104 research outputs found
Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels
We study two fundamental problems related to finding subgraphs: (1) given graphs G and H, Subgraph Test asks if H is isomorphic to a subgraph of G, (2) given graphs G, H, and an integer t, PACKING asks if G contains t vertex-disjoint subgraphs isomorphic to H. For every graph class F, let F-Subgraph Test and F-Packing be the special cases of the two problems where H is restricted to be in F. Our goal is to study which classes F make the two problems tractable in one of the following senses:
- (randomized) polynomial-time solvable,
- admits a polynomial (many-one) kernel (that is, has a polynomial-time preprocessing procedure that creates an equivalent instance whose size is polynomially bounded by the size of the solution), or
- admits a polynomial Turing kernel (that is, has an adaptive polynomial-time procedure that reduces the problem to a polynomial number of instances, each of which has size bounded polynomially by the size of the solution).
To obtain a more robust setting, we restrict our attention to hereditary classes F.
It is known that if every component of every graph in F has at most two vertices, then F-Packing is polynomial-time solvable, and NP-hard otherwise. We identify a simple combinatorial property (every component of every graph in F either has bounded size or is a bipartite graph with one of the sides having bounded size) such that if a hereditary class F has this property, then F-Packing admits a polynomial kernel, and has no polynomial (many-one) kernel otherwise, unless the polynomial hierarchy collapses. Furthermore, if F does not have this property, then F-Packing is either WK[1]-hard, W[1]-hard, or Long Path-hard, giving evidence that it does not admit polynomial Turing kernels either.
For F-Subgraph Test, we show that if every graph of a hereditary class F satisfies the property that it is possible to delete a bounded number of vertices such that every remaining component has size at most two, then F-Subgraph Test is solvable in randomized polynomial time and it is NP-hard otherwise. We introduce a combinatorial property called (a, b, c, d)-splittability and show that if every graph in a hereditary class F has this property, then F-Subgraph Test admits a polynomial Turing kernel and it is WK[1]-hard, W[1]-hard, or Long Path-hard otherwise. We do not give a complete characterization of the cases when F-Subgraph Test admits polynomial many-one kernels, but show examples that this question is much more fragile than the characterization for Turing kernels
Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels
We study two fundamental problems related to finding subgraphs: (1) given
graphs G and H, Subgraph Test asks if H is isomorphic to a subgraph of G, (2)
given graphs G, H, and an integer t, Packing asks if G contains t
vertex-disjoint subgraphs isomorphic to H. For every graph class F, let
F-Subgraph Test and F-Packing be the special cases of the two problems where H
is restricted to be in F. Our goal is to study which classes F make the two
problems tractable in one of the following senses:
* (randomized) polynomial-time solvable,
* admits a polynomial (many-one) kernel, or
* admits a polynomial Turing kernel (that is, has an adaptive polynomial-time
procedure that reduces the problem to a polynomial number of instances, each of
which has size bounded polynomially by the size of the solution).
We identify a simple combinatorial property such that if a hereditary class F
has this property, then F-Packing admits a polynomial kernel, and has no
polynomial (many-one) kernel otherwise, unless the polynomial hierarchy
collapses. Furthermore, if F does not have this property, then F-Packing is
either WK[1]-hard, W[1]-hard, or Long Path-hard, giving evidence that it does
not admit polynomial Turing kernels either.
For F-Subgraph Test, we show that if every graph of a hereditary class F
satisfies the property that it is possible to delete a bounded number of
vertices such that every remaining component has size at most two, then
F-Subgraph Test is solvable in randomized polynomial time and it is NP-hard
otherwise. We introduce a combinatorial property called (a,b,c,d)-splittability
and show that if every graph in a hereditary class F has this property, then
F-Subgraph Test admits a polynomial Turing kernel and it is WK[1]-hard,
W[1]-hard, or Long Path-hard, otherwise.Comment: 69 pages, extended abstract to appear in the proceedings of SODA 201
Alternative parameterizations of Metric Dimension
A set of vertices in a graph is called resolving if for any two
distinct , there is such that , where denotes the length of a shortest path
between and in the graph . The metric dimension of
is the minimum cardinality of a resolving set. The Metric Dimension problem,
i.e. deciding whether , is NP-complete even for interval
graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs)
from the lens of parameterized complexity. The problem parameterized by was
proved to be -hard by Hartung and Nichterlein (2013) and we study the
dual parameterization, i.e., the problem of whether
where is the order of . We prove that the dual parameterization admits
(a) a kernel with at most vertices and (b) an algorithm of runtime
Hartung and Nichterlein (2013) also observed that Metric
Dimension is fixed-parameter tractable when parameterized by the vertex cover
number of the input graph. We complement this observation by showing
that it does not admit a polynomial kernel even when parameterized by . Our reduction also gives evidence for non-existence of polynomial Turing
kernels
Parameterised and Fine-Grained Subgraph Counting, Modulo 2
Given a class of graphs ?, the problem ?Sub(?) is defined as follows. The input is a graph H ? ? together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs of G that are isomorphic to H. The goal of this research is to determine for which classes ? the problem ?Sub(?) is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|)?|G|^O(1).
Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that ?Sub(?) is FPT if and only if the class of allowed patterns ? is matching splittable, which means that for some fixed B, every H ? ? can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at most B vertices.
Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes ?, and (II) all tree pattern classes, i.e., all classes ? such that every H ? ? is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I)
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
The Parameterized Complexity of Degree Constrained Editing Problems
This thesis examines degree constrained editing problems within the framework of parameterized complexity. A degree constrained editing problem takes as input a graph and a set of constraints and asks whether the graph can be altered in at most k editing steps such that the degrees of the remaining vertices are within the given constraints. Parameterized complexity gives a framework for examining
problems that are traditionally considered intractable and developing efficient exact algorithms for them, or showing that it is unlikely that they have such algorithms, by introducing an additional component to the input, the parameter, which gives additional information about the structure of the problem. If the problem has an algorithm that is exponential in the parameter, but polynomial, with constant degree, in the size of the input, then it is considered to be fixed-parameter tractable.
Parameterized complexity also provides an intractability framework for identifying problems that are likely to not have such an algorithm.
Degree constrained editing problems provide natural parameterizations in terms of the total cost k of vertex deletions, edge deletions and edge additions allowed, and
the upper bound r on the degree of the vertices remaining after editing. We define a class of degree constrained editing problems, WDCE, which generalises several well know problems, such as Degree r Deletion, Cubic Subgraph, r-Regular Subgraph, f-Factor and General Factor. We show that in general if both k and r are part of the parameter, problems in the WDCE class are fixed-parameter tractable, and if parameterized by k or r alone, the problems are intractable in a parameterized sense.
We further show cases of WDCE that have polynomial time kernelizations, and in particular when all the degree constraints are a single number and the editing
operations include vertex deletion and edge deletion we show that there is a kernel with at most O(kr(k + r)) vertices. If we allow vertex deletion and edge addition,
we show that despite remaining fixed-parameter tractable when parameterized by k and r together, the problems are unlikely to have polynomial sized kernelizations, or
polynomial time kernelizations of a certain form, under certain complexity theoretic assumptions.
We also examine a more general case where given an input graph the question is whether with at most k deletions the graph can be made r-degenerate. We show that in this case the problems are intractable, even when r is a constant
Algorithm design techniques for parameterized graph modification problems
Diese Arbeit beschaeftigt sich mit dem Entwurf parametrisierter Algorithmen fuer Graphmodifikationsprobleme wie Feedback Vertex Set, Multicut in Trees, Cluster Editing und Closest 3-Leaf Powers. Anbei wird die Anwendbarkeit von vier Technicken zur Entwicklung parametrisierter Algorithmen, naemlich, Datenreduktion, Suchbaum, Iterative Kompression und Dynamische Programmierung, fuer solche Graphmodifikationsprobleme untersucht
Generalized Set and Graph Packing Problems
Many complex systems that exist in nature and society can be expressed in terms of networks (e.g.,
social networks, communication networks, biological networks, Web graph, among others). Usually a node
represents an entity while an edge represents an interaction between two entities. A community arises in a
network when two or more entities have common interests, e.g., related proteins, industrial sectors, groups
of people, documents of a collection. There exist applications that model a community as a fixed graph
H [98, 10, 119, 2, 142, 136]. Additionally, it is not expected that an entity of the network belongs to only
one community; that is, communities tend to share their members.
The community discovering or community detection problem consists on finding all communities in a
given network. This problem has been extensively studied from a practical perspective [61, 137, 122, 116].
