Fast Parallel Fixed-Parameter Algorithms via Color Coding

Fixed-parameter algorithms have been successfully applied to solve numerous difficult problems within acceptable time bounds on large inputs. However, most fixed-parameter algorithms are inherently \emph{sequential} and, thus, make no use of the parallel hardware present in modern computers. We show that parallel fixed-parameter algorithms do not only exist for numerous parameterized problems from the literature -- including vertex cover, packing problems, cluster editing, cutting vertices, finding embeddings, or finding matchings -- but that there are parallel algorithms working in \emph{constant} time or at least in time \emph{depending only on the parameter} (and not on the size of the input) for these problems. Phrased in terms of complexity classes, we place numerous natural parameterized problems in parameterized versions of AC$^0$. On a more technical level, we show how the \emph{color coding} method can be implemented in constant time and apply it to embedding problems for graphs of bounded tree-width or tree-depth and to model checking first-order formulas in graphs of bounded degree

Finding k-secluded trees faster

We revisit the k-SECLUDED TREE problem. Given a vertex-weighted undirected graph G, its objective is to find a maximum-weight induced subtree T whose open neighborhood has size at most k. We present a fixed-parameter tractable algorithm that solves the problem in time 2 O(klog⁡k)⋅n O(1), improving on a double-exponential running time from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from a single vertex, our algorithm grows a k-secluded tree by branching on vertices in the open neighborhood of the current tree T. To bound the branching depth, we prove a structural result that can be used to identify a vertex that belongs to the neighborhood of any k-secluded supertree T ′⊇T once the open neighborhood of T becomes sufficiently large. We extend the algorithm to enumerate compact descriptions of all maximum-weight k-secluded trees, which allows us to count them as well.</p

Finding k-secluded trees faster

We revisit the k-SECLUDED TREE problem. Given a vertex-weighted undirected graph G, its objective is to find a maximum-weight induced subtree T whose open neighborhood has size at most k. We present a fixed-parameter tractable algorithm that solves the problem in time 2 O(klog⁡k)⋅n O(1), improving on a double-exponential running time from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from a single vertex, our algorithm grows a k-secluded tree by branching on vertices in the open neighborhood of the current tree T. To bound the branching depth, we prove a structural result that can be used to identify a vertex that belongs to the neighborhood of any k-secluded supertree T ′⊇T once the open neighborhood of T becomes sufficiently large. We extend the algorithm to enumerate compact descriptions of all maximum-weight k-secluded trees, which allows us to count them as well.</p

Graph and Hypergraph Decompositions for Exact Algorithms

This thesis studies exact exponential and fixed-parameter algorithms for hard graph and hypergraph problems. Specifically, we study two techniques that can be used in the development of such algorithms: (i) combinatorial decompositions of both the input instance and the solution, and (ii) evaluation of multilinear forms over semirings. In the first part of the thesis we develop new algorithms for graph and hypergraph problems based on techniques (i) and (ii). While these techniques are independently both useful, the work presented in this part is largely characterised by their joint application. That is, combining results from different pieces of the decompositions often takes the from of multilinear form evaluation task, and on the other hand, decompositions offer the basic structure for dynamic-programming-style algorithms for the evaluation of multilinear forms. As main positive results of the first part, we give algorithms for three different problem families. First, we give a fast evaluation algorithm for linear forms defined by a disjointness matrix of small sets. This can be applied to obtain faster algorithms for counting maximum-weight objects of small size, such as k-paths in graphs. Second, we give a general framework for exponential-time algorithms for finding maximum-weight subgraphs of bounded tree-width, based on the theory of tree decompositions. Besides basic combinatorial problems, this framework has applications in learning Bayesian network structures. Third, we give a fixed-parameter algorithm for finding unbalanced vertex cuts, that is, vertex cuts that separate a small number of vertices from the rest of the graph. In the second part of the thesis we consider aspects of the complexity theory of linear forms over semirings, in order to better understand technique (ii). Specifically, we study how the presence of different algebraic catalysts in the ground semiring affects the complexity. As the main result, we show that there are linear forms that are easy to compute over semirings with idempotent addition, but difficult to compute over rings, unless the strong exponential time hypothesis fails.Yksi tietojenkäsittelytieteen perustavista tavoitteista on tehokkaiden algoritmien kehittäminen. Teoreettisesta näkökulmasta algoritmia yleensä pidetään tehokkaana mikäli sen ajoaika riippuu polynomisesti syötteen koosta. On kuitenkin laskennallisia ongelmia, joihin ei ole olemassa polynomiaikaisia algoritmeja. Esimerkiksi NP-kovia ongelmia ei voi ratkaista polynomisessa ajassa, mikäli yleinen vaativuusolettamus P ≠ NP pitää paikkansa. Tästä huolimatta haluaisimme kuitenkin usein ratkaista tällaisia vaikeita ongelmia. Kaksi yleistä lähestymistapaa vaikeiden, polynomisessa ajassa ratkeamattomien ongelmien tarkkaan ratkaisemiseen on (i) eksponentiaalinen algoritmiikka ja (ii) parametrisoitu algoritmiikka. Eksponentiaaliaikaisessa algoritmiikassa kehitetään algoritmeja, joiden ajoaika on edelleen eksponentiaalinen syötteen koon suhteen, mutta jotka välttävät koko ratkaisuavaruuden läpikäynnin; toisin sanoen, kyse on vähemmän eksponentiaalisten algoritmien kehittämisestä. Parametrisoitu algoritmiikka puolestaan pyrkii eristämään eksponentiaaliaikaisen riippuvuuden ajoajassa syötteen koosta riippumattomaan parametriin. Tässä väitöstyössä esitetään eksponentiaaliaikaisia ja parametrisoituja algoritmeja erinäisten vaikeiden verkko- ja hyperverkko-ongelmien tarkkaan ratkaisemiseen. Esitetyt algoritmit perustuvat kahteen algoritmiseen tekniikkaan: (i) monilineaarimuotojen evaluoiminen yli erilaisten puolirengaiden ja (ii) kombinatoristen hajotelmien käyttö. Algoritmien lisäksi työssä tarkastellaan näihin tekniikoihin liittyviä vaativuusteoreettisia kysymyksiä, mikä auttaa ymmärtämään tekniikoiden rajoituksia ja toistaiseksi hyödyntämättömiä mahdollisuuksia