Exponentially hard problems are sometimes polynomial, a large deviation analysis of search algorithms for the random Satisfiability problem, and its application to stop-and-restart resolutions

A large deviation analysis of the solving complexity of random 3-Satisfiability instances slightly below threshold is presented. While finding a solution for such instances demands an exponential effort with high probability, we show that an exponentially small fraction of resolutions require a computation scaling linearly in the size of the instance only. This exponentially small probability of easy resolutions is analytically calculated, and the corresponding exponent shown to be smaller (in absolute value) than the growth exponent of the typical resolution time. Our study therefore gives some theoretical basis to heuristic stop-and-restart solving procedures, and suggests a natural cut-off (the size of the instance) for the restart.Comment: Revtex file, 4 figure

On the freezing of variables in random constraint satisfaction problems

The set of solutions of random constraint satisfaction problems (zero energy groundstates of mean-field diluted spin glasses) undergoes several structural phase transitions as the amount of constraints is increased. This set first breaks down into a large number of well separated clusters. At the freezing transition, which is in general distinct from the clustering one, some variables (spins) take the same value in all solutions of a given cluster. In this paper we study the critical behavior around the freezing transition, which appears in the unfrozen phase as the divergence of the sizes of the rearrangements induced in response to the modification of a variable. The formalism is developed on generic constraint satisfaction problems and applied in particular to the random satisfiability of boolean formulas and to the coloring of random graphs. The computation is first performed in random tree ensembles, for which we underline a connection with percolation models and with the reconstruction problem of information theory. The validity of these results for the original random ensembles is then discussed in the framework of the cavity method.Comment: 32 pages, 7 figure

Interleukin 6 increases production of cytokines by colonic innate lymphoid cells in mice and patients with chronic intestinal inflammation

Background & Aims: Innate lymphoid cells (ILCs) are a heterogeneous group of mucosal inflammatory cells that participate in chronic intestinal inflammation. We investigated the role of interleukin 6 (IL6) in inducing activation of ILCs in mice and in human beings with chronic intestinal inflammation. Methods: ILCs were isolated from colons of Tbx21-/- × Rag2-/- mice (TRUC), which develop colitis; patients with inflammatory bowel disease (IBD); and patients without colon inflammation (controls). ILCs were characterized by flow cytometry; cytokine production was measured by enzyme-linked immunosorbent assay and cytokine bead arrays. Mice were given intraperitoneal injections of depleting (CD4, CD90), neutralizing (IL6), or control antibodies. Isolated colon tissues were analyzed by histology, explant organ culture, and cell culture. Bacterial DNA was extracted from mouse fecal samples to assess the intestinal microbiota. Results: IL17A- and IL22-producing, natural cytotoxicity receptor-negative, ILC3 were the major subset of ILCs detected in colons of TRUC mice. Combinations of IL23 and IL1α induced production of cytokines by these cells, which increased further after administration of IL6. Antibodies against IL6 reduced colitis in TRUC mice without significantly affecting the structure of their intestinal microbiota. Addition of IL6 increased production of IL17A, IL22, and interferon-γ by human intestinal CD3-negative, IL7-receptor-positive cells, in a dose-dependent manner. Conclusions: IL6 contributes to activation of colonic natural cytotoxicity receptor-negative, CD4-negative, ILC3s in mice with chronic intestinal inflammation (TRUC mice) by increasing IL23- and IL1α-induced production of IL17A and IL22. This pathway might be targeted to treat patients with IBD because IL6, which is highly produced in colonic tissue by some IBD patients, also increased the production of IL17A, IL22, and interferon-γ by cultured human colon CD3-negative, IL7-receptor-positive cells

Proof of a Hypercontractive Estimate via Entropy

with the uniform (=product) measure