329 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Clique‐width: Harnessing the power of atoms
Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class if they are so on the atoms (graphs with no clique cut-set) of . Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph is -free if is not an induced subgraph of , and it is -free if it is both -free and -free. A class of -free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for -free graphs, as evidenced by one known example. We prove the existence of another such pair and classify the boundedness of clique-width on -free atoms for all but 18 cases
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
Direct Access for Conjunctive Queries with Negation
Given a conjunctive query and a database , a direct access to
the answers of over is the operation of returning, given an
index , the answer for some order on its answers. While
this problem is -hard in general with respect to combined
complexity, many conjunctive queries have an underlying structure that allows
for a direct access to their answers for some lexicographical ordering that
takes polylogarithmic time in the size of the database after a polynomial time
precomputation. Previous work has precisely characterised the tractable classes
and given fine-grained lower bounds on the precomputation time needed depending
on the structure of the query.
In this paper, we generalise these tractability results to the case of signed
conjunctive queries, that is, conjunctive queries that may contain negative
atoms. Our technique is based on a class of circuits that can represent
relational data. We first show that this class supports tractable direct access
after a polynomial time preprocessing. We then give bounds on the size of the
circuit needed to represent the answer set of signed conjunctive queries
depending on their structure. Both results combined together allow us to prove
the tractability of direct access for a large class of conjunctive queries. On
the one hand, we recover the known tractable classes from the literature in the
case of positive conjunctive queries. On the other hand, we generalise and
unify known tractability results about negative conjunctive queries -- that is,
queries having only negated atoms. In particular, we show that the class of
-acyclic negative conjunctive queries and the class of bounded nest set
width negative conjunctive queries admit tractable direct access
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Diversity of Answers to Conjunctive Queries
Enumeration problems aim at outputting, without repetition, the set of
solutions to a given problem instance. However, outputting the entire solution
set may be prohibitively expensive if it is too big. In this case, outputting a
small, sufficiently diverse subset of the solutions would be preferable. This
leads to the Diverse-version of the original enumeration problem, where the
goal is to achieve a certain level d of diversity by selecting k solutions. In
this paper, we look at the Diverse-version of the query answering problem for
Conjunctive Queries and extensions thereof. That is, we study the problem if it
is possible to achieve a certain level d of diversity by selecting k answers to
the given query and, in the positive case, to actually compute such k answers.Comment: 34 pages, accepted to ICDT 202
An Improved Parameterized Algorithm for Treewidth
We give an algorithm that takes as input an -vertex graph and an
integer , runs in time , and outputs a tree
decomposition of of width at most , if such a decomposition exists. This
resolves the long-standing open problem of whether there is a time algorithm for treewidth. In particular, our algorithm is the
first improvement on the dependency on in algorithms for treewidth since
the time algorithm given by Bodlaender and Kloks [ICALP
1991] and Lagergren and Arnborg [ICALP 1991].
We also give an algorithm that given an -vertex graph , an integer ,
and a rational , in time
either outputs a tree decomposition of of width at most
or determines that the treewidth of is larger than . Prior to our work,
no approximation algorithms for treewidth with approximation ratio less than
, other than the exact algorithms, were known. Both of our algorithms work
in polynomial space.Comment: 57 pages, 2 figures. STOC 2023. In version v2 added a conclusion
sectio
Typical Sequences Revisited — Computing Width Parameters of Graphs
In this work, we give a structural lemma on merges of typical sequences, a notion that was introduced in 1991 [Lagergren and Arnborg, Bodlaender and Kloks, both ICALP 1991] to obtain constructive linear time parameterized algorithms for treewidth and pathwidth. The lemma addresses a runtime bottleneck in those algorithms but so far it does not lead to asymptotically faster algorithms. However, we apply the lemma to show that the cutwidth and the modified cutwidth of series parallel digraphs can be computed in polynomial time
- …