if p(x) denotes mandatorya polynomial of degree n such that P(k)=1/k for k=1,2,....,n+1,determine p(n+2).

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Detecting High Log-Densities – anO(n 1/4) Approximation for Densestk-Subgraph Aditya Bhaskara ? Moses Charikar ? Eden Chlamtac ? Uriel Feige § Aravindan Vijayaraghavan ? Abstract In the Densestk-Subgraph problem, given a graphGand a parameterk,
one needs to ?nd a subgraph ofGinduced onkvertices that contains the largest number of edges. There is a signi?cant gap between the best known upper and lower bounds for this problem. It is NP-hard, and does not have a PTAS unless NP has subexponential time algorithms. On the other hand, the current best known algorithm of Feige, Kortsarz and Peleg [FKP01], gives an approximation ratio ofn 1/3?εfor some speci?cε&0 (estimated by those authors at aroundε= 1/60). We present an algorithm that for everyε&0 approximates the Densestk-Subgraph problem within a ratio ofn 1/4+εin timen O(1/ε).
If allowed to run for timen O(logn),
our algorithm achieves an approximation ratio ofO(n 1/4). Our algorithm is inspired by studying an average- case version of the problem where the goal is to distinguish random graphs from random graphs with planted dense subgraphs – the approximation ratio we achieve for the general case matches the “distinguishing ratio” we obtain for this planted problem. Achieving a distinguishing ratio ofo(n 1/4)
for the planted problem (in polynomial time) is beyond the reach of our current techniques. At a high level, our algorithms involve cleverly counting appropriately de?ned trees of con- stant size inG,
and using these counts to identify the vertices of the dense subgraph. Our algorithm is based on the following principle. We say that a graphG(V, E) has log-densityαif its average degree is Θ(|V| α). The algorithmic core of our result is a family of algorithms that outputk-subgraphs of nontrivial density whenever the log-density of the densestk-subgraph is larger than the log-density of the host graph. Finally, we extend this algorithm to obtain anO(n 1/4?ε)-approximation algorithm which runs in timeO(2 n )
and also explore various approaches to obtain better approximation algorithms in restricted parameter settings for random instances. ?Department puter Science, Princeton University, supported by NSF awards MSPA-MCS 0528414, CCF 0832797, and AF 0916218. Email:bhaskara@cs.princeton.edu ?Department puter Science, Princeton University, supported by NSF awards MSPA-MCS 0528414, CCF 0832797, and AF 0916218. Email:moses@cs.princeton.edu ?Weizmann Institute of Science, Rehovot, Israel, supported by a Sir Charles Clore Postdoctoral Fellowship. Email: eden.chlamtac@weizmann.ac.il §Weizmann Institute of Science, Rehovot, Israel. Email:uriel.feige@weizmann.ac.il.
The author holds the Lawrence G. Horowitz Professorial Chair at the Weizmann Institute. Work supported in part by The Israel Science Foundation (grant No. 873/08). ?Department puter Science, Princeton University, supported by NSF awards MSPA-MCS 0528414, CCF 0832797, and AF1
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A Dual Polynomial for OR
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&&We reprove that the approximate degree of the OR function on n bits is Omega(sqrt(n)). We consider a linear program which is feasible if and only if there is an approximate polynomial for a given function, and apply the duality theory. The duality theory s
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