Planar Subgraph Isomorphism Revisited

The problem of Subgraph Isomorphism is defined as follows: Given a pattern H and a host graph G on n vertices, does G contain a subgraph that is isomorphic to H? Eppstein [SODA 95, J'GAA 99] gives the first linear time algorithm for subgraph isomorphism for a fixed-size pattern, say of order k, and arbitrary planar host graph, improving upon the O(n^\sqrt{k})-time algorithm when using the ``Color-coding'' technique of Alon et al [J'ACM 95]. Eppstein's algorithm runs in time k^O(k) n, that is, the dependency on k is superexponential. We solve an open problem posed in Eppstein's paper and improve the running time to 2^O(k) n, that is, single exponential in k while keeping the term in n linear. Next to deciding subgraph isomorphism, we can construct a solution and enumerate all solutions in the same asymptotic running time. We may list w subgraphs with an additive term O(w k) in the running time of our algorithm. We introduce the technique of "embedded dynamic programming" on a suitably structured graph decomposition, which exploits the topology of the underlying embeddings of the subgraph pattern (rather than of the host graph). To achieve our results, we give an upper bound on the number of partial solutions in each dynamic programming step as a function of pattern size--as it turns out, for the planar subgraph isomorphism problem, that function is single exponential in the number of vertices in the pattern.Comment: 13 pages, 4 figure

An FPT Algorithm for Directed Spanning k-Leaf

An out-branching of a directed graph is a rooted spanning tree with all arcs directed outwards from the root. We consider the problem of deciding whether a given directed graph D has an out-branching with at least k leaves (Directed Spanning k-Leaf). We prove that this problem is fixed parameter tractable, when k is chosen as the parameter. Previously this was only known for restricted classes of directed graphs. The main new ingredient in our approach is a lemma that shows that given a locally optimal out-branching of a directed graph in which every arc is part of at least one out-branching, either an out-branching with at least k leaves exists, or a path decomposition with width O(k^3) can be found. This enables a dynamic programming based algorithm of running time 2^{O(k^3 \log k)} n^{O(1)}, where n=|V(D)|.Comment: 17 pages, 8 figure

Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs

We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc $k$-Leaf Out-Branching}, which is to find an oriented spanning tree with at least $k$ leaves, we obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} n+ n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed graph $H$ as a minor. For the special case when the input directed graph is planar, the running time can be improved to $2^{O(\sqrt{k})}n + n^{O(1)}$. The second example is a generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc $k$-Internal Out-Branching}, which is to find an oriented spanning tree with at least $k$ internal vertices. We obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} + n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed apex graph $H$ as a minor. Finally, we observe that for any $\epsilon>0$, the {\sc $k$-Directed Path} problem is solvable in time $O((1+\epsilon)^k n^{f(\epsilon)})$, where $f$ is some function of \ve. Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs

Carcinoma and multiple lymphomas in one patient: establishing the diagnoses and analyzing risk factors

Multiple malignancies may occur in the same patient, and a few reports describe cases with multiple hematologic and non-hematologic neoplasms. We report the case of a patient who showed the sequential occurrence of four different lymphoid neoplasms together with a squamous cell carcinoma of the lung. A 62-year-old man with adenopathy was admitted to the hospital, and lymph node biopsy was positive for low-grade follicular lymphoma. He achieved a partial remission with chemotherapy. Two years later, a PET-CT scan showed a left hilar mass in the lung; biopsy showed a squamous cell carcinoma. Simultaneously, he was diagnosed with diffuse large B cell lymphoma in a neck lymph node; after chemo- and radiotherapy, he achieved a complete response. A restaging PET-CT scan 2 years later revealed a retroperitoneal nodule, and biopsy again showed a low-grade follicular lymphoma, while a biopsy of a cutaneous scalp lesion showed a CD30-positive peripheral T cell lymphoma. After some months, a liver biopsy and a right cervical lymph node biopsy showed a CD30-positive peripheral T cell lymphoma consistent with anaplastic lymphoma kinase-negative anaplastic large cell lymphoma. Flow cytometry and cytogenetic and molecular genetic analysis performed at diagnosis and during the patient’s follow-up confirmed the presence of two clonally distinct B cell lymphomas, while the two T cell neoplasms were confirmed to be clonally related. We discuss the relationship between multiple neoplasms occurring in the same patient and the various possible risk factors involved in their development

