179 research outputs found
Counting edge-injective homomorphisms and matchings on restricted graph classes
We consider the -hard problem of counting all matchings with
exactly edges in a given input graph ; we prove that it remains
-hard on graphs that are line graphs or bipartite graphs
with degree on one side. In our proofs, we use that -matchings in line
graphs can be equivalently viewed as edge-injective homomorphisms from the
disjoint union of length- paths into (arbitrary) host graphs. Here, a
homomorphism from to is edge-injective if it maps any two distinct
edges of to distinct edges in . We show that edge-injective
homomorphisms from a pattern graph can be counted in polynomial time if
has bounded vertex-cover number after removing isolated edges. For hereditary
classes of pattern graphs, we complement this result: If the
graphs in have unbounded vertex-cover number even after deleting
isolated edges, then counting edge-injective homomorphisms with patterns from
is -hard. Our proofs rely on an edge-colored
variant of Holant problems and a delicate interpolation argument; both may be
of independent interest.Comment: 35 pages, 9 figure
Homomorphisms are a good basis for counting small subgraphs
We introduce graph motif parameters, a class of graph parameters that depend
only on the frequencies of constant-size induced subgraphs. Classical works by
Lov\'asz show that many interesting quantities have this form, including, for
fixed graphs , the number of -copies (induced or not) in an input graph
, and the number of homomorphisms from to .
Using the framework of graph motif parameters, we obtain faster algorithms
for counting subgraph copies of fixed graphs in host graphs : For graphs
on edges, we show how to count subgraph copies of in time
by a surprisingly simple algorithm. This
improves upon previously known running times, such as time
for -edge matchings or time for -cycles.
Furthermore, we prove a general complexity dichotomy for evaluating graph
motif parameters: Given a class of such parameters, we consider
the problem of evaluating on input graphs , parameterized
by the number of induced subgraphs that depends upon. For every recursively
enumerable class , we prove the above problem to be either FPT or
#W[1]-hard, with an explicit dichotomy criterion. This allows us to recover
known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms
in a uniform and simplified way, together with improved lower bounds.
Finally, we extend graph motif parameters to colored subgraphs and prove a
complexity trichotomy: For vertex-colored graphs and , where is from
a fixed class , we want to count color-preserving -copies in
. We show that this problem is either polynomial-time solvable or FPT or
#W[1]-hard, and that the FPT cases indeed need FPT time under reasonable
assumptions.Comment: An extended abstract of this paper appears at STOC 201
Counting Problems on Quantum Graphs: Parameterized and Exact Complexity Classifications
Quantum graphs, as defined by LovĂĄsz in the late 60s, are formal linear combinations of simple graphs with finite support. They allow for the complexity analysis of the problem of computing finite linear combinations of homomorphism counts, the latter of which constitute the foundation of the structural hardness theory for parameterized counting problems: The framework of parameterized counting complexity was introduced by Flum and Grohe, and McCartin in 2002 and forms a hybrid between the classical field of computational counting as founded by Valiant in the late 70s and the paradigm of parameterized complexity theory due to Downey and Fellows which originated in the early 90s.
The problem of computing homomorphism numbers of quantum graphs subsumes general motif counting problems and the complexity theoretic implications have only turned out recently in a breakthrough regarding the parameterized subgraph counting problem by Curticapean, Dell and Marx in 2017.
We study the problems of counting partially injective and edge-injective homomorphisms, counting induced subgraphs, as well as counting answers to existential first-order queries. We establish novel combinatorial, algebraic and even topological properties of quantum graphs that allow us to provide exhaustive parameterized and exact complexity classifications, including necessary, sufficient and mostly explicit tractability criteria, for all of the previous problems.Diese Arbeit befasst sich mit der Komplexit atsanalyse von mathematischen Problemen die als Linearkombinationen von Graphhomomorphismenzahlen darstellbar sind. Dazu wird sich sogenannter Quantengraphen bedient, bei denen es sich um formale Linearkombinationen von Graphen handelt und welche von Lov asz Ende der 60er eingef uhrt wurden. Die Bestimmung der Komplexit at solcher Probleme erfolgt unter dem von Flum, Grohe und McCartin im Jahre 2002 vorgestellten Paradigma der parametrisierten Z ahlkomplexit atstheorie, die als Hybrid der von Valiant Ende der 70er begr undeten klassischen Z ahlkomplexit atstheorie und der von Downey und Fellows Anfang der 90er eingef uhrten parametrisierten Analyse zu verstehen ist. Die Berechnung von Homomorphismenzahlen zwischen Quantengraphen und Graphen subsumiert im weitesten Sinne all jene Probleme, die das Z ahlen von kleinen Mustern in gro en Strukturen erfordern. Aufbauend auf dem daraus resultierenden Durchbruch von Curticapean, Dell und Marx, das Subgraphz ahlproblem betre end, behandelt diese Arbeit die Analyse der Probleme des Z ahlens von partiell- und kanteninjektiven Homomorphismen, induzierten Subgraphen, und Tre ern von relationalen Datenbankabfragen die sich als existentielle Formeln ausdr ucken lassen. Insbesondere werden dabei neue kombinatorische, algebraische und topologische Eigenschaften von Quantengraphen etabliert, die hinreichende, notwendige und meist explizite Kriterien f ur die Existenz e zienter Algorithmen liefern
Counting Problems in Parameterized Complexity
This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity. After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large graphs, and counting perfect matchings and Hamiltonian cycles in well-structured graphs.
