Integer Linear Programming (ILP) has a broad range of applications in various areas of artificial intelligence. Yet in spite of recent advances, we still lack a thorough understanding of which structural restrictions make ILP tractable. Here we study ILP instances consisting of a small number of "global" variables and/or constraints such that the remaining part of the instance consists of small and otherwise independent components; this is captured in terms of a structural measure we call fracture backdoors which generalizes, for instance, the well-studied class of N -fold ILP instances. Our main contributions can be divided into three parts. First, we formally develop fracture backdoors and obtain exact and approximation algorithms for computing these. Second, we exploit these backdoors to develop several new parameterized algorithms for ILP; the performance of these algorithms will naturally scale based on the number of global variables or constraints in the instance. Finally, we complement the developed algorithms with matching lower bounds. Altogether, our results paint a near-complete complexity landscape of ILP with respect to fracture backdoors.

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BackgroundThe role of adjuvant treatment in the intermediate-risk group of patients with early-stage cervical cancer is controversial and is supported by a single randomized Gynecologic Oncology Group (GOG) 92 study performed more than 20 years ago. Recent retrospective studies have shown excellent local control in this group of patients after radical surgery with no additional adjuvant treatment.Primary ObjectiveTo evaluate if adjuvant (chemo)radiation is associated with a survival benefit after radical surgery in patients with intermediate-risk cervical cancer.Study HypothesisRadical surgery alone is non-inferior to the combined treatment of radical surgery followed by adjuvant (chemo)radiation in disease-free survival in patients with intermediate-risk cervical cancer.Trial DesignThis is a phase III, international, multicenter, randomized, non-inferiority trial in which patients with intermediate-risk cervical cancer will be randomized 1:1 into arm A, with no additional treatment after radical surgery, and arm B, receiving adjuvant external beam radiotherapy±brachytherapy ± concomitant chemotherapy. Patient data will be collected over 3 years post-randomization of the last enrolled patient for primary endpoint analysis or for 6 years for the overall survival analysis.Major Inclusion/Exclusion CriteriaPatients with intermediate-risk early-stage cervical cancer (IB1–IIA), defined as lymph node-negative patients with a combination of negative prognostic factors (tumor size >4 cm; tumor size >2 cm and lymphovascular space invasion; deep stromal invasion >2/3; or tumor-free distance <3 mm) with squamous cell carcinoma or human papillomavirus (HPV)-related adenocarcinoma, are eligible for the trial.Primary EndpointDisease-free survival defined as time from randomization to recurrence diagnosis.Sample Size514 patients from up to 90 sites will be randomized.Estimated Dates for Completing Accrual and Presenting ResultsIt is estimated that the accrual will be completed by 2027 (with 3 additional years of follow-up) and primary endpoint results will be published by 2031. Estimated trial completion is by 2034.Trial RegistrationNCT04989647.

We show that the Minimal Length-Bounded L-But problem can be computed in linear time with respect to L and the tree-width of the input graph as parameters. In this problem the task is to find a set of edges of a graph such that after removal of this set, the shortest path between two prescribed vertices is at least L long. We derive an FPT algorithm for a more general multi-commodity length bounded cut problem when parameterized by the number of terminals also. For the former problem we show a W[1]-hardness result when the parameterization is done by the path-width only (instead of the tree-width) and that this problem does not admit polynomial kernel when parameterized by tree-width and L. We also derive an FPT algorithm for the Minimal Length-Bounded Cut problem when parameterized by the tree-depth. Thus showing an interesting paradigm for this problem and parameters tree-depth and path-width.