Restricted Max-Min Allocation: Approximation and Integrality Gap

Asadpour, Feige, and Saberi proved that the integrality gap of the configuration LP for the restricted max-min allocation problem is at most 4. However, their proof does not give a polynomial-time approximation algorithm. A lot of efforts have been devoted to designing an efficient algorithm whose approximation ratio can match this upper bound for the integrality gap. In ICALP 2018, we present a (6 + delta)-approximation algorithm where delta can be any positive constant, and there is still a gap of roughly 2. In this paper, we narrow the gap significantly by proposing a (4+delta)-approximation algorithm where delta can be any positive constant. The approximation ratio is with respect to the optimal value of the configuration LP, and the running time is poly(m,n)* n^{poly(1/(delta))} where n is the number of players and m is the number of resources. We also improve the upper bound for the integrality gap of the configuration LP to 3 + 21/26 =~ 3.808

Restricted Max-Min Fair Allocation

The restricted max-min fair allocation problem seeks an allocation of resources to players that maximizes the minimum total value obtained by any player. It is NP-hard to approximate the problem to a ratio less than 2. Comparing the current best algorithm for estimating the optimal value with the current best for constructing an allocation, there is quite a gap between the ratios that can be achieved in polynomial time: 4+delta for estimation and 6 + 2 sqrt{10} + delta ~~ 12.325 + delta for construction, where delta is an arbitrarily small constant greater than 0. We propose an algorithm that constructs an allocation with value within a factor 6 + delta from the optimum for any constant delta > 0. The running time is polynomial in the input size for any constant delta chosen

Research on Experiential Marketing Strategy Based on the Sale of Baking Products

To understand the consumer is the prerequisite for the enterprise to enter the consumer market. With the market competition intensified, Traditional marketing strategy is difficult to achieve the expected goal of the enterprise. It needs new marketing theory to guide the market and satisfy the demand. In this paper, the sales of baking products are taken for an example in which 425 questionnaires analyze five models of emotion, culture, service, environment and personality in detail and the paper suppllies some corresponding suggestions & measure to tackle the problems as the lack of large-scale leading enterprises and Enterprise homogeneity

THE HYDRAULICS OF NATURE-LIKE FISHWAYS

Nature-like fishway arrangements are commonly used because these structures imitate the characteristics of natural rivers and effectively allow fish to migrate past river sections blocked by hydraulic structures. In this paper, physical models were analyzed, and the velocity distributions of two different fishway structures (Types I and II) were compared. Results showed that the maximum mainstream velocity of the Type I structure was 5.3% lower than that of the Type II structure. However, the average mainstream velocity of the Type I structure was 21.1% greater than that of the Type II structure. The total per-cycle length of the mainstream path in the Type II structure was 2.1 times greater than that of the Type I structure, which indicated that the length of the mainstream path was somewhat proportional to the average velocity of the mainstream. When the flow rate was kept constant, increases in the velocity of the main flow associated with changes in the internal structure of the fishway decreased the average velocity of the main flow, while decreases in the total length of the flow path led to increases in the average velocity of the main flow. Due to frictional head loss along the fishway and local head loss, as well as the overlaps between these factors, the overall flow rate gradually decreased every cycle, despite periodic fluctuations

Commercial Janus Fabrics as Reusable Facemask Materials: A Balance of Water Repellency, Filtration Efficiency, Breathability, and Reusability

Facemasks as personal protective equipment play a significant role in helping prevent the spread of viruses during the COVID-19 pandemic. A desired reusable fabric facemask should strike a balance of water repellency, good filtration efficiency (FE), breathability, and mechanical robustness against washing cycles. Despite significant efforts in testing various commercial fabric materials for filtration efficiency, few have investigated fabric performance as a function of the fiber/yarn morphology and wettability of the fabric itself. In this study, we examine commercial fabrics with Janus-like behaviors to determine the best reusable fabric facemask materials by understanding the roles of morphology, porosity, and wettability of the fabric on its overall performance. We find that the outer layer of the diaper fabric consisted of laminated polyurethane, which is hydrophobic, has low porosity (∼5%) and tightly woven yarn structures, and shows the highest overall FE (up to 54%) in the submicron particle size range (0.03-0.6 μm) among the fabrics tested. Fabric layers with higher porosity lead to lower-pressure drops, indicating higher breathability but lower FE. Tightly woven waterproof rainwear fabrics perform the best after 10 washing cycles, remaining intact morphologically with only a 2-5% drop in the overall FE in the submicron particle size range, whereas other knitted fabric layers become loosened and the laminated polyurethane thin film on the diaper fabric is wrinkled. In comparison, the surgical masks and N95 respirators made from nonwoven polypropylene (PP) fibers see over a 30% decline in the overall FE after 10 washing cycles. Overall, we find that tightly woven Janus fabrics consisting of a low porosity, a hydrophobic outer layer, and a high porosity and hydrophilic inner layer offer the best performance among the fabrics tested as they can generate a high overall FE, achieve good breathability, and maintain fabric morphology and performance over multiple washing cycles