148 research outputs found
Parameterizing edge modification problems above lower bounds
We study the parameterized complexity of a variant of the -free Editing
problem: Given a graph and a natural number , is it possible to modify
at most edges in so that the resulting graph contains no induced
subgraph isomorphic to ? In our variant, the input additionally contains a
vertex-disjoint packing of induced subgraphs of , which
provides a lower bound on the number of edge modifications
required to transform into an -free graph. While earlier works used the
number as parameter or structural parameters of the input graph , we
consider instead the parameter , that is, the number of
edge modifications above the lower bound . We develop a
framework of generic data reduction rules to show fixed-parameter tractability
with respect to for -Free Editing, Feedback Arc Set in Tournaments,
and Cluster Editing when the packing contains subgraphs with
bounded solution size. For -Free Editing, we also prove NP-hardness in
case of edge-disjoint packings of s and , while for -Free
Editing and , NP-hardness for even holds for vertex-disjoint
packings of s. In addition, we provide NP-hardness results for -free
Vertex Deletion, were the aim is to delete a minimum number of vertices to make
the input graph -free.Comment: Version accepted to Theory of Computing Systems, CSR'16 special issu
Lower bounds for the parameterized complexity of Minimum Fill-in and other completion problems
In this work, we focus on several completion problems for subclasses of
chordal graphs: Minimum Fill-In, Interval Completion, Proper Interval
Completion, Threshold Completion, and Trivially Perfect Completion. In these
problems, the task is to add at most k edges to a given graph in order to
obtain a chordal, interval, proper interval, threshold, or trivially perfect
graph, respectively. We prove the following lower bounds for all these
problems, as well as for the related Chain Completion problem: Assuming the
Exponential Time Hypothesis, none of these problems can be solved in time
2^O(n^(1/2) / log^c n) or 2^O(k^(1/4) / log^c k) n^O(1), for some integer c.
Assuming the non-existence of a subexponential-time approximation scheme for
Min Bisection on d-regular graphs, for some constant d, none of these problems
can be solved in time 2^o(n) or 2^o(sqrt(k)) n^O(1).
For all the aforementioned completion problems, apart from Proper Interval
Completion, FPT algorithms with running time of the form 2^O(sqrt(k) log k)
n^O(1) are known. Thus, the second result proves that a significant improvement
of any of these algorithms would lead to a surprising breakthrough in the
design of approximation algorithms for Min Bisection.
To prove our results, we use a reduction methodology based on combining the
classic approach of starting with a sparse instance of 3-Sat, prepared using
the Sparsification Lemma, with the existence of almost linear-size
Probabilistically Checkable Proofs (PCPs). Apart from our main results, we also
obtain lower bounds excluding the existence of subexponential algorithms for
the Optimum Linear Arrangement problem, as well as improved, yet still not
tight, lower bounds for Feedback Arc Set in Tournaments.Comment: Accepted to SODA 201
On explicit random-like tournaments
We give a new theorem describing a relation between the quasi-random property
of regular tournaments and their spectra. This provides many solutions to a
constructing problem mentioned by Erd\H{o}s and Moon (1965) and Spencer (1985).Comment: 11 pages, Section 5 in the previous verison was deleted, and a new
construction was adde
A Note on Arc-Disjoint Cycles in Bipartite Tournaments
We show that for each non-negative integer k, every bipartite tournament
either contains k arc-disjoint cycles or has a feedback arc set of size at most
7(k - 1)
k-Majority Digraphs and the Hardness of Voting with a Constant Number of Voters
Many hardness results in computational social choice make use of the fact
that every directed graph may be induced as the pairwise majority relation of
some preference profile. However, this fact requires a number of voters that is
almost linear in the number of alternatives. It is therefore unclear whether
these results remain intact when the number of voters is bounded, as is, for
example, typically the case in search engine aggregation settings. In this
paper, we provide a systematic study of majority digraphs for a constant number
of voters resulting in analytical, experimental, and complexity-theoretic
insights. First, we characterize the set of digraphs that can be induced by two
and three voters, respectively, and give sufficient conditions for larger
numbers of voters. Second, we present a surprisingly efficient implementation
via SAT solving for computing the minimal number of voters that is required to
induce a given digraph and experimentally evaluate how many voters are required
to induce the majority digraphs of real-world and generated preference
profiles. Finally, we leverage our sufficient conditions to show that the
winner determination problem of various well-known voting rules remains hard
even when there is only a small constant number of voters. In particular, we
show that Kemeny's rule is hard to evaluate for 7 voters, while previous
methods could only establish such a result for constant even numbers of voters.Comment: 45 page
Conflict Packing: an unifying technique to obtain polynomial kernels for editing problems on dense instances
We develop a technique that we call Conflict Packing in the context of
kernelization, obtaining (and improving) several polynomial kernels for editing
problems on dense instances. We apply this technique on several well-studied
problems: Feedback Arc Set in (Bipartite) Tournaments, Dense Rooted Triplet
Inconsistency and Betweenness in Tournaments. For the former, one is given a
(bipartite) tournament and seeks a set of at most arcs whose
reversal in results in an acyclic (bipartite) tournament. While a linear
vertex-kernel is already known for the first problem, using the Conflict
Packing allows us to find a so-called safe partition, the central tool of the
kernelization algorithm in, with simpler arguments. For the case of bipartite
tournaments, the same technique allows us to obtain a quadratic vertex-kernel.
