A new way to use semidefinite programming with applications to linear equations mod p

Abstract

We introduce a new method to construct approximation algorithms for combinatorial optimization problems using semidefinite programming. It consists of expressing each combinatorial object in the original problem as a constellation of vectors in the semidefinite program. When we apply this technique to systems of linear equations mod p with at most two variables in each equation, we can show that the problem is approximable within (1 − κ(p))p, where κ(p) > 0 for all p. Using standard techniques, we also show that it is NP-hard to approximate the problem within a constant ratio, independent of p. Warning: Essentially this paper has been published in Journal of Algorithms and is subject to copyright restrictions. In particular it is for personal use only. New Use of Semidefinite Programming 4 List of symbols used: α lower case Greek letter alpha. δ lower case Greek letter delta. e lower case Greek letter epsilon. κ lower case Greek letter kappa. λ lower case Greek letter lambda. μ lower case Greek letter mu. π lower case Greek letter pi. σ lower case Greek letter sigma. θ lower case Greek letter theta. Θ upper case Greek letter Theta. φ lower case Greek letter phi. Φ upper case Greek letter Phi. ζ lower case Greek letter zeta. ξ lower case Greek letter xi. lower case Latin letter ell. ∫ integral sign. ∑ summation sign. R bold face upper case Roman letter ar (denotes the set of reals). Z bold face upper case Roman letter zed (denotes the set of integers). ⊥ perpendicular sign. ← left arrow. → right arrow with small bar (denotes “maps to”). =⇒ double right arrow (denotes implication). ⇐⇒ double left-right arrow (denotes equivalence). ∅ the empty set. ∪ set union. ∩ set intersection. ∈ belongs to sign. | · | absolute value. ‖ · ‖ Euclidean norm. 〈·, ·〉 angular brackets (denotes inner product) ∀ universal quantifier. ∃ existential quantifier. √· square root sign. × multiplication sign. New Use of Semidefinite Programming 5