112 research outputs found
Approximate kernel clustering
In the kernel clustering problem we are given a large positive
semi-definite matrix with and a small
positive semi-definite matrix . The goal is to find a
partition of which maximizes the quantity We study the
computational complexity of this generic clustering problem which originates in
the theory of machine learning. We design a constant factor polynomial time
approximation algorithm for this problem, answering a question posed by Song,
Smola, Gretton and Borgwardt. In some cases we manage to compute the sharp
approximation threshold for this problem assuming the Unique Games Conjecture
(UGC). In particular, when is the identity matrix the UGC
hardness threshold of this problem is exactly . We present
and study a geometric conjecture of independent interest which we show would
imply that the UGC threshold when is the identity matrix is
for every
A Characterization of Approximation Resistance for Even -Partite CSPs
A constraint satisfaction problem (CSP) is said to be \emph{approximation
resistant} if it is hard to approximate better than the trivial algorithm which
picks a uniformly random assignment. Assuming the Unique Games Conjecture, we
give a characterization of approximation resistance for -partite CSPs
defined by an even predicate
Hardness of Bipartite Expansion
We study the natural problem of estimating the expansion of subsets of vertices on one side of a bipartite graph. More precisely, given a bipartite graph G(U,V,E) and a parameter beta, the goal is to find a subset V\u27 subseteq V containing beta fraction of the vertices of V which minimizes the size of N(V\u27), the neighborhood of V\u27. This problem, which we call Bipartite Expansion, is a special case of submodular minimization subject to a cardinality constraint, and is also related to other problems in graph partitioning and expansion. Previous to this work, there was no hardness of approximation known for Bipartite Expansion.
In this paper we show the following strong inapproximability for Bipartite Expansion: for any constants tau, gamma > 0
there is no algorithm which, given a constant beta > 0 and a bipartite graph G(U,V,E), runs in polynomial time and decides whether
- (YES case) There is a subset S^* subseteq V s.t. |S^*| >= beta*|V| satisfying |N(S^*)| <= gamma |U|, or
- (NO case) Any subset S subseteq V s.t. |S| >= tau*beta*|V| satisfies |N(S)| >= (1 - gamma)|U|, unless
NP subseteq intersect_{epsilon > 0}{DTIME}(2^{n^epsi;on}) i.e. NP has subexponential time algorithms.
We note that our hardness result stated above is a vertex expansion analogue of the Small Set (Edge) Expansion Conjecture of
Raghavendra and Steurer 2010
A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover
Given a -uniform hyper-graph, the E-Vertex-Cover problem is to find the
smallest subset of vertices that intersects every hyper-edge. We present a new
multilayered PCP construction that extends the Raz verifier. This enables us to
prove that E-Vertex-Cover is NP-hard to approximate within factor
for any and any . The result is
essentially tight as this problem can be easily approximated within factor .
Our construction makes use of the biased Long-Code and is analyzed using
combinatorial properties of -wise -intersecting families of subsets
On the hardness of learning intersections of two halfspaces
AbstractWe show that unless NP=RP, it is hard to (even) weakly PAC-learn intersection of two halfspaces in Rn using a hypothesis which is a function of up to β halfspaces (linear threshold functions) for any integer β. Specifically, we show that for every integer β and an arbitrarily small constant Ξ΅>0, unless NP=RP, no polynomial time algorithm can distinguish whether there is an intersection of two halfspaces that correctly classifies a given set of labeled points in Rn, or whether any function of β halfspaces can correctly classify at most 12+Ξ΅ fraction of the points
- β¦