3,132 research outputs found
Approximating Hereditary Discrepancy via Small Width Ellipsoids
The Discrepancy of a hypergraph is the minimum attainable value, over
two-colorings of its vertices, of the maximum absolute imbalance of any
hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum
discrepancy of a restriction of the hypergraph to a subset of its vertices, is
a measure of its complexity. Lovasz, Spencer and Vesztergombi (1986) related
the natural extension of this quantity to matrices to rounding algorithms for
linear programs, and gave a determinant based lower bound on the hereditary
discrepancy. Matousek (2011) showed that this bound is tight up to a
polylogarithmic factor, leaving open the question of actually computing this
bound. Recent work by Nikolov, Talwar and Zhang (2013) showed a polynomial time
-approximation to hereditary discrepancy, as a by-product
of their work in differential privacy. In this paper, we give a direct simple
-approximation algorithm for this problem. We show that up to
this approximation factor, the hereditary discrepancy of a matrix is
characterized by the optimal value of simple geometric convex program that
seeks to minimize the largest norm of any point in a ellipsoid
containing the columns of . This characterization promises to be a useful
tool in discrepancy theory
On Generalizations of Network Design Problems with Degree Bounds
Iterative rounding and relaxation have arguably become the method of choice
in dealing with unconstrained and constrained network design problems. In this
paper we extend the scope of the iterative relaxation method in two directions:
(1) by handling more complex degree constraints in the minimum spanning tree
problem (namely, laminar crossing spanning tree), and (2) by incorporating
`degree bounds' in other combinatorial optimization problems such as matroid
intersection and lattice polyhedra. We give new or improved approximation
algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure
On Computing Centroids According to the p-Norms of Hamming Distance Vectors
In this paper we consider the p-Norm Hamming Centroid problem which asks to determine whether some given strings have a centroid with a bound on the p-norm of its Hamming distances to the strings. Specifically, given a set S of strings and a real k, we consider the problem of determining whether there exists a string s^* with (sum_{s in S} d^{p}(s^*,s))^(1/p) <=k, where d(,) denotes the Hamming distance metric. This problem has important applications in data clustering and multi-winner committee elections, and is a generalization of the well-known polynomial-time solvable Consensus String (p=1) problem, as well as the NP-hard Closest String (p=infty) problem.
Our main result shows that the problem is NP-hard for all fixed rational p > 1, closing the gap for all rational values of p between 1 and infty. Under standard complexity assumptions the reduction also implies that the problem has no 2^o(n+m)-time or 2^o(k^(p/(p+1)))-time algorithm, where m denotes the number of input strings and n denotes the length of each string, for any fixed p > 1. The first bound matches a straightforward brute-force algorithm. The second bound is tight in the sense that for each fixed epsilon > 0, we provide a 2^(k^(p/((p+1))+epsilon))-time algorithm. In the last part of the paper, we complement our hardness result by presenting a fixed-parameter algorithm and a factor-2 approximation algorithm for the problem
Scheduling over Scenarios on Two Machines
We consider scheduling problems over scenarios where the goal is to find a
single assignment of the jobs to the machines which performs well over all
possible scenarios. Each scenario is a subset of jobs that must be executed in
that scenario and all scenarios are given explicitly. The two objectives that
we consider are minimizing the maximum makespan over all scenarios and
minimizing the sum of the makespans of all scenarios. For both versions, we
give several approximation algorithms and lower bounds on their
approximability. With this research into optimization problems over scenarios,
we have opened a new and rich field of interesting problems.Comment: To appear in COCOON 2014. The final publication is available at
link.springer.co
Hardness of Rainbow Coloring Hypergraphs
A hypergraph is k-rainbow colorable if there exists a vertex coloring using k colors such that each hyperedge has all the k colors. Unlike usual hypergraph coloring, rainbow coloring becomes harder as the number of colors increases. This work studies the rainbow colorability of hypergraphs which are guaranteed to be nearly balanced rainbow colorable. Specifically, we show that for any Q,k >= 2 and ell <= k/2, given a Qk-uniform hypergraph which admits a k-rainbow coloring satisfying:
- in each hyperedge e, for some ell_e <= ell all but 2ell_e colors occur exactly Q times and the rest (Q +/- 1) times,
it is NP-hard to compute an independent set of (1 - (ell+1)/k + eps)-fraction of vertices, for any constant eps > 0. In particular, this implies the hardness of even (k/ell)-rainbow coloring such hypergraphs.
The result is based on a novel long code PCP test that ensures the strong balancedness property desired of the k-rainbow coloring in the completeness case. The soundness analysis relies on a mixing bound based on uniform reverse hypercontractivity due to Mossel, Oleszkiewicz, and Sen, which was also used in earlier proofs of the hardness of omega(1)-coloring 2-colorable 4-uniform hypergraphs due to Saket, and k-rainbow colorable 2k-uniform hypergraphs due to Guruswami and Lee
The Geometry of Differential Privacy: the Sparse and Approximate Cases
In this work, we study trade-offs between accuracy and privacy in the context
of linear queries over histograms. This is a rich class of queries that
includes contingency tables and range queries, and has been a focus of a long
line of work. For a set of linear queries over a database , we
seek to find the differentially private mechanism that has the minimum mean
squared error. For pure differential privacy, an approximation to
the optimal mechanism is known. Our first contribution is to give an approximation guarantee for the case of (\eps,\delta)-differential
privacy. Our mechanism is simple, efficient and adds correlated Gaussian noise
to the answers. We prove its approximation guarantee relative to the hereditary
discrepancy lower bound of Muthukrishnan and Nikolov, using tools from convex
geometry.
We next consider this question in the case when the number of queries exceeds
the number of individuals in the database, i.e. when . It is known that better mechanisms exist in this setting. Our second
main contribution is to give an (\eps,\delta)-differentially private
mechanism which is optimal up to a \polylog(d,N) factor for any given query
set and any given upper bound on . This approximation is
achieved by coupling the Gaussian noise addition approach with a linear
regression step. We give an analogous result for the \eps-differential
privacy setting. We also improve on the mean squared error upper bound for
answering counting queries on a database of size by Blum, Ligett, and Roth,
and match the lower bound implied by the work of Dinur and Nissim up to
logarithmic factors.
The connection between hereditary discrepancy and the privacy mechanism
enables us to derive the first polylogarithmic approximation to the hereditary
discrepancy of a matrix
Information-theoretic lower bounds on the oracle complexity of stochastic convex optimization
Relative to the large literature on upper bounds on complexity of convex
optimization, lesser attention has been paid to the fundamental hardness of
these problems. Given the extensive use of convex optimization in machine
learning and statistics, gaining an understanding of these complexity-theoretic
issues is important. In this paper, we study the complexity of stochastic
convex optimization in an oracle model of computation. We improve upon known
results and obtain tight minimax complexity estimates for various function
classes
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