While this survey presupposes familiarity with parameterized algorithms and complexity, we aim at explaining all relevant notions from counting complexity in a self-contained way
Counting Subgraphs in Somewhere Dense Graphs
We study the problems of counting copies and induced copies of a small
pattern graph in a large host graph . Recent work fully classified the
complexity of those problems according to structural restrictions on the
patterns . In this work, we address the more challenging task of analysing
the complexity for restricted patterns and restricted hosts. Specifically we
ask which families of allowed patterns and hosts imply fixed-parameter
tractability, i.e., the existence of an algorithm running in time for some computable function . Our main results present
exhaustive and explicit complexity classifications for families that satisfy
natural closure properties. Among others, we identify the problems of counting
small matchings and independent sets in subgraph-closed graph classes
as our central objects of study and establish the following crisp
dichotomies as consequences of the Exponential Time Hypothesis: (1) Counting
-matchings in a graph is fixed-parameter tractable if and
only if is nowhere dense. (2) Counting -independent sets in a
graph is fixed-parameter tractable if and only if
is nowhere dense. Moreover, we obtain almost tight conditional
lower bounds if is somewhere dense, i.e., not nowhere dense.
These base cases of our classifications subsume a wide variety of previous
results on the matching and independent set problem, such as counting
-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in
-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs
(Bressan, Roth; FOCS 21), as well as counting -independent sets in bipartite
graphs (Curticapean et al.; Algorithmica 19).Comment: 35 pages, 3 figures, 4 tables, abstract shortened due to ArXiv
requirement
Counting Subgraphs in Somewhere Dense Graphs
We study the problems of counting copies and induced copies of a small pattern graph H in a large host graph G. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns H. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time f(H)?|G|^O(1) for some computable function f. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes ? as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis:
- Counting k-matchings in a graph G ? ? is fixed-parameter tractable if and only if ? is nowhere dense.
- Counting k-independent sets in a graph G ? ? is fixed-parameter tractable if and only if ? is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if ? is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting k-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in F-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting k-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19).
At the same time our proofs are much simpler: using structural characterisations of somewhere dense graphs, we show that a colourful version of a recent breakthrough technique for analysing pattern counting problems (Curticapean, Dell, Marx; STOC 17) applies to any subgraph-closed somewhere dense class of graphs, yielding a unified view of our current understanding of the complexity of subgraph counting
Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths
We systematically investigate the complexity of counting subgraph patterns
modulo fixed integers. For example, it is known that the parity of the number
of -matchings can be determined in polynomial time by a simple reduction to
the determinant. We generalize this to an -time algorithm to
compute modulo the number of subgraph occurrences of patterns that are
vertices away from being matchings. This shows that the known
polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry
over into the setting of counting modulo .
Complementing our algorithm, we also give a simple and self-contained proof
that counting -matchings modulo odd integers is Mod_q-W[1]-complete and
prove that counting -paths modulo is Parity-W[1]-complete, answering an
open question by Bj\"orklund, Dell, and Husfeldt (ICALP 2015).Comment: 23 pages, to appear at ESA 202
Limits of dense graph sequences
We show that if a sequence of dense graphs has the property that for every
fixed graph F, the density of copies of F in these graphs tends to a limit,
then there is a natural ``limit object'', namely a symmetric measurable
2-variable function on [0,1]. This limit object determines all the limits of
subgraph densities. We also show that the graph parameters obtained as limits
of subgraph densities can be characterized by ``reflection positivity'',
semidefiniteness of an associated matrix. Conversely, every such function
arises as a limit object. Along the lines we introduce a rather general model
of random graphs, which seems to be interesting on its own right.Comment: 27 pages; added extension of result (Sept 22, 2004
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