NP = PCP(log n, 1) and related results crucially depend upon
the close connection between the probability with which a
function passes a ``low degree test'' and the distance of
this function to the nearest degree d polynomial. In this
paper we study a test ... more >>>

We study dense instances of several covering problems. An instance of
the set cover problem with $m$ sets is dense if there is $\epsilon>0$
such that any element belongs to at least $\epsilon m$ sets. We show
that the dense set cover problem can be approximated with ... more >>>

We study small degree graph problems such as Maximum Independent Set
and Minimum Node Cover and improve approximation lower bounds for
them and for a number of related problems, like Max-B-Set Packing,
Min-B-Set Cover, Max-Matching in B-uniform 2-regular hypergraphs.
For example, we prove NP-hardness factor of 95/94
more >>>

In the set cover problem we are given a collection of $m$ sets whose union covers $[n] = \{1,\ldots,n\}$ and must find a minimum-sized subcollection whose union still covers $[n]$. We investigate the approximability of set cover by an approximation ratio that depends only on $m$ and observe that, for ... more >>>

Given a finite set of straight line segments $S$ in $R^{2}$ and some $k\in N$, is there a subset $V$ of points on segments in $S$ with $\vert V \vert \leq k$ such that each segment of $S$ contains at least one point in $V$? This is a special case ... more >>>

In this paper we put forward a conjecture: an instantiation of the Sliding Scale Conjecture of Bellare, Goldwasser, Lund and Russell to projection games. We refer to this conjecture as the Projection Games Conjecture.

We further suggest the research agenda of establishing new hardness of approximation results based on the ... more >>>

Set cover, over a universe of size $n$, may be modelled as a
data-streaming problem, where the $m$ sets that comprise the instance
are to be read one by one. A semi-streaming algorithm is allowed only
$O(n \text{ poly}\{\log n, \log m\})$ space to process this ... more >>>

A famous conjecture of Tuza states that the minimum number of edges needed to cover all the triangles in a graph is at most twice the maximum number of edge-disjoint triangles. This conjecture was couched in a broader setting by Aharoni and Zerbib who proposed a hypergraph version of this ... more >>>

k-median and k-means are the two most popular objectives for clustering algorithms. Despite intensive effort, a good understanding of the approximability of these objectives, particularly in $\ell_p$-metrics, remains a major open problem. In this paper, we significantly improve upon the hardness of approximation factors known in literature for these objectives ... more >>>