On Quadratic Programming with a Ratio Objective

@inproceedings{Bhaskara2011OnQP,
  title={On Quadratic Programming with a Ratio Objective},
  author={Aditya Bhaskara and Moses Charikar and Rajsekar Manokaran and Aravindan Vijayaraghavan},
  booktitle={International Colloquium on Automata, Languages and Programming},
  year={2011}
}
Quadratic Programming (QP) is the well-studied problem of maximizing over {−1,1} values the quadratic form ∑i≠jaijxixj. QP captures many known combinatorial optimization problems, and assuming the Unique Games conjecture, Semidefinite Programming (SDP) techniques give optimal approximation algorithms. We extend this body of work by initiating the study of Quadratic Programming problems where the variables take values in the domain {−1,0,1}. The specific problem we study is $$\begin{aligned… 

Sum of squares lower bounds for refuting any CSP

This work shows that if P is δ-close to supporting a t-wise uniform distribution on satisfying assignments, then the degree-Θ(n/Δ2/(t - 1) logΔ) SOS algorithm cannot (δ+o(1))-refute a random instance of CSP(P).

Refutation of random constraint satisfaction problems using the sum of squares proof system

This thesis studies refutation using sum-of-squares (SOS) proof systems using a sequence of increasingly powerful proof systems parameterized by degree: the higher the degree, the more powerful the proof system.

Finding dense structures in graphs and matrices

The best known algorithm for the so-called densest k-subgraph problem is presented, which gives roughly an n1/4 factor approximation, and it is proved that without such a restriction, the problem is hard to approximate to an almost polynomial factor.

M ar 2 01 3 Testing Microscopic Discretization

What can we say about the spectra of a collection of microscopic variables when only their coarse-grained sums are experimentally accessible? In this paper, using the tools and methodology from the

Testing microscopic discretization

What can we say about the spectra of a collection of microscopic variables when only their coarse-grained sums are experimentally accessible? In this paper, using the tools and methodology from the

Discovering conflicting groups in signed networks

This paper derives a novel formulation in which each conflicting group is naturally characterized by the solution to the maximum discrete Rayleigh’s quotient ( M AX -DRQ ) problem, and presents two spectral methods for approximate solutions to the problem.

NEW APPROACH FOR WOLFE'S MODIFIED SIMPLEX METHOD TO SOLVE QUADRATIC PROGRAMMING PROBLEMS

In this paper, an alternative method for Wolfe’s modified simplex method is introduced. This method is easy to solve quadratic programming problem (QPP) concern with non-linear programming problem

Copositivity-based approximations for mixed-integer fractional quadratic optimization

This paper adds to the rich evidence for the versatility of copositive optimization approaches, and hints at possible novel approximation strategies combining continuous and discrete optimization techniques in the domain of (fractional) polynomial optimization.

References

SHOWING 1-10 OF 29 REFERENCES

On maximization of quadratic form over intersection of ellipsoids with common center

It is shown that this bound is sharp in order, as far as the dependence on m is concerned, and that a~feasible solution x to (P) with x^TAx\ge \frac{{\Opt(\hbox{{\rm SDP}})}}{{2\ln(2m^2)}} \eqno{(*)}$$ can be found efficiently.

Zero-One Rounding of Singular Vectors

We propose a generic and simple technique called dyadic rounding for rounding real vectors to zero-one vectors, and show its several applications in approximating singular vectors of matrices by

Subexponential Algorithms for Unique Games and Related Problems

A sub exponential time approximation algorithm for the Unique Games problem that is exponential in an arbitrarily small polynomial of the input size, n, and shows that for every $\epsilon>0$ and every regular $n$-vertex graph~$G, one can break into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most n eigenvalues larger than $1-\eta.

Maximizing quadratic programs: extending Grothendieck's inequality

  • M. CharikarA. Wirth
  • Computer Science
    45th Annual IEEE Symposium on Foundations of Computer Science
  • 2004
The approximation algorithm for this type of quadratic programming problem uses the canonical semidefinite relaxation and returns a solution whose ratio to the optimum is in /spl Omega/(1/ logn), which can be seen as an extension to that of maximizing x/sup T/Ay.

On non-approximability for quadratic programs

These techniques yield an explicit instance for which the integrality gap is /spl Omega/ (log n/log log n), essentially answering one of the open problems of Alon et al.

Towards Sharp Inapproximability For Any 2-CSP

  • Per Austrin
  • Computer Science
    48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
  • 2007
It is shown how to reduce the search for a good inapproximability result to a certain numeric minimization problem, and conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly.

Optimal algorithms and inapproximability results for every CSP?

A generic conversion from SDP integrality gaps to UGC hardness results for every CSP is shown, which achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut and Unique Games.

Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming

This algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.

Approximating the cut-norm via Grothendieck's inequality

The problem of approximating the cut-norm of a given real matrix is MAX SNP hard, and the algorithm combines semidefinite programming with a rounding technique based on Grothendieck's Inequality to provide an efficient approximation algorithm.

The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into 1 (Extended Abstract)

In this paper we disprove the following conjecture due to Goemans [16] and Linial [24] (also see [5, 26]): “Every negative type metric embeds into � 1 with constant distortion.” We show that for