Designing Subexponential Algorithms: Problems, Techniques & Structures

In this thesis we focus on subexponential algorithms for NP-hard graph problems: exact and parameterized algorithms that have a truly subexponential running time behavior. For input instances of size n we study exact algorithms with running time 2O(√n) and parameterized algorithms with running time 2O(√k) ·nO(1) with parameter k, respectively. We study a class of problems for which we design such algorithms for three different types of graph classes: planar graphs, graphs of bounded genus, and H-minor-free graphs. We distinguish between unconnected and connected problems, and discuss how to conceive parameterized and exact algorithms for such problems. We improve upon existing dynamic programming techniques used in algorithms solving those problems. We compare tree-decomposition and branch-decomposition based dynamic programming algorithms and show how to unify both algorithms to one single algorithm. Then we give a dynamic programming technique that reduces much of the computation involved to fast matrix multiplication. In this manner, we obtain branch-decomposition based algorithms on numerous problems, such as Vertex Cover and Dominating Set. We also show how to exploit planarity for obtaining faster dynamic programming approaches, a) in connection with fast matrix multiplication and b) for tree-decompositions. Furthermore, we focus on connected problems in particular, and their relation to the input graph structure. We state the basis for how the latter problems can be attacked for graph classes that inherit the Catalan structure. Truly subexponential algorithms for edge-subset problems such as k-Longest Path and Planar Graph TSP are derived by employing the planar graph structure. Moreover, we investigate how to obtain truly subexponential algorithms for torus-embedded graphs, bounded genus graph and Hminor- free graphs, by first using planarization techniques, and then proving the Catalan structure for the planarized instances

Designing Subexponential Algorithms: Problems, Techniques & Structures

In this thesis we focus on subexponential algorithms for NP-hard graph problems: exact and parameterized algorithms that have a truly subexponential running time behavior. For input instances of size n we study exact algorithms with running time 2 O( √ n) and parameterized algorithms with running time 2 O( √ k) ·n O(1) with parameter k, respectively. We study a class of problems for which we design such algorithms for three different types of graph classes: planar graphs, graphs of bounded genus, and H-minor-free graphs. We distinguish between unconnected and connected problems, and discuss how to conceive parameterized and exact algorithms for such problems. We improve upon existing dynamic programming techniques used in algorithms solving those problems. We compare tree-decomposition and branch-decomposition based dynamic programming algorithms and show how to unify both algorithms to one single algorithm. Then we give a dynamic programming technique that reduces much of the computation involved to fast matrix multiplication. In this manner, we obtain branch-decomposition based algorithms on numerous problems, such as Vertex Cover and Dominating Set. We also show how to exploit planarity for obtaining faster dynamic programming approaches, a) in connection with fast matrix multiplication and b) for tree-decompositions. Furthermore, we focus on connected problems in particular, and their relation to the input graph structure. We state the basis for how the latter problems can be attacked for graph classes that inherit the Catalan structure. Truly subexponential algorithms for edge-subset problems such as k-Longest Path and Planar Graph TSP are derived by employing the planar graph structure. Moreover, we investigate how to obtain truly subexponential algorithms for torus-embedded graphs, bounded genus graph and Hminor-free graphs, by first using planarization techniques, and then proving the Catalan structure for the planarized instances. Preface “So Long, and Thanks for All the Fish” All work on this thesis has been funded mainly by Exact Algorithms for Hard Problem

UNIVERSITAS

How to use planarity efficiently: new tree-decomposition based algorithm

Modelling Minimum Pressure Height in Short-term Hydropower Production Planning

-When planning the production for certain hydropower plants, minimum pressure is one of the major critical points. Violation of the minimum pressure causes the power plant to automatically shut down, hence violating the obligations of the plant. Automatic pressure switches and pressure constraints are difficult to model in particular when embedded in a complex water way. This problem is expected to increase when retrofitting hydro installations with new parallel units and increased exploitation of inflow resources. From a scheduling point of view, however, such switches become hard to integrate in an optimal operation plan as the constraint depends on the system state. This paper introduces a novelty in short-term production planning, namely a solution for modelling minimum pressure height in regulated watercourses when optimizing the energy production of hydropower plants. This solution is integrated in the short-term hydropower scheduling tool SHOP. The tool finds an optimal strategy to run a power station with such minimum pressure restrictions and the state dependent topological couplings within the water system. We apply the model on a complex topology, the Sira-Kvina water system, where Norway's largest hydropower station Tonstad Kraftstajon is operationally subject to this rigorous pressure constraint. First, in order to illustrate the concepts of the model, we apply the model on a simplified water course including one reservoir. Next, the outcome and tests are demonstrated on the final model of two reservoirs whose respective outflows are joining together above the pressure gauge, as found in the Sira-Kvina water system