Weak designs were defined by Raz, Reingold and Vadhan (1999) and are
used in constructions of extractors. Roughly speaking, a weak design
is a collection of subsets satisfying some near-disjointness
properties. Constructions of weak designs with certain parameters are
given in [RRV99]. These constructions are explicit in the sense that
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We design a $0.795$ approximation algorithm for the Max-Bisection problem
restricted to regular graphs. In the case of three regular graphs our
results imply an approximation ratio of $0.834$.

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We show that the k-CSP problem over a finite Abelian group G
cannot be approximated within |G|^{k-O(sqrt{k})}-epsilon, for
any constant epsilon>0, unless P=NP. This lower bound matches
well with the best known upper bound, |G|^{k-1}, of Serna,
Trevisan and Xhafa. The proof uses a combination of PCP
techniques---most notably a ... more >>>