its LPS molecules that possessed O antigen was only 40% of the wild-type value. Its LPS also differed from the wild type by lacking quinovosamine (2-amino-2,6-dideoxyglucose). Both of these defects were due to a single genetic locus carrying a Tn5 insertion. The deficiency in O-antigen amount, but not the absence of quinovosamine, was suppressed by transferring into this strain recombinant plasmids that shared a 7.8-kb stretch of the R. etli CE3 lps genetic region α, even though this suppressing DNA did not carry the genetic region mutated in strain CE166. Strain CE166 gave rise to pseudonodules on legume host Phaseolus vulgaris, whereas the mutant suppressed by DNA from lps region α elicited nitrogen-fixing nodules. However, the nodules in the latter case developed slowly and were widely dispersed. Two other R. etli mutants that had one-half or less of the normal amount of O antigen also gave rise to pseudonodules on P. vulgaris. The latter strains were mutated in lps region α and could be restored to normal LPS content and normal symbiosis by complementation with wild-type DNA from this region. Hence, the symbiotic role of LPS requires near-normal abundance of O antigen and may require a structural feature conferred by quinovosamin