The Firefighter Problem: A Structural Analysis

We consider the complexity of the firefighter problem where b>=1 firefighters are available at each time step. This problem is proved NP-complete even on trees of degree at most three and budget one (Finbow et al.,2007) and on trees of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this paper, we provide further insight into the complexity landscape of the problem by showing that the pathwidth and the maximum degree of the input graph govern its complexity. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any fixed budget b>=1. We then show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter "pathwidth" and "maximum degree" of the input graph

Estrogens promote misfolded proinsulin degradation to protect insulin production and delay diabetes

Summary: Conjugated estrogens (CE) delay the onset of type 2 diabetes (T2D) in postmenopausal women, but the mechanism is unclear. In T2D, the endoplasmic reticulum (ER) fails to promote proinsulin folding and, in failing to do so, promotes ER stress and β cell dysfunction. We show that CE prevent insulin-deficient diabetes in male and in female Akita mice using a model of misfolded proinsulin. CE stabilize the ER-associated protein degradation (ERAD) system and promote misfolded proinsulin proteasomal degradation. This involves activation of nuclear and membrane estrogen receptor-α (ERα), promoting transcriptional repression and proteasomal degradation of the ubiquitin-conjugating enzyme and ERAD degrader, UBC6e. The selective ERα modulator bazedoxifene mimics CE protection of β cells in females but not in males. : Estrogens prevent diabetes in women, but the mechanism is poorly understood. Xu et al. report that estrogens activate the endoplasmic-reticulum-associated protein degradation pathway, which promotes misfolded proinsulin degradation, suppresses endoplasmic reticulum stress, and protects insulin secretion in mice and in human pancreatic β cells. Keywords: estrogens, beta cell, islet, endoplasmic reticulum stress, proinsulin misfolding, diabetes, bazedoxifene, sex dimorphism, ERAD, SER