Single-Elimination Brackets Fail to Approximate Copeland Winner

Single-elimination (SE) brackets appear commonly in both sports tournaments and the voting theory literature. In certain tournament models, they have been shown to select the unambiguously-strongest competitor with optimum probability. By contrast, we reevaluate SE brackets through the lens of approximation, where the goal is to select a winner who would beat the most other competitors in a round robin (i.e., maximize the Copeland score), and find them lacking. Our primary result establishes the approximation ratio of a randomly-seeded SE bracket is 2^{- Theta(sqrt{log n})}; this is underwhelming considering a 1/2 ratio is achieved by choosing a winner uniformly at random. We also establish that a generalized version of the SE bracket performs nearly as poorly, with an approximation ratio of 2^{- Omega(sqrt[4]{log n})}, addressing a decade-old open question in the voting tree literature

Normalizing Microbiota-Induced Retinoic Acid Deficiency Stimulates Protective CD8+ T Cell-Mediated Immunity in Colorectal Cancer

Although all-trans retinoic acid (atRA) is a key regulator of intestinal immunity, its role in colorectal cancer (CRC) is unknown. We found that mice with colitis-associated CRC had a marked deficiency in colonic atRA due to alterations in atRA metabolism mediated by microbiota-induced intestinal inflammation. Human ulcerative colitis (UC), UC-associated CRC, and sporadic CRC specimens have similar alterations in atRA metabolic enzymes, consistent with reduced colonic atRA. Inhibition of atRA signaling promoted tumorigenesis whereas atRA supplementation reduced tumor burden. The benefit of atRA treatment was mediated by cytotoxic CD8(+) T cells, activated due to MHCI upregulation on tumor cells. Consistent with these findings, increased colonic expression of the atRA-catabolizing enzyme, CYP26A1, correlated with reduced frequencies of tumoral cytotoxic CD8(+) T cells and with worse disease prognosis in human CRC. These results reveal a mechanism by which microbiota drive colon carcinogenesis and highlight atRA metabolism as a therapeutic target for CRC