We consider a geometric programming problem consisting in minimizing a function given by the supremum of finitely many log-Laplace transforms of discrete nonnegative measures on a Euclidean space. Under a coerciveness assumption, we show that a $\varepsilon$-minimizer can be computed in a time that is polynomial in the input size and in $|\log\varepsilon|$. This is obtained by establishing bit-size estimates on approximate minimizers and by applying the ellipsoid method. We also derive polynomial iteration complexity bounds for the interior point method applied to the same class of problems. We deduce that the spectral radius of a partially symmetric, weakly irreducible nonnegative tensor can be approximated within $\varepsilon$ error in poly-time. For strongly irreducible tensors, we also show that the logarithm of the positive eigenvector is poly-time computable. Our results also yield that the the maximum of a nonnegative homogeneous $d$-form in the unit ball with respect to $d$-H\"older norm can be approximated in poly-time. In particular, the spectral radius of uniform weighted hypergraphs and some known upper bounds for the clique number of uniform hypergraphs are poly-time computable.

翻译：考虑一类几何规划问题，其目标函数由欧氏空间上离散非负测度对数-拉普拉斯变换的上确界构成。在强制性假设下，我们证明可在输入规模与$|\log\varepsilon|$的多项式时间内计算$\varepsilon$-近似极小值点。该结论通过建立近似极小值点的比特长度估计并应用椭球法获得。针对同一问题类，我们还推导出内点法的多项式迭代复杂度界。我们据此推论：部分对称弱不可约非负张量的谱半径可在多项式时间内达到$\varepsilon$误差逼近。对强不可约张量，我们进一步证明正特征向量的对数具有多项式时间可计算性。研究结果同时表明：在$d$-Hölder范数单位球内，非负齐次$d$次型的最大值可在多项式时间内逼近。特别地，一致加权超图的谱半径以及一致超图团数的若干已知上界均具有多项式时间可计算性。