Extremal Optimization: Methods derived from Co-Evolution

We describe a general-purpose method for finding high-quality solutions to hard optimization problems, inspired by self-organized critical models of co-evolution such as the Bak-Sneppen model. The method, called Extremal Optimization, successively eliminates extremely undesirable components of sub-optimal solutions, rather than ``breeding'' better components. In contrast to Genetic Algorithms which operate on an entire ``gene-pool'' of possible solutions, Extremal Optimization improves on a single candidate solution by treating each of its components as species co-evolving according to Darwinian principles. Unlike Simulated Annealing, its non-equilibrium approach effects an algorithm requiring few parameters to tune. With only one adjustable parameter, its performance proves competitive with, and often superior to, more elaborate stochastic optimization procedures. We demonstrate it here on two classic hard optimization problems: graph partitioning and the traveling salesman problem.Comment: 8 pages, Latex, 5 ps-figures included. To appear in ``GECCO-99: Proceedings of the Genetic and Evolutionary Computation Conference,'' (Morgan Kaufmann, San Francisco, 1999

The Peculiar Phase Structure of Random Graph Bisection

The mincut graph bisection problem involves partitioning the n vertices of a graph into disjoint subsets, each containing exactly n/2 vertices, while minimizing the number of "cut" edges with an endpoint in each subset. When considered over sparse random graphs, the phase structure of the graph bisection problem displays certain familiar properties, but also some surprises. It is known that when the mean degree is below the critical value of 2 log 2, the cutsize is zero with high probability. We study how the minimum cutsize increases with mean degree above this critical threshold, finding a new analytical upper bound that improves considerably upon previous bounds. Combined with recent results on expander graphs, our bound suggests the unusual scenario that random graph bisection is replica symmetric up to and beyond the critical threshold, with a replica symmetry breaking transition possibly taking place above the threshold. An intriguing algorithmic consequence is that although the problem is NP-hard, we can find near-optimal cutsizes (whose ratio to the optimal value approaches 1 asymptotically) in polynomial time for typical instances near the phase transition.Comment: substantially revised section 2, changed figures 3, 4 and 6, made minor stylistic changes and added reference

Influence of dietary protein on glomerular filtration before and after bariatric surgery: a cohort study

BACKGROUND: Obesity-associated elevations in glomerular filtration rate (GFR) are common and may play a role in the development of kidney disease, so identifying the underlying mechanism is important. We therefore studied whether reductions in dietary protein intake, which is known to modulate GFR, explain why GFR decreases after bariatric surgery-induced weight loss. STUDY DESIGN: Cohort study with participants as their own controls. SETTING & PARTICIPANTS: 8 severely obese patients with normal kidney function were recruited from bariatric surgery centers in Indianapolis, IN. All participants were placed on a fixed-protein (50-g/d) diet for 1 week before and after a minimum of a 20-kg weight loss by bariatric surgery and were followed up closely by dieticians for adherence. PREDICTOR: Ad lib versus low-protein diet before versus after bariatric surgery. OUTCOME: Measured GFR, using repeated-measures analysis, was used to estimate the independent effects of diet and surgery. MEASUREMENT: GFR was measured using plasma iohexol clearance. RESULTS: A median of 32.9 (range, 19.5-54.4)kg was lost between the first presurgery visit and first postsurgery visit. Dietetic evaluations and urinary urea excretion confirmed that patients generally adhered to the study diet. GFRs on an ad lib diet were significantly higher before compared to after surgery (GFR medians were 144 (range, 114-178) and 107 (range, 85-147) mL/min, respectively; P=0.01). Although bariatric surgery (-26mL/min; P=0.005) and dietary sodium intake (+7.5mL/min per 100mg of dietary sodium; P=0.001) both influenced GFR, consuming a low-protein diet did not (P=0.7). LIMITATIONS: Small sample size; mostly white women; possible lack of generalizability. CONCLUSIONS: The decrease in GFR observed after bariatric surgery is explained at least in part by the effects of surgery and/or dietary sodium intake, but not by low dietary protein consumption