We propose a simple game for modeling containment of the spread of viruses in a graph of n nodes. Each node must choose to either install anti-virus software at some known cost C, or risk infection and a loss L if a virus that starts at a random initial point in the graph can reach it without being stopped by some intermediate node. We prove many game theoretic properties of the model, including an easily applied characterization of Nash equilibria, culminating in our showing that a centralized solution can give a much better total cost than an equilibrium solution. Though it is NP-hard to compute such a social optimum, we show that the problem can be reduced to a previously unconsidered combinatorial problem that we call the sum-of-squares partition problem. Using a greedy algorithm based on sparse cuts, we show that this problem can be approximated to within a factor of O(log 1.5 n).

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Immune stimulating agents like Toll-like receptor 7 (TLR7) agonists induce potent antitumor immunity but are limited in their therapeutic window due to off-target immune activation. Here, we developed a polymeric delivery platform that binds excess unpaired cysteines on tumor cell surfaces and debris to adjuvant tumor neoantigens as an in situ vaccine. The metabolic and enzymatic dysregulation in the tumor microenvironment produces these exofacial free thiols, which can undergo efficient disulfide exchange with thiol-reactive pyridyl disulfide moieties upon intratumoral injection. These functional monomers are incorporated into a copolymer with pendant mannose groups and TLR7 agonists to target both antigen and adjuvant to antigen presenting cells. When tethered in the tumor, the polymeric glyco-adjuvant induces a robust antitumor response and prolongs survival of tumor-bearing mice, including in checkpoint-resistant B16F10 melanoma. The construct additionally reduces systemic toxicity associated with clinically relevant small molecule TLR7 agonists.

We develop a quasi-polynomial time approximation scheme for the Euclidean version of the Degree-Restricted MST Problem by adapting techniques used previously by Arora for approximating TSP. Given n points in the plane, d = 3 or 4, and ε > 0, the scheme finds an approximation with cost within 1 + ε of the lowest cost spanning tree with the property that all nodes have degree at most d.We also develop a polynomial time approximation scheme for the Euclidean version of the Red-Blue Separation Problem, again extending Arora's techniques. Given ε > 0, the scheme finds an approximation with cost within 1 + ε of the cost of the optimum separating polygon of the input nodes, in nearly linear time. Introduction.In the Degree-Restricted Minimum Spanning Tree (DRMST) problem we are given n points in R 2 (more generally, R k ) and a degree bound d ≥ 2, and have to find the spanning tree of lowest cost in which every node has degree at most d. The case d = 2 is equivalent to traveling salesman and hence NP-hard. Papadimitriou and Vazirani [15] showed NP-hardness for d = 3 in the plane, and conjectured that the problem remains NP-hard for d = 4. The problem can be solved in polynomial time for d = 5 since there always exists a minimum spanning tree with degree at most 5. We are interested in approximation algorithms for the difficult cases, both in R 2 and R k . (In R k the degree bound on the optimum spanning tree is of the form exp( (k)), so all degrees less than that are interesting cases for the problem.) This problem is the most basic of a family of well-studied problems about finding degree-constrained structures; see Raghavachari's survey [16].An approximation scheme for an NP-minimization problem is an algorithm that can, for every ε > 0, compute a solution whose cost is at most (1+ε) times the optimum. If the running time is polynomial for every fixed ε, then we say that the approximation scheme is a Polynomial Time Approximation Scheme (PTAS), and if the running time is not quite polynomial but n poly(log n) , then we say the approximation scheme is a Quasi-Polynomial Time Approximation Scheme (QPTAS).

The frequency content of the heart rate (HR) series contains information regarding the state of the autonomic nervous system. Of particular importance is respiratory sinus arrhythmia (RSA), the high-frequency fluctuation in HR attributable to respiration. The unevenly sampled nature of heart rate data, however, presents a problem for the discrete Fourier transform. Interpolation of the HR series allows even sampling, but filters high-frequency content. The Lomb periodogram (LP) is a regression-based method that addresses these issues. To evaluate the efficacy of the LP and Fourier techniques in detecting RSA, we compared the spectrum of intervals, the spectrum of HR samples, and the LP of simulated and clinical neonatal time series. We found the LP was superior to the spectrum of intervals and the spectrum of HR samples in analysis near the critical frequency of one half the average sampling rate. Applying the LP to clinical data, we found (1) evidence of stochastic resonance, an enhancement of periodicity with the addition of small amounts of noise, and (2) reduced power at all frequencies prior to clinical diagnosis of neonatal sepsis.

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