Kernelization for Counting Problems on Graphs: Preserving the Number of Minimum Solutions

A kernelization for a parameterized decision problem $\mathcal{Q}$ is a polynomial-time preprocessing algorithm that reduces any parameterized instance $(x,k)$ into an instance $(x',k')$ whose size is bounded by a function of $k$ alone and which has the same yes/no answer for $\mathcal{Q}$. Such preprocessing algorithms cannot exist in the context of counting problems, when the answer to be preserved is the number of solutions, since this number can be arbitrarily large compared to $k$. However, we show that for counting minimum feedback vertex sets of size at most $k$, and for counting minimum dominating sets of size at most $k$ in a planar graph, there is a polynomial-time algorithm that either outputs the answer or reduces to an instance $(G',k')$ of size polynomial in $k$ with the same number of minimum solutions. This shows that a meaningful theory of kernelization for counting problems is possible and opens the door for future developments. Our algorithms exploit that if the number of solutions exceeds $2^{\mathsf{poly}(k)}$, the size of the input is exponential in terms of $k$ so that the running time of a parameterized counting algorithm can be bounded by $\mathsf{poly}(n)$. Otherwise, we can use gadgets that slightly increase $k$ to represent choices among $2^{O(k)}$ options by only $\mathsf{poly}(k)$ vertices.Comment: Extended abstract appears in the proceedings of IPEC 202

Simply Realising an Imprecise Polyline is NP-hard

We consider the problem of deciding, given a sequence of regions, if there is a choice of points, one for each region, such that the induced polyline is simple or weakly simple, meaning that it can touch but not cross itself. Specifically, we consider the case where each region is a translate of the same shape. We show that the problem is NP-hard when the shape is a unit-disk or unit-square. We argue that the problem is is NP-complete when the shape is a vertical unit-segment

Simply Realising an Imprecise Polyline is NP-hard

We consider the problem of deciding, given a sequence of regions, if there is a choice of points, one for each region, such that the induced polyline is simple or weakly simple, meaning that it can touch but not cross itself. Specifically, we consider the case where each region is a translate of the same shape. We show that the problem is NP-hard when the shape is a unit-disk or unit-square. We argue that the problem is is NP-complete when the shape is a vertical unit-segment

Simply Realising an Imprecise Polyline is NP-hard

We consider the problem of deciding, given a sequence of regions, if there is a choice of points, one for each region, such that the induced polyline is simple or weakly simple, meaning that it can touch but not cross itself. Specifically, we consider the case where each region is a translate of the same shape. We show that the problem is NP-hard when the shape is a unit-disk or unit-square. We argue that the problem is is NP-complete when the shape is a vertical unit-segment

Aortic Customize: A new alternative endovascular approach to aortic aneurysm repair using injectable biocompatible elastomer. An in vitro study

PurposeAortic Customize is a new concept for endovascular aortic aneurysm repair in which a non polymerized elastomer is injected to fill the aneurysm sac around a balloon catheter. The aim of this in vitro study was to investigate the extent of aneurysm wall stress reduction by the presence of a noncompliant elastomer cuff.MethodsA thin-walled latex aneurysm (inner radius sac 18 mm, inner radius neck 8 mm), equipped with 12 tantalum markers, was attached to an in vitro circulation model. Fluoroscopic roentgenographic stereo photogrammetric analysis (FRSA) was used to measure marker movement during six cardiac cycles. The radius of three circles drawn through the markers was measured before and after sac filling. Wall movement was measured at different systemic pressures. Wall stress was calculated from the measured radius (σ = pr/2t).ResultsThe calculated wall stress was 7.5-15.6 N/cm2 before sac filling and was diminished to 0.43-1.1 N/cm2 after sac filling. Before sac filling, there was a clear increase (P < .001) in radius of the proximal (range, 7.9%-33.5%), middle (range, 3.3%-25.2%), and distal (range, 10.5%-184.3%) rings with increasing systemic pressure. After sac filling with the elastomer, there remained a small, significant (P < .001) increase in the radius of the circles (ranges: 6.8%-8.8%; 0.7%-1.1%; 5.3%-6.7%). The sac filling reduced the extent of radius increase. The treated aneurysm withstood systemic pressures up to 220/140 mm Hg without noticeable wall movement. After the sac filling, there was no pulsation visible in the aneurysm wall.ConclusionsFilling the aneurysm sac of a simplified in vitro latex model with a biocompatible elastomer leads to successful exclusion of the aneurysm sac from the circulation. Wall movement and calculated wall stress are diminished noticeably by the injection of biocompatible elastomer.Clinical RelevanceFilling the aneurysm sac with an elastomer has a lot of potential advantages, compared with the current endovascular treatment options. To fill the sac with the biocompatible elastomer, only a fill catheter with diameter of minimal 7 F and endovascular balloons need to be introduced transfemorally to the aneurysm sac. Most stent grafts need a minimal diameter of 14 F-22 F for access to the bulky delivery sheath, which makes many aneurysms with strong tortuosity or occlusive disease of the iliac arteries ineligible for treatment. In theory, any abdominal aortic aneurysm with a deviant anatomy will become treatable, as endovascular balloons will be available in different kinds of shape and configurations. As stated above, future research must take place before this treatment option can be applied in vivo. Animal experiments will take place to prevent embolic complications during the filling process and to investigate the short- and long-term effects of the presence of the elastomer in the aorta. Research on this novel treatment concept is in full progress and will be reported in the near future