Dynamic programming and fast matrix multiplication

Abstract. We give a novel general approach for solving NP-hard optimization problems that combines dynamic programming and fast matrix multiplication. The technique is based on reducing much of the computation involved to matrix multiplication. We exemplify our approach on problems like Vertex Cover, Dominating Set and Longest Path. Our approach works faster than the usual dynamic programming solution for any vertex subset problem on graphs of bounded branchwidth. In particular, we obtain the currently fastest algorithms for Planar Vertex Cover of runtime O(2 2.52 √ n), for Planar Dominating Set of runtime exact O(2 3.99 √ n) and parameterized O(2 11.98 √ k) · n O(1) , and for Planar Longest Path of runtime O(2 5.58 √ n). The exponent of the running time is depending heavily on the running time of the fastest matrix multiplication algorithm that is currently o(n 2.376).

Prioritizing Species by Conservation Value and Vulnerability: A New Index Applied to Species Threatened by Sea-Level Rise and Other Risks in Florida

Land-use change, climate change, and sea-level rise (SLR) pose substantial threats to biodiversity. Conservation resources are limited and must be directed toward the species and ecosystems that are most vulnerable, biologically distinct, likely to respond favorably to conservation interventions, and valuable ecologically, socially, or economically. Many prioritization and vulnerability assessment schemes exist, each emphasizing different types of vulnerabilities and values and often yielding disparate evaluations of the same species. We developed an integrative and flexible framework that incorporates existing assessments and is useful for illuminating the differences between systems such as the IUCN Red List, the US Endangered Species Act, and NatureServe\u27s Conservation Status Assessment and Climate Change Vulnerability Index. The Standardized Index of Vulnerability and Value Assessment (SIVVA) includes five advancements over existing tools: (1) the ability to import criteria and data from previous assessments, (2) explicit attention to SLR, (3) a flexible system of scoring, (4) metrics for both vulnerability and conservation value, and (5) quantitative and transparent accounting of multiple sources of uncertainty. We apply this system to 40 species in Florida previously identified as being vulnerable to SLR by the year 2100, describe the influence of different types of uncertainty on the resulting prioritizations, and explore the power of SIVVA to evaluate alternative prioritization schemes. This type of assessment is particularly relevant in low-lying coastal regions where vulnerability to SLR is predictable, severe, and likely to interact synergistically with other threats such as coastal development