Again, such a kernel was already known to exist, using the concept of so-called
bimodules. We believe however that providing an unifying technique to cope with
such problems is interesting. Regarding Dense Rooted Triplet Inconsistency, one
is given a set of vertices and a dense collection of rooted
binary trees over three vertices of and seeks a rooted tree over
containing all but at most triplets from . As a main
consequence of our technique, we prove that the Dense Rooted Triplet
Inconsistency problem admits a linear vertex-kernel. This result improves the
best known bound of vertices for this problem. Finally, we use this
technique to obtain a linear vertex-kernel for Betweenness in Tournaments,
where one is given a set of vertices and a dense collection
of so-called betweenness triplets and seeks a linear ordering of the vertices
containing all but at most triplets from
Minimum length RNA folding trajectories
The Kinfold and KFOLD programs for RNA folding kinetics implement the
Gillespie algorithm to generate stochastic folding trajectories from an initial
structure s to a target structure t, in which each intermediate secondary
structure is obtained from its predecessor by the addition, removal or shift of
a single base pair. Define MS2 distance between secondary structures s and t to
be the minimum path length to refold s to t, where a move from MS2 is applied
in each step. We describe algorithms to compute the shortest MS2 folding
trajectory between any two given RNA secondary structures. These algorithms
include an optimal integer programming (IP) algorithm, an accurate and
efficient near-optimal algorithm, a greedy algorithm, a branch-and-bound
algorithm, and an optimal algorithm if one allows intermediate structures to
contain pseudoknots. Our optimal IP [resp. near-optimal IP] algorithm maximizes
[resp. approximately maximizes] the number of shifts and minimizes [resp.
approximately minimizes] the number of base pair additions and removals by
applying integer programming to (essentially) solve the minimum feedback vertex
set (FVS) problem for the RNA conflict digraph, then applies topological sort
to tether subtrajectories into the final optimal folding trajectory. We prove
NP-hardness of the problem to determine the minimum barrier energy over all
possible MS2 folding pathways, and conjecture that computing the MS2 distance
between arbitrary secondary structures is NP-hard. Since our optimal IP
algorithm relies on the FVS, known to be NP-complete for arbitrary digraphs, we
compare the family of RNA conflict digraphs with the following classes of
digraphs (planar, reducible flow graph, Eulerian, and tournament) for which FVS
is known to be either polynomial time computable or NP-hard. Source code
available at http://bioinformatics.bc.edu/clotelab/MS2distance/.Comment: 38 pages with 26 figures and additional 11 page appendix containing 3
tables and supplementary figure
Safe sets in digraphs
A non-empty subset of the vertices of a digraph is called a {\it safe
set} if \begin{itemize}
\item[(i)] for every strongly connected component of , there exists
a strongly connected component of such that there exists an arc from
to ; and \item[(ii)] for every strongly connected component of
and every strongly connected component of , we have
whenever there exists an arc from to . \end{itemize} In the case of
acyclic digraphs a set of vertices is a safe set precisely when is an
{\it in-dominating set}, that is, every vertex not in has at least one arc
to . We prove that, even for acyclic digraphs which are traceable (have a
hamiltonian path) it is NP-hard to find a minimum cardinality in-dominating
set. Then we show that the problem is also NP-hard for tournaments and give,
for every positive constant , a polynomial algorithm for finding a minimum
cardinality safe set in a tournament on vertices in which no strong
component has size more than . Under the so called Exponential Time
Hypothesis (ETH) this is close to best possible in the following sense: If ETH
holds, then, for every there is no polynomial time algorithm for
finding a minimum cardinality safe set for the class of tournaments in which
the largest strong component has size at most .
We also discuss bounds on the cardinality of safe sets in tournaments
Feedback arc set problem and NP-hardness of minimum recurrent configuration problem of Chip-firing game on directed graphs
In this paper we present further studies of recurrent configurations of
Chip-firing games on Eulerian directed graphs (simple digraphs), a class on the
way from undirected graphs to general directed graphs. A computational problem
that arises naturally from this model is to find the minimum number of chips of
a recurrent configuration, which we call the minimum recurrent configuration
(MINREC) problem. We point out a close relationship between MINREC and the
minimum feedback arc set (MINFAS) problem on Eulerian directed graphs, and
prove that both problems are NP-hard.Comment: 18 pages, 6 figure
(Arc-disjoint) cycle packing in tournament: classical and parameterized complexity
Given a tournament , the problem MaxCT consists of finding a maximum
(arc-disjoint) cycle packing of . In the same way, MaxTT corresponds to the
specific case where the collection of cycles are triangles (i.e. directed
3-cycles). Although MaxCT can be seen as the LP dual of minimum feedback arc
set in tournaments which have been widely studied, surprisingly no algorithmic
results seem to exist concerning the former.
In this paper, we prove the NP-hardness of both MaxCT and MaxTT. We also show
that deciding if a tournament has a cycle packing and a feedback arc set with
the same size is an NP-complete problem. In light of this, we show that MaxTT
admits a vertex linear-kernel when parameterized with the size of the solution.
Finally, we provide polynomial algorithms for MaxTT and MaxCT when the
tournament is sparse, that is when it admits a FAS which is a matching.Comment: 17 pages, 2 figure
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