The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations

Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the l norm. Also, for some of these problems we can prove the same result as above, but for a larger factor such as 2 1 & = n or n. Improving the factor 2 0.5 & = n to dimension for either of the lattice problems would imply the hardness of the Shortest Vector Problem in l2 norm; an old open problem. Our proofs use reductions from few-prover, one-round interactive proof systems [FL], BG+], either directly, or through a set-cover problem. ] 1997 Academic Press

[1]  Elwyn R. Berlekamp,et al.  On the inherent intractability of certain coding problems (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[2]  Franco P. Preparata,et al.  The Densest Hemisphere Problem , 1978, Theor. Comput. Sci..

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[5]  Jeffrey C. Lagarias,et al.  Solving low density subset sum problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[6]  András Frank,et al.  An application of simultaneous approximation in combinatorial optimization , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[7]  László Lovász,et al.  Algorithmic theory of numbers, graphs and convexity , 1986, CBMS-NSF regional conference series in applied mathematics.

[8]  László Babai,et al.  On Lovász’ lattice reduction and the nearest lattice point problem , 1986, Comb..

[9]  R. Kannan ALGORITHMIC GEOMETRY OF NUMBERS , 1987 .

[10]  Ravi Kannan,et al.  Minkowski's Convex Body Theorem and Integer Programming , 1987, Math. Oper. Res..

[11]  Avi Wigderson,et al.  Multi-prover interactive proofs: how to remove intractability assumptions , 2019, STOC '88.

[12]  Manuel Blum,et al.  Designing programs that check their work , 1989, STOC '89.

[13]  Nondeterministic exponential time has two-prover interactive protocols , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[14]  Moni Naor,et al.  The hardness of decoding linear codes with preprocessing , 1990, IEEE Trans. Inf. Theory.

[15]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[16]  Jeffrey C. Lagarias,et al.  Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice , 1990, Comb..

[17]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[18]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[19]  Leonid A. Levin,et al.  Checking computations in polylogarithmic time , 1991, STOC '91.

[20]  Hans Ulrich Simon,et al.  Robust Trainability of Single Neurons , 1995, J. Comput. Syst. Sci..

[21]  David Zuckerman,et al.  NP-complete problems have a version that's hard to approximate , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[22]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1994, JACM.

[23]  Edoardo Amaldi,et al.  The Complexity and Approximability of Finding Maximum Feasible Subsystems of Linear Relations , 1995, Theor. Comput. Sci..

[24]  Edoardo Amaldi,et al.  On the approximability of some NP-hard minimization problems for linear systems , 1996, Electron. Colloquium Comput. Complex..

[25]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[26]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[27]  Kenneth J. Giuliani Factoring Polynomials with Rational Coeecients , 1998 .