143 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Complexity Framework for Forbidden Subgraphs II: When Hardness Is Not Preserved under Edge Subdivision
For a fixed set of graphs, a graph is -subgraph-free
if does not contain any as a (not necessarily induced)
subgraph. A recently proposed framework gives a complete classification on
-subgraph-free graphs (for finite sets ) 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 -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: -Induced Disjoint Paths,
-Colouring, Hamilton Cycle and Star -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 -subgraph-free graph classes
Efficient parameterized algorithms on structured graphs
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
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
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 WL for any . We analyze the
power of Local WL and prove that it is more expressive than WL and at
most as expressive as WL. We give a characterization of patterns whose
count as a subgraph and induced subgraph are invariant if two graphs are Local
WL equivalent. We also introduce two variants of WL: Layer WL and
recursive WL. These methods are more time and space efficient than applying
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 WL. The same idea can be extended further for larger patterns using
. We also compare the expressive power of Local 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]
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Hardness Transitions of Star Colouring and Restricted Star Colouring
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 , a -colouring of a graph is a function
such that for every edge of
. A -colouring of is called a -star colouring of if there is
no path in with and . A -colouring of
is called a -rs colouring of if there is no path in with
. For , the problem -STAR COLOURABILITY
takes a graph as input and asks whether admits a -star colouring.
The problem -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 -star colouring
and -rs colouring with respect to the maximum degree for all . For
, let us denote the least integer such that -STAR COLOURABILITY
(resp. -RS COLOURABILITY) is NP-complete for graphs of maximum degree by
(resp. ).
We prove that for and , -STAR COLOURABILITY is NP-complete
for graphs of maximum degree . We also show that -RS COLOURABILITY is
NP-complete for planar 3-regular graphs of girth 5 and -RS COLOURABILITY is
NP-complete for triangle-free graphs of maximum degree for .
Using these results, we prove the following: (i) for and ,
-STAR COLOURABILITY is NP-complete for -regular graphs if and only if
; and (ii) for , -RS COLOURABILITY is NP-complete
for -regular graphs if and only if
Complexity Framework for Forbidden Subgraphs {III:}: When Problems are Tractable on Subcubic Graphs
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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