Randomized rounding: a technique for provably good algorithms and algorithmic proofs
Combinatorica - Theory of Computing
Probabilistic construction of deterministic algorithms: approximating packing integer programs
Journal of Computer and System Sciences - 27th IEEE Conference on Foundations of Computer Science October 27-29, 1986
Linear programming
Polynomial time approximation schemes for dense instances of NP-hard problems
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
MAX-CUT has a randomized approximation scheme in dense graphs
Random Structures & Algorithms
Matrix computations (3rd ed.)
Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
Polynomial time approximation of dense weighted instances of MAX-CUT
Random Structures & Algorithms
Random sampling and approximation of MAX-CSP problems
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
A Polynomial Time Approximation Scheme for Subdense MAX-CUT
A Polynomial Time Approximation Scheme for Subdense MAX-CUT
On Non-Approximability for Quadratic Programs
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Proof verification and hardness of approximation problems
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
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Let A be a real symmetric n× n-matrix with eigenvalues λ1,⋯,λn ordered after decreasing absolute value, and b an n× 1-vector. We present an algorithm finding approximate solutions to min x*(Ax+b) and max x*(Ax+b) over x∈ {–1,1}n, with an absolute error of at most $(c_{1}|{\rm \lambda_{1}}|+|{\rm \lambda}_{\lceil c_{2} {\rm log}n\rceil}|)2n+O((\alpha n+\beta)\sqrt{n {\rm log}n})$, where α and β are the largest absolute values of the entries in A and b, respectively, for any positive constants c1 and c2, in time polynomial in n. We demonstrate that the algorithm yields a PTAS for MAXCUT in regular graphs on n vertices of degree d of $\omega(\sqrt{n{\rm log}n})$, as long as they contain O(d4log n) 4-cycles. The strongest previous result showed that Ω(n/log n) average degree graphs admit a PTAS. We also show that smooth n-variate polynomial integer programs of constant degree k, always can be approximated in polynomial time leaving an absolute error of o(nk), answering in the affirmative a suspicion of Arora, Karger, and Karpinski in STOC 1995.