38 research outputs found
Parameterized Complexity Dichotomy for Steiner Multicut
The Steiner Multicut problem asks, given an undirected graph G, terminals
sets T1,...,Tt V(G) of size at most p, and an integer k, whether
there is a set S of at most k edges or nodes s.t. of each set Ti at least one
pair of terminals is in different connected components of G \ S. This problem
generalizes several graph cut problems, in particular the Multicut problem (the
case p = 2), which is fixed-parameter tractable for the parameter k [Marx and
Razgon, Bousquet et al., STOC 2011].
We provide a dichotomy of the parameterized complexity of Steiner Multicut.
That is, for any combination of k, t, p, and the treewidth tw(G) as constant,
parameter, or unbounded, and for all versions of the problem (edge deletion and
node deletion with and without deletable terminals), we prove either that the
problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or
even (para-)NP-complete). We highlight that:
- The edge deletion version of Steiner Multicut is fixed-parameter tractable
for the parameter k+t on general graphs (but has no polynomial kernel, even on
trees). We present two proofs: one using the randomized contractions technique
of Chitnis et al, and one relying on new structural lemmas that decompose the
Steiner cut into important separators and minimal s-t cuts.
- In contrast, both node deletion versions of Steiner Multicut are W[1]-hard
for the parameter k+t on general graphs.
- All versions of Steiner Multicut are W[1]-hard for the parameter k, even
when p=3 and the graph is a tree plus one node. Hence, the results of Marx and
Razgon, and Bousquet et al. do not generalize to Steiner Multicut.
Since we allow k, t, p, and tw(G) to be any constants, our characterization
includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1), and a
polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to
constant or unbounded).Comment: As submitted to journal. This version also adds a proof of
fixed-parameter tractability for parameter k+t using the technique of
randomized contraction
Parameterized Complexity of Equality MinCSP
We study the parameterized complexity of MinCSP for so-called equality languages, i.e., for finite languages over an infinite domain such as ?, where the relations are defined via first-order formulas whose only predicate is =. This is an important class of languages that forms the starting point of all study of infinite-domain CSPs under the commonly used approach pioneered by Bodirsky, i.e., languages defined as reducts of finitely bounded homogeneous structures. Moreover, MinCSP over equality languages forms a natural class of optimisation problems in its own right, covering such problems as Edge Multicut, Steiner Multicut and (under singleton expansion) Edge Multiway Cut. We classify MinCSP(?) for every finite equality language ?, under the natural parameter, as either FPT, W[1]-hard but admitting a constant-factor FPT-approximation, or not admitting a constant-factor FPT-approximation unless FPT=W[2]. In particular, we describe an FPT case that slightly generalises Multicut, and show a constant-factor FPT-approximation for Disjunctive Multicut, the generalisation of Multicut where the "cut requests" come as disjunctions over O(1) individual cut requests s_i ? t_i. We also consider singleton expansions of equality languages, enriching an equality language with the capability for assignment constraints (x = i) for either a finite or infinitely many constants i, and fully characterize the complexity of the resulting MinCSP
Parameterized Complexity of Critical Node Cuts
We consider the following natural graph cut problem called Critical Node Cut
(CNC): Given a graph on vertices, and two positive integers and
, determine whether has a set of vertices whose removal leaves
with at most connected pairs of vertices. We analyze this problem in the
framework of parameterized complexity. That is, we are interested in whether or
not this problem is solvable in time (i.e., whether
or not it is fixed-parameter tractable), for various natural parameters
. We consider four such parameters:
- The size of the required cut.
- The upper bound on the number of remaining connected pairs.
- The lower bound on the number of connected pairs to be removed.
- The treewidth of .
We determine whether or not CNC is fixed-parameter tractable for each of
these parameters. We determine this also for all possible aggregations of these
four parameters, apart from . Moreover, we also determine whether or not
CNC admits a polynomial kernel for all these parameterizations. That is,
whether or not there is an algorithm that reduces each instance of CNC in
polynomial time to an equivalent instance of size , where
is the given parameter
Efficient Enumerations for Minimal Multicuts and Multiway Cuts
Let be an undirected graph and let be a
set of terminal pairs. A node/edge multicut is a subset of vertices/edges of
whose removal destroys all the paths between every terminal pair in .
