Resolution with Symmetry Rule Applied to Linear Equations

This paper considers the length of resolution proofs when using Krishnamurthy's classic symmetry rules. We show that inconsistent linear equation systems of bounded width over a fixed finite field $\mathbb{F}_p$ with $p$ a prime have, in their standard encoding as CNFs, polynomial length resolutions when using the local symmetry rule (SRC-II). As a consequence it follows that the multipede instances for the graph isomorphism problem encoded as CNF formula have polynomial length resolution proofs. This contrasts exponential lower bounds for individualization-refinement algorithms on these graphs. For the Cai-F\"urer-Immerman graphs, for which Tor\'an showed exponential lower bounds for resolution proofs (SAT 2013), we also show that already the global symmetry rule (SRC-I) suffices to allow for polynomial length proofs.Comment: 18 pages, to be published in STACS 202

Algorithms and Hardness for Non-Pool-Based Line Planning

Line planning, i.e. choosing paths which are operated by one vehicle end-to-end, is an important aspect of public transport planning. While there exist heuristic procedures for generating lines from scratch, most theoretical observations consider the problem of choosing lines from a predefined line pool. In this paper, we consider the complexity of the line planning problem when all simple paths can be used as lines. Depending on the cost structure, we show that the problem can be NP-hard even for paths and stars, and that no polynomial time approximation of sub-linear performance is possible. Additionally, we identify polynomially solvable cases and present a pseudo-polynomial solution approach for trees

Non-Pool-Based Line Planning on Graphs of Bounded Treewidth

Line planning, i.e. choosing routes which are to be serviced by vehicles in order to satisfy network demands, is an important aspect of public transport planning. While there exist heuristic procedures for generating lines from scratch, most theoretical investigations consider the problem of choosing lines only from a predefined line pool. We consider the line planning problem when all simple paths can be used as lines and present an algorithm which is fixed-parameter tractable, i.e. it is efficient on instances with small parameter. As a parameter we consider the treewidth of the public transport network, along with its maximum degree as well as the maximum allowed frequency

Improving mandibular reconstruction by using topology optimization, patient specific design and additive manufacturing?-A biomechanical comparison against miniplates on human specimen.

In this study, topology optimized, patient specific osteosynthesis plates (TOPOS-implants) are evaluated for the mandibular reconstruction using fibula segments. These shape optimized implants are compared to a standard treatment with miniplates (thickness: 1.0 mm, titanium grade 4) in biomechanical testing using human cadaveric specimen. Mandible and fibula of 21 body donors were used. Geometrical models were created based on automated segmentation of CT-scans of all specimens. All reconstructions, including cutting guides for osteotomy as well as TOPOS-implants, were planned using a custom-made software tool. The TOPOS-implants were produced by electron beam melting (thickness: 1.0 mm, titanium grade 5). The fibula-reconstructed mandibles were tested in static and dynamic testing in a multi-axial test system, which can adapt to the donor anatomy and apply side-specific loads. Static testing was used to confirm mechanical similarity between the reconstruction groups. Force-controlled dynamic testing was performed with a sinusoidal loading between 60 and 240 N (reconstructed side: 30% reduction to consider resected muscles) at 5 Hz for up to 5 · 105 cycles. There was a significant difference between the groups for dynamic testing: All TOPOS-implants stayed intact during all cycles, while miniplate failure occurred after 26.4% of the planned loading (1.32 · 105 ± 1.46 · 105 cycles). Bone fracture occurred in both groups (miniplates: n = 3, TOPOS-implants: n = 2). A correlation between bone failure and cortical bone thickness in mandible angle as well as the number of bicortical screws used was demonstrated. For both groups no screw failure was detected. In conclusion, the topology optimized, patient specific implants showed superior fatigue properties compared to miniplates in mandibular reconstruction. Additionally, the patient specific shape comes with intrinsic guiding properties to support the reconstruction process during surgery. This demonstrates that the combination of additive manufacturing and topology optimization can be beneficial for future maxillofacial surgery