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Abstract Given graph$$G=(V,E)$$ $G=(V,E)$with vertex setVand edge setE, the maxk-cut problem seeks to partition the vertex setVinto at mostksubsets that maximize the weight (number) of edges with endpoints in different parts. This paper proposes a graph folding procedure (i.e., a procedure that reduces the number of the vertices and edges of graphG) for the weighted maxk-cut problem that may help reduce the problem’s dimensionality. While our theoretical results hold for any$$k \ge 2$$ $k\ge 2$, our computational results show the effectiveness of the proposed preprocessonlyfor$$k=2$$ $k=2$and on two sets of instances. Furthermore, we observe that the preprocess improves the performance of a MIP solver on a set of large-scale instances of the max cut problem. 

Motivated by the importance of user engagement as a crucial element in cascading leaving of users from a social network, we study identifying a largest relaxed variant of a degree-based cohesive subgraph: the maximum anchored k-core problem. Given graph [Formula: see text] and integers k and b, the maximum anchored k-core problem seeks to find a largest subset of vertices [Formula: see text] that induces a subgraph with at least [Formula: see text] vertices of degree at least k. We introduce a new integer programming (IP) formulation for the maximum anchored k-core problem and conduct a polyhedral study on the polytope of the problem. We show the linear programming relaxation of the proposed IP model is at least as strong as that of a naïve formulation. We also identify facet-defining inequalities of the IP formulation. Furthermore, we develop inequalities and fixing procedures to improve the computational performance of our IP model. We use benchmark instances to compare the computational performance of the IP model with (i) the naïve IP formulation and (ii) two existing heuristic algorithms. Our proposed IP model can optimally solve half of the benchmark instances that cannot be solved to optimality either by the naïve model or the existing heuristic approaches. Funding: This work is funded by the National Science Foundation (NSF) [Grant DMS-2318790] titled AMPS: Novel Combinatorial Optimization Techniques for Smartgrids and Power Networks. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2022.0024 . 

Distributed networked systems form an essential resource for computation and applications ranging from commercial, military, scientific, and research communities. Allocation of resources on a given infrastructure is realized through various mapping systems that are tailored towards specific use cases of the requesting applications. While HPC system requests demand compute resources heavy on processor and memory, cloud applications may demand distributed web services that are composed of networked processing and some memory. All resource requests allocate on the infrastructure with some form of network connectivity. However, during mapping of resources, the features and topology constraints of network components are typically handled indirectly through abstractions of user requests. This paper is on a novel graph representation that enables precise mapping methods for distributed networked systems. The proposed graph representations are demonstrated to allocate specific network components and adjacency requirements of a requested graph on a given infrastructure. Furthermore, we report on application of business policy requirements that resulted in increased utilization and a gradual decrease in idle node count as requests are mapped using our proposed methods. 

Beginning in the 1960s, techniques from operations research began to be used to generate political districting plans. A classical example is the integer programming model of Hess et al. [Hess SW, Weaver JB, Siegfeldt HJ, Whelan JN, Zitlau PA ( 1965 ) Oper. Res. 13(6):998–1006.]. Because of the model’s compactness-seeking objective, it tends to generate contiguous or nearly contiguous districts, although none of the model’s constraints explicitly impose contiguity. Consequently, Hess et al. had to manually adjust their solutions to make them contiguous. Since then, there have been several attempts to adjust the Hess model and other models so that contiguity is explicitly ensured. In this paper, we review two existing models for imposing contiguity, propose two new ones, and analytically compare them in terms of their strength and size. We conduct an extensive set of numerical experiments to evaluate their performance. Although many believe that contiguity constraints are particularly difficult to deal with, we find that the districting problem considered by Hess et al. does not become harder when contiguity is imposed. In fact, a branch-and-cut implementation of a cut-based model generates, for the first time, optimally compact districting plans for 21 different U.S. states at the census tract level. To encourage future research in this area, and for purposes of transparency, we make our test instances and source code publicly available. 

Two nodes of a wireless network may not be able to communicate with each other directly, perhaps because of obstacles or insufficient signal strength. This necessitates the use of intermediate nodes to relay information. Often, one designates a (preferably small) subset of them to relay these messages (i.e., to serve as a virtual backbone for the wireless network), which can be seen as a connected dominating set (CDS) of the associated graph. Ideally, these communication paths should be short, leading to the notion of a latency-constrained CDS. In this paper, we point out several shortcomings of a previously studied formalization of a latency-constrained CDS and propose an alternative one. We introduce an integer programming formulation for the problem that has a variable for each node and imposes the latency constraints via an exponential number of cut-like inequalities. Two nice properties of this formulation are that (1) it applies when distances are hop-based and when they are weighted and (2) it easily generalizes to ensure fault tolerance. We provide a branch-and-cut implementation of this formulation and compare it with a new polynomial-size formulation. Computational experiments demonstrate the superiority of the cut-like formulation. We also study related questions from computational complexity, such as approximation hardness, and answer an open problem regarding the fault diameter of graphs. 

In the article “A linear‐size zero‐one programming model for the minimum spanning tree problem in planar graphs” (Networks39(1) (2002), 53‐60), Williams introduced an extended formulation for the spanning tree polytope of a planar graph. This formulation is remarkably small (using onlyO(n) variables and constraints) and remarkably strong (defining an integral polytope). In this note, we point out that Williams' formulation, as originally stated, is incorrect. Specifically, we construct a binary feasible solution to Williams' formulation that does not represent a spanning tree. Fortunately, there is a simple fix, which is to restrict the choice of the root vertices in the primal and dual spanning trees, whereas Williams explicitly allowed them to be chosen arbitrarily. The same flaw and fix apply to a subsequent formulation of Williams (“A zero‐one programming model for contiguous land acquisition.” Geographical Analysis34(4) (2002), 330‐349).