Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable

Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical DP-complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+k clauses can be recognized in time . We improve this result and present an algorithm with time complexity ; hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, 1999). Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given set of clauses F, the number of clauses in each of its subsets exceeds the number of variables occurring in the subset at most by k, then we can decide in time whether F is satisfiable; k is called the maximum deficiency of F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails

Structural and functional studies of ligandin, a major renal organic anion-binding protein.

Sephadex gel filtration of the 1000,000 g supernate of homogenates of rat kidney revealed binding of various organic anions (penicillin, Bromsulphalein [BSP], bilirubin, phenolsulfonphthalein [PSP], phlorizin, glutathione [GSH], p-amino hippurate (PAH), probenecid, conjugated bilirubin, and BSP-GSH) to a nonalbumin-containing protein fraction (Y), which precipated on addition of monospecific anti-rat liver ligandin (Y protein)-IgG, but not control IgG. Quantitatively similar organic anion binding was observed in vivo after injection of BSP, BSP-GSH, phlorizin, probenecid, conjugated bilirubin, PAH, or penicillin. The binding protein was purified to apparent homogeneity and is a basic protein (pI 8.9) of 44,000 daltons with two apparently identical subunits of 22,000 daltons. Monospecific antibody was produced against the renal protein. The results of binding studies in vivo and in vitro and phsicochemical, immunologic, structural, and binding site investigations indicate that the renal protein is identical to hepatic ligandin. Immunofluorescent studies utilizing anti-ligandin IgG previously localized ligandin in the kidney to all proximal tubular cells. By quantitative radial immunodiffusion, the concentration of renal ligandin was 31.2 plus or minus 2.2 mug/mg supernatant protein and was increased 160% above basal values by pretreatment of rats with tetrachloro-dibenzo-p-dioxin. Pretreatment with phenobarbital, DDT, or pregnene-16alpha-carbonitrile did not increase renal ligandin concentration but doubled hepatic ligandin concentration. Circular dichroism studies of renal ligandin revealed percent helical structure similar to hepatic ligandin and primary association contrasts were derived for BSP (10-6 M-1) and PAH, probenecid, and penicillin (10-3 M-1). Administration of BSP or probenecid simultaneously with [C14] penicillin resulted in increased plasma retention and reduced kidney and urinary bladder content of [14C] penicillin and a correlation coefficient of -0.8 between total kidney/plasma radioactivity and percent of protein-bound radioactivity bound to ligandin in the kidney. These studies indicate that renal and hepatic ligandin are identical. Their response to drugs and chemicals varies. Competitive binding between several organic anions for ligandin correlated with their renal uptake from plasma, which suggests that ligandin may function in the proximal tubular cell as a component of the renal organic anion transport system