On the complexity of nonlinear mixed-integer optimization

This is a survey on the computational complexity of nonlinear mixed-integer optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained fully polynomial time approximation schemes in fixed dimension, and to strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear Optimization, IMA Volumes, Springer-Verla

Root finding with threshold circuits

We show that for any constant d, complex roots of degree d univariate rational (or Gaussian rational) polynomials---given by a list of coefficients in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a uniform family of constant-depth polynomial-size threshold circuits). The basic idea is to compute the inverse function of the polynomial by a power series. We also discuss an application to the theory VTC^0 of bounded arithmetic.Comment: 19 pages, 1 figur

Syntactic Separation of Subset Satisfiability Problems

Variants of the Exponential Time Hypothesis (ETH) have been used to derive lower bounds on the time complexity for certain problems, so that the hardness results match long-standing algorithmic results. In this paper, we consider a syntactically defined class of problems, and give conditions for when problems in this class require strongly exponential time to approximate to within a factor of (1-epsilon) for some constant epsilon > 0, assuming the Gap Exponential Time Hypothesis (Gap-ETH), versus when they admit a PTAS. Our class includes a rich set of problems from additive combinatorics, computational geometry, and graph theory. Our hardness results also match the best known algorithmic results for these problems