Almost Linear Time Algorithms for Some Problems on Dynamic Flow Networks

Motivated by evacuation planning, several problems regarding dynamic flow networks have been studied in recent years. A dynamic flow network consists of an undirected graph with positive edge lengths, positive edge capacities, and positive vertex weights. The road network in an area can be treated as a graph where the edge lengths are the distances along the roads and the vertex weights are the number of people at each site. An edge capacity limits the number of people that can enter the edge per unit time. In a dynamic flow network, when particular points on edges or vertices called sinks are given, all of the people are required to evacuate from the vertices to the sinks as quickly as possible. This chapter gives an overview of two of our recent results on the problem of locating multiple sinks in a dynamic flow path network such that the max/sum of evacuation times for all the people to sinks is minimized, and we focus on techniques that enable the problems to be solved in almost linear time.

W[1]-hardness of the k-center problem parameterized by the skeleton dimension

We study the k-Center problem, where the input is a graph $$G=(V,E)$$ G = ( V , E ) with positive edge weights and an integer k, and the goal is to select k center vertices $$C \subseteq V$$ C ⊆ V such that the maximum distance from any vertex to the closest center vertex is minimized. In general, this problem is $$\mathsf {NP}$$ NP -hard and cannot be approximated within a factor less than 2. Typical applications of the k-Center problem can be found in logistics or urban planning and hence, it is natural to study the problem on transportation networks. Common characterizations of such networks are graphs that are (almost) planar or have low doubling dimension, highway dimension or skeleton dimension. It was shown by Feldmann and Marx that k-Center is $$\mathsf {W[1]}$$ W [ 1 ] -hard on planar graphs of constant doubling dimension when parameterized by the number of centers k, the highway dimension $$hd$$ hd and the pathwidth $$pw$$ pw (Feldmann and Marx 2020). We extend their result and show that even if we additionally parameterize by the skeleton dimension $$\kappa $$ κ , the k-Center problem remains $$\mathsf {W[1]}$$ W [ 1 ] -hard. Moreover, we prove that under the Exponential Time Hypothesis there is no exact algorithm for k-Center that has runtime $$f(k,hd,pw,\kappa ) \cdot \vert V \vert ^{o(pw+ \kappa + \sqrt{k+hd})}$$ f ( k , h d , p w , κ ) · | V | o ( p w + κ + k + h d ) for any computable function f.

Equilibrium measure for a nonlocal dislocation energy with physical confinement

In this paper, we characterize the equilibrium measure for a family of nonlocal and anisotropic energies I α I_{\alpha} that describe the interaction of particles confined in an elliptic subset of the plane. The case α = 0 \alpha=0 corresponds to purely Coulomb interactions, while the case α = 1 \alpha=1 describes interactions of positive edge dislocations in the plane. The anisotropy into the energy is tuned by the parameter 𝛼 and favors the alignment of particles. We show that the equilibrium measure is completely unaffected by the anisotropy and always coincides with the optimal distribution in the case α = 0 \alpha=0 of purely Coulomb interactions, which is given by an explicit measure supported on the boundary of the elliptic confining domain. Our result does not seem to agree with the mechanical conjecture that positive edge dislocations at equilibrium tend to arrange themselves along “wall-like” structures. Moreover, this is one of the very few examples of explicit characterization of the equilibrium measure for nonlocal interaction energies outside the radially symmetric case.

TREATMENT OPTIONS OF BREAST CANCER PATIENTS AFTER BREAST-CONSERVING SURGERY WITH POSITIVE MARGINS R1

Introduction: in organ-preserving operations for breast cancer, the risk of recurrence is associated with many factors, including positive resection margins. The article presents data from the literature, which considers the risk of relapse depending on the positive, close and negative edges of resection. The purpose of this study was to increase the effectiveness of treatment of breast cancer patients after organ-preserving operations with positive resection edges. Materials and methods: the study included 1219 patients with breast cancer who underwent organ-preserving and on-coplastic resections of the breast. Urgent cytological and histological intraoperative examination of the resection edges is analyzed in detail, and the marking of the resection edges is presented. The clinical and morphological characteristics of patients with breast cancer at R0 and R1 are presented. Results: positive edge of R1 resection was diagnosed in 53 cases, which was 4,3±2,8%, in oncoplastic resections in 4,1±1,1%, in classical breast resections in 4,6±0,7% (p>0,05). In the group of patients with R1, multicentricity of the tumor was diagnosed in 11,1±5,3%, and monocentric tumor was detected in 4,1±0,5%. Further tactics in the case of R1 detection were as follows: in 21 cases, radiation therapy was performed on the breast, in 32 cases, re - operation: resection of the edges - in 14 patients, radical mastectomy - in 9 patients, subcutaneous mastectomy with simultaneous reconstruction with autologous flaps or endoprostheses - in 9 patients. In the group of patients with re-operation, 43,7% of the planned study showed no signs of malignancy, 56,3% showed a residual tumor, while 31.3% were diagnosed with cancer in situ. So in the case of resection of the edges in 5 cases, a residual tumor was diagnosed in the resected edges, which was 35,7%, and in the case of mastectomy, a residual tumor was detected in 68,4%. Re-operation in R1 after oncoplastic resections was performed in 71,4%, and in classical resections in 56,4%, which correlates with the literature data. Conclusion: in two groups of patients after organ-preserving operations with positive resection margins, no local recurrence was detected during the follow-up period from 1 to 60 months, and distant metastases, namely, lesions of the bones of the skeleton, were diagnosed in 2 patients. Thus, with a positive edge of resection after organ-preserving surgery, both surgical treatment and radiation therapy can be performed. Although the presence of a tumor in the colored edges of resection is clearly associated with a high frequency of local relapses, but the relapse is also influenced by the biological characteristics of the tumor and the body. Individual characteristics are the basis of tumor biology, and the extent of their influence on long-term results is not reduced due to the wide surgical margins of resection.