# Linear equations modulo 2 and the L1 diameter of convex bodies

Subhash Khot, Assaf Naor

Research output: Contribution to journalArticle

### Abstract

We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A = {aijk}n i, j, k=1 such that for all i, j, k zε {1,...,n} we have aijk = aikj = akji = ajik = akij = ajki and a iik = aijj = aiji = 0, computes a number Alg(A) which satisfies with probability at least 1/2, ω(√ log n/n t) - maxxzε{-1, 1} nn i, j, k=1 aijkxixjxk ≤ Alg(A) ≤ max xzε{-1, 1} nn i, j, k=1 aijkxixjxk. On the other hand, we show via a simple reduction from a result of Håstad and Venkatesh [Random Structures Algorithms, 25 (2004), pp. 117-149] that under the assumption NP ⊈ DTIME(n(log n)O(1)), for every zε > 0 there is no algorithm that approx imates maxxzε{-1, 1} nn i, j, k=1 aijkx ixjxk within a factor of 2 (log n)1-zε in time 2(log n)O(1). Our algorithm is based on a reduction to the problem of computing the diameter of a convex body in &Rn with respect to the L1 norm. We show that it is possible to do so up to a multiplicative error of O(√n/log n), while no randomized polynomial time algorithm can achieve accuracy O(√n/log n). This resolves a question posed by Brieden et al. in [Mathematika, 48 (2001), pp. 63-105]. We apply our new algorithm to improve the algorithm of Håstad and Venkatesh for the Max-E3-Lin-2 problem. Given an overdetermined system ε of N linear equations modulo 2 in n ≤ N Boolean variables such that in each equation only three distinct variables appear, the goal is to approximate in polynomial time the maximum number of satisfiable equations in ε minus N/2 (i.e., we subtract the expected number of satisfied equations in a random assignment). Håstad and Venkatesh obtained an algorithm which approximates this value up to a factor of O(√N). We obtain an O(√ n/log n) approximation algorithm. By relating this problem to the refutation problem for random 3 - CNF formulas, we give evidence that obtaining a significant improvement over this approximation factor is likely to be difficult.

Original language English (US) 1448-1463 16 SIAM Journal on Computing 38 4 https://doi.org/10.1137/070691140 Published - 2008

### Fingerprint

Convex Body
Linear equations
Modulo
Linear equation
Randomized Algorithms
Polynomial-time Algorithm
Polynomials
Random Structure
Overdetermined Systems
L1-norm
Approximate Algorithm
Approximation Algorithms
Resolve
Multiplicative
Polynomial time
Assignment
Tensor
Likely
Approximation algorithms
Distinct

### Keywords

• Computational convex geometry
• Grothendieck's inequality
• Max-E3-Lin-2
• Refutation of random SAT
• Semidefinite programming

### ASJC Scopus subject areas

• Mathematics(all)
• Computer Science(all)

### Cite this

In: SIAM Journal on Computing, Vol. 38, No. 4, 2008, p. 1448-1463.

Research output: Contribution to journalArticle

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N2 - We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A = {aijk}n i, j, k=1 such that for all i, j, k zε {1,...,n} we have aijk = aikj = akji = ajik = akij = ajki and a iik = aijj = aiji = 0, computes a number Alg(A) which satisfies with probability at least 1/2, ω(√ log n/n t) - maxxzε{-1, 1} n ∑n i, j, k=1 aijkxixjxk ≤ Alg(A) ≤ max xzε{-1, 1} n ∑n i, j, k=1 aijkxixjxk. On the other hand, we show via a simple reduction from a result of Håstad and Venkatesh [Random Structures Algorithms, 25 (2004), pp. 117-149] that under the assumption NP ⊈ DTIME(n(log n)O(1)), for every zε > 0 there is no algorithm that approx imates maxxzε{-1, 1} n ∑n i, j, k=1 aijkx ixjxk within a factor of 2 (log n)1-zε in time 2(log n)O(1). Our algorithm is based on a reduction to the problem of computing the diameter of a convex body in &Rn with respect to the L1 norm. We show that it is possible to do so up to a multiplicative error of O(√n/log n), while no randomized polynomial time algorithm can achieve accuracy O(√n/log n). This resolves a question posed by Brieden et al. in [Mathematika, 48 (2001), pp. 63-105]. We apply our new algorithm to improve the algorithm of Håstad and Venkatesh for the Max-E3-Lin-2 problem. Given an overdetermined system ε of N linear equations modulo 2 in n ≤ N Boolean variables such that in each equation only three distinct variables appear, the goal is to approximate in polynomial time the maximum number of satisfiable equations in ε minus N/2 (i.e., we subtract the expected number of satisfied equations in a random assignment). Håstad and Venkatesh obtained an algorithm which approximates this value up to a factor of O(√N). We obtain an O(√ n/log n) approximation algorithm. By relating this problem to the refutation problem for random 3 - CNF formulas, we give evidence that obtaining a significant improvement over this approximation factor is likely to be difficult.

AB - We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A = {aijk}n i, j, k=1 such that for all i, j, k zε {1,...,n} we have aijk = aikj = akji = ajik = akij = ajki and a iik = aijj = aiji = 0, computes a number Alg(A) which satisfies with probability at least 1/2, ω(√ log n/n t) - maxxzε{-1, 1} n ∑n i, j, k=1 aijkxixjxk ≤ Alg(A) ≤ max xzε{-1, 1} n ∑n i, j, k=1 aijkxixjxk. On the other hand, we show via a simple reduction from a result of Håstad and Venkatesh [Random Structures Algorithms, 25 (2004), pp. 117-149] that under the assumption NP ⊈ DTIME(n(log n)O(1)), for every zε > 0 there is no algorithm that approx imates maxxzε{-1, 1} n ∑n i, j, k=1 aijkx ixjxk within a factor of 2 (log n)1-zε in time 2(log n)O(1). Our algorithm is based on a reduction to the problem of computing the diameter of a convex body in &Rn with respect to the L1 norm. We show that it is possible to do so up to a multiplicative error of O(√n/log n), while no randomized polynomial time algorithm can achieve accuracy O(√n/log n). This resolves a question posed by Brieden et al. in [Mathematika, 48 (2001), pp. 63-105]. We apply our new algorithm to improve the algorithm of Håstad and Venkatesh for the Max-E3-Lin-2 problem. Given an overdetermined system ε of N linear equations modulo 2 in n ≤ N Boolean variables such that in each equation only three distinct variables appear, the goal is to approximate in polynomial time the maximum number of satisfiable equations in ε minus N/2 (i.e., we subtract the expected number of satisfied equations in a random assignment). Håstad and Venkatesh obtained an algorithm which approximates this value up to a factor of O(√N). We obtain an O(√ n/log n) approximation algorithm. By relating this problem to the refutation problem for random 3 - CNF formulas, we give evidence that obtaining a significant improvement over this approximation factor is likely to be difficult.

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