143 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Complexity Framework for Forbidden Subgraphs II: When Hardness Is Not Preserved under Edge Subdivision

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    For a fixed set H{\cal H} of graphs, a graph GG is H{\cal H}-subgraph-free if GG does not contain any HHH \in {\cal H} as a (not necessarily induced) subgraph. A recently proposed framework gives a complete classification on H{\cal H}-subgraph-free graphs (for finite sets H{\cal H}) for problems that are solvable in polynomial time on graph classes of bounded treewidth, NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge subdivision. While a lot of problems satisfy these conditions, there are also many problems that do not satisfy all three conditions and for which the complexity H{\cal H}-subgraph-free graphs is unknown. In this paper, we study problems for which only the first two conditions of the framework hold (they are solvable in polynomial time on classes of bounded treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved under edge subdivision). In particular, we make inroads into the classification of the complexity of four such problems: kk-Induced Disjoint Paths, C5C_5-Colouring, Hamilton Cycle and Star 33-Colouring. Although we do not complete the classifications, we show that the boundary between polynomial time and NP-complete differs among our problems and differs from problems that do satisfy all three conditions of the framework. Hence, we exhibit a rich complexity landscape among problems for H{\cal H}-subgraph-free graph classes

    Efficient parameterized algorithms on structured graphs

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    In der klassischen Komplexitätstheorie werden worst-case Laufzeiten von Algorithmen typischerweise einzig abhängig von der Eingabegröße angegeben. In dem Kontext der parametrisierten Komplexitätstheorie versucht man die Analyse der Laufzeit dahingehend zu verfeinern, dass man zusätzlich zu der Eingabengröße noch einen Parameter berücksichtigt, welcher angibt, wie strukturiert die Eingabe bezüglich einer gewissen Eigenschaft ist. Ein parametrisierter Algorithmus nutzt dann diese beschriebene Struktur aus und erreicht so eine Laufzeit, welche schneller ist als die eines besten unparametrisierten Algorithmus, falls der Parameter klein ist. Der erste Hauptteil dieser Arbeit führt die Forschung in diese Richtung weiter aus und untersucht den Einfluss von verschieden Parametern auf die Laufzeit von bekannten effizient lösbaren Problemen. Einige vorgestellte Algorithmen sind dabei adaptive Algorithmen, was bedeutet, dass die Laufzeit von diesen Algorithmen mit der Laufzeit des besten unparametrisierten Algorithm für den größtmöglichen Parameterwert übereinstimmt und damit theoretisch niemals schlechter als die besten unparametrisierten Algorithmen und übertreffen diese bereits für leicht nichttriviale Parameterwerte. Motiviert durch den allgemeinen Erfolg und der Vielzahl solcher parametrisierten Algorithmen, welche eine vielzahl verschiedener Strukturen ausnutzen, untersuchen wir im zweiten Hauptteil dieser Arbeit, wie man solche unterschiedliche homogene Strukturen zu mehr heterogenen Strukturen vereinen kann. Ausgehend von algebraischen Ausdrücken, welche benutzt werden können, um von Parametern beschriebene Strukturen zu definieren, charakterisieren wir klar und robust heterogene Strukturen und zeigen exemplarisch, wie sich die Parameter tree-depth und modular-width heterogen verbinden lassen. Wir beschreiben dazu effiziente Algorithmen auf heterogenen Strukturen mit Laufzeiten, welche im Spezialfall mit den homogenen Algorithmen übereinstimmen.In classical complexity theory, the worst-case running times of algorithms depend solely on the size of the input. In parameterized complexity the goal is to refine the analysis of the running time of an algorithm by additionally considering a parameter that measures some kind of structure in the input. A parameterized algorithm then utilizes the structure described by the parameter and achieves a running time that is faster than the best general (unparameterized) algorithm for instances of low parameter value. In the first part of this thesis, we carry forward in this direction and investigate the influence of several parameters on the running times of well-known tractable problems. Several presented algorithms are adaptive algorithms, meaning that they match the running time of a best unparameterized algorithm for worst-case parameter values. Thus, an adaptive parameterized algorithm is asymptotically never worse than the best unparameterized algorithm, while it outperforms the best general algorithm already for slightly non-trivial parameter values. As illustrated in the first part of this thesis, for many problems there exist efficient parameterized algorithms regarding multiple parameters, each describing a different kind of structure. In the second part of this thesis, we explore how to combine such homogeneous structures to more general and heterogeneous structures. Using algebraic expressions, we define new combined graph classes of heterogeneous structure in a clean and robust way, and we showcase this for the heterogeneous merge of the parameters tree-depth and modular-width, by presenting parameterized algorithms on such heterogeneous graph classes and getting running times that match the homogeneous cases throughout

    Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs

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    For any finite set H = {H1,. .. , Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1,. .. , Hp as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of H-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set

    Improving Expressivity of Graph Neural Networks using Localization

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    In this paper, we propose localized versions of Weisfeiler-Leman (WL) algorithms in an effort to both increase the expressivity, as well as decrease the computational overhead. We focus on the specific problem of subgraph counting and give localized versions of kk-WL for any kk. We analyze the power of Local kk-WL and prove that it is more expressive than kk-WL and at most as expressive as (k+1)(k+1)-WL. We give a characterization of patterns whose count as a subgraph and induced subgraph are invariant if two graphs are Local kk-WL equivalent. We also introduce two variants of kk-WL: Layer kk-WL and recursive kk-WL. These methods are more time and space efficient than applying kk-WL on the whole graph. We also propose a fragmentation technique that guarantees the exact count of all induced subgraphs of size at most 4 using just 11-WL. The same idea can be extended further for larger patterns using k>1k>1. We also compare the expressive power of Local kk-WL with other GNN hierarchies and show that given a bound on the time-complexity, our methods are more expressive than the ones mentioned in Papp and Wattenhofer[2022a]

    Parameterized Graph Modification Beyond the Natural Parameter

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    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Parameterized Graph Modification Beyond the Natural Parameter

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    Hardness Transitions of Star Colouring and Restricted Star Colouring

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    We study how the complexity of the graph colouring problems star colouring and restricted star colouring vary with the maximum degree of the graph. Restricted star colouring (in short, rs colouring) is a variant of star colouring. For kNk\in \mathbb{N}, a kk-colouring of a graph GG is a function f ⁣:V(G)Zkf\colon V(G)\to \mathbb{Z}_k such that f(u)f(v)f(u)\neq f(v) for every edge uvuv of GG. A kk-colouring of GG is called a kk-star colouring of GG if there is no path u,v,w,xu,v,w,x in GG with f(u)=f(w)f(u)=f(w) and f(v)=f(x)f(v)=f(x). A kk-colouring of GG is called a kk-rs colouring of GG if there is no path u,v,wu,v,w in GG with f(v)>f(u)=f(w)f(v)>f(u)=f(w). For kNk\in \mathbb{N}, the problem kk-STAR COLOURABILITY takes a graph GG as input and asks whether GG admits a kk-star colouring. The problem kk-RS COLOURABILITY is defined similarly. Recently, Brause et al. (Electron. J. Comb., 2022) investigated the complexity of 3-star colouring with respect to the graph diameter. We study the complexity of kk-star colouring and kk-rs colouring with respect to the maximum degree for all k3k\geq 3. For k3k\geq 3, let us denote the least integer dd such that kk-STAR COLOURABILITY (resp. kk-RS COLOURABILITY) is NP-complete for graphs of maximum degree dd by Ls(k)L_s^{(k)} (resp. Lrs(k)L_{rs}^{(k)}). We prove that for k=5k=5 and k7k\geq 7, kk-STAR COLOURABILITY is NP-complete for graphs of maximum degree k1k-1. We also show that 44-RS COLOURABILITY is NP-complete for planar 3-regular graphs of girth 5 and kk-RS COLOURABILITY is NP-complete for triangle-free graphs of maximum degree k1k-1 for k5k\geq 5. Using these results, we prove the following: (i) for k4k\geq 4 and dk1d\leq k-1, kk-STAR COLOURABILITY is NP-complete for dd-regular graphs if and only if dLs(k)d\geq L_s^{(k)}; and (ii) for k4k\geq 4, kk-RS COLOURABILITY is NP-complete for dd-regular graphs if and only if Lrs(k)dk1L_{rs}^{(k)}\leq d\leq k-1

    Complexity Framework for Forbidden Subgraphs {III:}: When Problems are Tractable on Subcubic Graphs

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    For any finite set H={H1,…,Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1,…,Hp as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity is determined on classes of H-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most~3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set
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