However, we believe that this problem also brings many interesting theoretical questions. Thus in this
thesis, we will address this problem using a more rigorous approach. To that end, we first introduce
graph problems that we consider capture well the community discovering problem. These graph problems
generalize the classical H-Packing problem [88] in two different ways.
In the H-Packing with t-Overlap problem, the goal is to find in a given graph G (the network) at
least k subgraphs (the communities) isomorphic to a member of a family of graphs H (the community
models) such that each pair of subgraphs overlaps in at most t vertices (the shared members). On the
other hand, in the H-Packing with t-Membership problem instead of limiting the pairwise overlap, each
vertex of G is contained in at most t subgraphs of the solution. For both problems each member of H has
at most r vertices and m edges. An instance of the H-Packing with t-Overlap and t-Membership problems
corresponds to an instance of the H-Packing problem for t = 0 and t = 1, respectively. We also restrict
the overlap between the edges of the subgraphs in the solution instead of the vertices (called H-Packing
with t-Edge Overlap and t-Edge Membership problems).
Given the closeness of the r-Set Packing problem [87] to the H-Packing problem, we also consider
overlap in the problem of packing disjoint sets of size at most r. As usual for set packing problems, given
a collection S drawn from a universe U, we seek a sub-collection S'âS consisting of at least k sets
subject to certain disjointness restrictions. In the r-Set Packing with t-Membership, each element of U
belongs to at most t sets of S' while in the r-Set Packing with t-Overlap each pair of sets in S' overlaps
in at most t elements. For both problems, each set of S has at most r elements. We refer to all the
problems introduced in this thesis simply as packing problems with overlap. Also, we group as the family
of t-Overlap problems: H-Packing with t-Overlap, H-Packing with t-Edge Overlap, and r-Set Packing with
t-Overlap. While we call the family of t-Membership problems: H-Packing with t-Membership, H-Packing
with t-Edge Membership, and r-Set Packing with t-Membership.
The classical H-Packing and r-Set Packing problems are NP-complete [87, 88]. We will show in this
thesis that allowing overlap in a packing does not make the problems "easier". More precisely, we show
that the H-Packing with t-Membership and the r-Set Packing with t-Membership are NP-complete when
H = {H'} and H' is an arbitrary connected graph with at least three vertices and râ„3, respectively.
Parameterized complexity, introduced by Downey and Fellows [44], is an exciting and interesting approach
to deal with NP-complete problems. The underlying idea of this approach is to isolate some aspects
or parts of the input (known as the parameters) to investigate whether these parameters make the problem
tractable or intractable. The main goal of this thesis is to study the parameterized complexity of our packing
problems with overlap. We set up as a parameter k the size of the solution (number of communities),
and we consider as fixed-constants r, m and t.
We show that our problems admit polynomial kernels via two types of techniques: polynomial parametric
transformations (PPTs) [16] and classical reduction algorithms [43]. PPTs are mainly used to show
lower bounds and as far as we know they have not been used as extensively to obtain kernel results as
classical kernelization techniques [96, 42]. Thus, we believe that employing PPTs is a promising approach
to obtain kernel reductions for other problems as well. On the other hand, with non-trivial generalizations
of kernelization algorithms for the classical H-Packing problem [114], we are able to improve our kernel
sizes obtained via PPTs. These improved kernel sizes are equivalent to the kernel sizes for the disjoint
version when t = 0 and t = 1 for the t-Overlap and t-Membership problems, respectively.
We also obtain fixed-parameter algorithms for our packing problems with overlap (other than running
brute force on the kernel). Our algorithms combine a search tree and a greedy localization technique and
generalize a fixed-parameter algorithm for the problem of packing disjoint triangles [54]. In addition, we
obtain faster FPT-algorithms by transforming our overlapping problems into an instance of the disjoint
version of our problems.
Finally, we introduce the Π-Packing with α()-Overlap problem to allow for more complex overlap
constraints than the ones considered by the t-Overlap and t-Membership problems and also to include
more general communities definitions. This problem seeks at least k induced subgraphs in a graph G
subject to: each subgraph has at most r vertices and obeys a property Î (a community definition) and for
any pair of subgraphs Hi,Hj, with iâ j, we have that α(Hi,Hj) = 0 holds (an overlap constraint).
We show that the Î -Packing with α()-Overlap problem is fixed-parameter tractable provided that Î
is computable in polynomial time in n and α() obeys some natural conditions. Motivated by practical
applications we give several examples of α() functions which meet those conditions
- âŠ