Local approximability of max-min and min-max linear programs

In a max-min LP, the objective is to maximise ω subject to Ax ≤ 1, Cx ≥ ω1, and x ≥ 0. In a min-max LP, the objective is to minimise ρ subject to Ax ≤ ρ1, Cx ≥ 1, and x ≥ 0. The matrices A and C are nonnegative and sparse: each row ai of A has at most ΔI positive elements, and each row ck of C has at most ΔK positive elements. We study the approximability of max-min LPs and min-max LPs in a distributed setting; in particular, we focus on local algorithms (constant-time distributed algorithms). We show that for any ΔI ≥ 2, ΔK ≥ 2, and ε > 0 there exists a local algorithm that achieves the approximation ratio ΔI (1 − 1/ΔK) + ε. We also show that this result is the best possible: no local algorithm can achieve the approximation ratio ΔI (1 − 1/ΔK) for any ΔI ≥ 2 and ΔK ≥ 2.

Does the age of fine root carbon indicate the age of fine roots in boreal forests?

To test the reliability of the radiocarbon method for determining root age, we analyzed fine roots (originating from the years 1985 to 1993) from ingrowth cores with known maximum root age (1 to 6 years old). For this purpose, three Scots pine (Pinus sylvestris L.) stands were selected from boreal forests in Finland. We analyzed root 14C age by the radiocarbon method and compared it with the above-mentioned known maximum fine root age. In general, ages determined by the two methods (root 14C age and ingrowth core root maximum age) were in agreement with each other for roots of small diameter (<0.5mm). By contrast, in most of the samples of fine roots of larger diameter (1.5-2mm), the 14C age of root samples of 1987-89 exceeded the ingrowth core root maximum age by 1-10 years. This shows that these roots had received a large amount of older stored carbon from unknown sources in addition to atmospheric CO2 directly from photosynthesis. We conclude that the 14C signature of fine roots, especially those of larger diameter, may not always be indicative of root age, and that further studies are needed concerning the extent of possible root uptake of older carbon and its residence time in roots. Keywords: fine root age, Pinus sylvestris, radiocarbon, root carbon, ingrowth cores, tree ring