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While much of network design focuses mostly on cost (number or weight of edges), node degrees have also played an important role. They have traditionally either appeared as an objective, to minimize the maximum degree (e.g., the Minimum Degree Spanning Tree problem), or as constraints that might be violated to give bicriteria approximations (e.g., the Minimum Cost Degree Bounded Spanning Tree problem). We extend the study of degrees in network design in two ways. First, we introduce and study a new variant of the Survivable Network Design Problem where in addition to the traditional objective of minimizing the cost of the chosen edges, we add a constraint that the 𝓁_p-norm of the node degree vector is bounded by an input parameter. This interpolates between the classical settings of maximum degree (the 𝓁_∞-norm) and the number of edges (the 𝓁₁-degree), and has natural applications in distributed systems and VLSI design. We give a constant bicriteria approximation in both measures using convex programming. Second, we provide a polylogarithmic bicriteria approximation for the Degree Bounded Group Steiner problem on bounded treewidth graphs, solving an open problem from [Guy Kortsarz and Zeev Nutov, 2022] and [X. Guo et al., 2022]. 

Conjugated polymers have been widely investigated where ladder-type conjugated polymers receive more attention due to their rigid backbones and extraordinary properties. However, the understanding of how the rigid conformation of ladder polymers translates to material properties is still limited. Here, we systematically investigated the solution aggregation properties of a carbazole-derived conjugated ladder polymer (LP) and its analogous non-ladder control polymer (CP) via light scattering, neutron scattering, and UV-vis absorption spectroscopy characterization techniques, revealing a highly robust, temperature-insensitive aggregation behavior of the LP. The experimental findings were further validated by computational molecular dynamics simulations. We found that the peak positions and intensities of the UV spectra of the LP remained constant between 20 °C and 120 °C in chlorobenzene solution. The polymer also showed a stable hydrodynamic radius measured by dynamic light scattering from 20 °C to 70 °C in the chlorobenzene solution. Using small-angle neutron scattering, no Guinier region was reached in the measured q range down to 0.008 Å −1 , even at elevated temperature. In contrast, the non-ladder control polymer CP was fully soluble in the chlorobenzene solvent without the observation of any notable aggregates. The Brownian dynamics simulation showed that during polymer aggregation, the entropy change of the LP was significantly less negative than that of the non-ladder control polymer. These findings revealed the low entropy nature of rigid conjugated ladder polymers and the low entropy penalty for their aggregation, which is promising for highly robust intermolecular interactions at high temperatures. Such a unique thermodynamic feature of rigid ladder polymers can be leveraged in the design and application of next-generation electronic and optoelectronic devices that function under unconventional high temperature conditions.