Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in $\mathbb{R}^2$ can be done in polynomial time, while optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing that this problem can be solved in polynomial time for cubic polynomials. Furthermore, we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a bounded polyhedron in $\mathbb{R}^2$ is solvable in polynomial time. We show that this holds for polynomials that can be translated into homogeneous polynomials, even when the translation vector is unknown. We demonstrate that such problems in the unbounded case can have smallest optimal solutions of exponential size in the size of the input, thus requiring a compact representation of solutions for a general polynomial time algorithm for the unbounded case

Integer Polynomial Optimization in Fixed Dimension

The problem of optimizing multivariate scalar polynomial functions over mixed-integer points in polyhedra is a generalization of the well known Linear Programming (LP) problem. While LP problems have been shown to be polynomially solvable, their extension to mixed-integer points, known as Mixed-Integer Linear Programming (MILP), is NP-hard. However, if the number of integer variables is fixed, MILP is polynomially solvable. We develop algorithms for (mixed-)integer programming problems with a fixed number of variables for several classes of objective functions that are polynomials of fixed degree at least two, with some additional assumptions. With this, we are able to provide new complexity results for the given classes of problems. The first result deals with minimizing cubic polynomials over the integer points of a polyhedron in dimension two. We show that this problem is solvable in time polynomial in the input size by providing an explicit algorithm, thus extending a previous result that showed this for quadratic polynomials. We prove this for both bounded and unbounded polyhedra. The second result deals with minimizing homogeneous polynomials over the integer points of a polyhedron in dimension two. We provide an algorithm for solving this problem in time polynomial in the input size if the degree of the polynomial is fixed and the polyhedron is bounded. This result still holds when the function is the translation of a homogeneous function, even when the resulting function is not homogeneous and the translation vector is not known. We also show that if the polyhedron is unbounded, then a solution of minimal size to this problem can have exponential size in the input size. The third result deals with minimizing quadratic polynomials over the mixed-integer points of polyhedra in fixed dimension. We show that this problem is solvable in time that is polynomial in the size of the input and the maximum absolute values of the matrices defining the constraints and the objective function. We also combine this with previous results to derive a similar result for quadratic functions defined by a low-rank matrix in variable dimension. The fourth result also deals with minimizing quadratic polynomials over the integer points of a polyhedron in fixed dimension. We present a fully polynomial-time approximation scheme (FPTAS) for this problem when the objective function is homogeneous and the matrix defining it has at most one positive or at most one negative eigenvalue

Spatial scale and structure of complex life cycle trematode parasite communities in streams.

By considering the role of site-level factors and dispersal, metacommunity concepts have advanced our understanding of the processes that structure ecological communities. In dendritic systems, like streams and rivers, these processes may be impacted by network connectivity and unidirectional current. Streams and rivers are central to the dispersal of many pathogens, including parasites with complex, multi-host life cycles. Patterns in parasite distribution and diversity are often driven by host dispersal. We conducted two studies at different spatial scales (within and across stream networks) to investigate the importance of local and regional processes that structure trematode (parasitic flatworms) communities in streams. First, we examined trematode communities in first-intermediate host snails (Elimia proxima) in a survey of Appalachian headwater streams within the Upper New River Basin to assess regional turnover in community structure. We analyzed trematode communities based on both morphotype (visual identification) and haplotype (molecular identification), as cryptic diversity in larval trematodes could mask important community-level variation. Second, we examined communities at multiple sites (headwaters and main stem) within a stream network to assess potential roles of network position and downstream drift. Across stream networks, we found a broad scale spatial pattern in morphotype- and haplotype-defined communities due to regional turnover in the dominant parasite type. This pattern was correlated with elevation, but not with any other environmental factors. Additionally, we found evidence of multiple species within morphotypes, and greater genetic diversity in parasites with hosts limited to in-stream dispersal. Within network parasite prevalence, for at least some parasite taxa, was related to several site-level factors (elevation, snail density and stream depth), and total prevalence decreased from headwaters to main stem. Variation in the distribution and diversity of parasites at the regional scale may reflect differences in the abilities of hosts to disperse across the landscape. Within a stream network, species-environment relationships may counter the effects of downstream dispersal on community structure