The problem of computing a {\em minimum} node/edge multicut is NP-hard and
extensively studied from several viewpoints. In this paper, we study the
problem of enumerating all {\em minimal} node multicuts. We give an incremental
polynomial delay enumeration algorithm for minimal node multicuts, which
extends an enumeration algorithm due to Khachiyan et al. (Algorithmica, 2008)
for minimal edge multicuts. Important special cases of node/edge multicuts are
node/edge {\em multiway cuts}, where the set of terminal pairs contains every
pair of vertices in some subset , that is, . We
improve the running time bound for this special case: We devise a polynomial
delay and exponential space enumeration algorithm for minimal node multiway
cuts and a polynomial delay and space enumeration algorithm for minimal edge
multiway cuts
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
Parameterization and approximation are two popular ways of coping with
NP-hard problems. More recently, the two have also been combined to derive many
interesting results. We survey developments in the area both from the
algorithmic and hardness perspectives, with emphasis on new techniques and
potential future research directions
Parameterized Algorithms for Min-Max Multiway Cut and List Digraph Homomorphism
In this paper we design {\sf FPT}-algorithms for two parameterized problems.
The first is \textsc{List Digraph Homomorphism}: given two digraphs and
and a list of allowed vertices of for every vertex of , the question is
whether there exists a homomorphism from to respecting the list
constraints. The second problem is a variant of \textsc{Multiway Cut}, namely
\textsc{Min-Max Multiway Cut}: given a graph , a non-negative integer
, and a set of terminals, the question is whether we can
partition the vertices of into parts such that (a) each part contains
one terminal and (b) there are at most edges with only one endpoint in
this part. We parameterize \textsc{List Digraph Homomorphism} by the number
of edges of that are mapped to non-loop edges of and we give a time
algorithm, where is the order of the host graph . We also prove that
\textsc{Min-Max Multiway Cut} can be solved in time . Our approach introduces a general problem, called
{\sc List Allocation}, whose expressive power permits the design of
parameterized reductions of both aforementioned problems to it. Then our
results are based on an {\sf FPT}-algorithm for the {\sc List Allocation}
problem that is designed using a suitable adaptation of the {\em randomized
contractions} technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk,
and Pilipczuk, FOCS 2012]).Comment: An extended abstract of this work will appear in the Proceedings of
the 10th International Symposium on Parameterized and Exact Computation
(IPEC), Patras, Greece, September 201
On Computing the Maximum Parsimony Score of a Phylogenetic Network
Phylogenetic networks are used to display the relationship of different
species whose evolution is not treelike, which is the case, for instance, in
the presence of hybridization events or horizontal gene transfers. Tree
inference methods such as Maximum Parsimony need to be modified in order to be
applicable to networks. In this paper, we discuss two different definitions of
Maximum Parsimony on networks, "hardwired" and "softwired", and examine the
complexity of computing them given a network topology and a character. By
exploiting a link with the problem Multicut, we show that computing the
hardwired parsimony score for 2-state characters is polynomial-time solvable,
while for characters with more states this problem becomes NP-hard but is still
approximable and fixed parameter tractable in the parsimony score. On the other
hand we show that, for the softwired definition, obtaining even weak
approximation guarantees is already difficult for binary characters and
restricted network topologies, and fixed-parameter tractable algorithms in the
parsimony score are unlikely. On the positive side we show that computing the
softwired parsimony score is fixed-parameter tractable in the level of the
network, a natural parameter describing how tangled reticulate activity is in
the network. Finally, we show that both the hardwired and softwired parsimony
score can be computed efficiently using Integer Linear Programming. The
software has been made freely available
A survey of parameterized algorithms and the complexity of edge modification
The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio
Designing FPT algorithms for cut problems using randomized contractions
We introduce a new technique for designing fixed-parameter algorithms for cut problems, namely randomized contractions. We apply our framework to obtain the first FPT algorithm for the Unique Label Cover problem and new FPT algorithms with exponential speed up for the Steiner Cut and Node Multiway Cut-Uncut problems. More precisely, we show the following: • We prove that the parameterized version of the Unique Label Cover problem, which is the base of the Unique Games Conjecture, can be solved in 2O(k 2 log |Σ|)n4 log n deterministic time (even in the stronger, vertex-deletion variant) where k is the number of unsatisfied edges and |Σ | is the size of the alphabet. As a consequence, we show that one can in polynomial time solve instances of Unique Games where the number of edges allowed not to be satisfied is upper bounded by O( log n) to optimality, which improves over the trivial O(1) upper bound