3,000 research outputs found
Longest Common Separable Pattern between Permutations
In this article, we study the problem of finding the longest common separable
pattern between several permutations. We give a polynomial-time algorithm when
the number of input permutations is fixed and show that the problem is NP-hard
for an arbitrary number of input permutations even if these permutations are
separable. On the other hand, we show that the NP-hard problem of finding the
longest common pattern between two permutations cannot be approximated better
than within a ratio of (where is the size of an optimal
solution) when taking common patterns belonging to pattern-avoiding classes of
permutations.Comment: 15 page
Zero forcing number, constrained matchings and strong structural controllability
The zero forcing number is a graph invariant introduced to study the minimum
rank of the graph. In 2008, Aazami proved the NP-hardness of computing the zero
forcing number of a simple undirected graph. We complete this NP-hardness
result by showing that the non-equivalent problem of computing the zero forcing
number of a directed graph allowing loops is also NP-hard. The rest of the
paper is devoted to the strong controllability of a networked system. This kind
of controllability takes into account only the structure of the interconnection
graph, but not the interconnection strengths along the edges. We provide a
necessary and sufficient condition in terms of zero forcing sets for the strong
controllability of a system whose underlying graph is a directed graph allowing
loops. Moreover, we explain how our result differs from a recent related result
discovered by Monshizadeh et al. Finally, we show how to solve the problem of
finding efficiently a minimum-size input set for the strong controllability of
a self-damped system with a tree-structure.Comment: Submitted as a journal paper in May 201
Inapproximability of maximal strip recovery
In comparative genomic, the first step of sequence analysis is usually to
decompose two or more genomes into syntenic blocks that are segments of
homologous chromosomes. For the reliable recovery of syntenic blocks, noise and
ambiguities in the genomic maps need to be removed first. Maximal Strip
Recovery (MSR) is an optimization problem proposed by Zheng, Zhu, and Sankoff
for reliably recovering syntenic blocks from genomic maps in the midst of noise
and ambiguities. Given genomic maps as sequences of gene markers, the
objective of \msr{d} is to find subsequences, one subsequence of each
genomic map, such that the total length of syntenic blocks in these
subsequences is maximized. For any constant , a polynomial-time
2d-approximation for \msr{d} was previously known. In this paper, we show that
for any , \msr{d} is APX-hard, even for the most basic version of the
problem in which all gene markers are distinct and appear in positive
orientation in each genomic map. Moreover, we provide the first explicit lower
bounds on approximating \msr{d} for all . In particular, we show that
\msr{d} is NP-hard to approximate within . From the other
direction, we show that the previous 2d-approximation for \msr{d} can be
optimized into a polynomial-time algorithm even if is not a constant but is
part of the input. We then extend our inapproximability results to several
related problems including \cmsr{d}, \gapmsr{\delta}{d}, and
\gapcmsr{\delta}{d}.Comment: A preliminary version of this paper appeared in two parts in the
Proceedings of the 20th International Symposium on Algorithms and Computation
(ISAAC 2009) and the Proceedings of the 4th International Frontiers of
Algorithmics Workshop (FAW 2010
Complexity of Token Swapping and its Variants
In the Token Swapping problem we are given a graph with a token placed on
each vertex. Each token has exactly one destination vertex, and we try to move
all the tokens to their destinations, using the minimum number of swaps, i.e.,
operations of exchanging the tokens on two adjacent vertices. As the main
result of this paper, we show that Token Swapping is -hard parameterized
by the length of a shortest sequence of swaps. In fact, we prove that, for
any computable function , it cannot be solved in time where is the number of vertices of the input graph, unless the ETH
fails. This lower bound almost matches the trivial -time algorithm.
We also consider two generalizations of the Token Swapping, namely Colored
Token Swapping (where the tokens have different colors and tokens of the same
color are indistinguishable), and Subset Token Swapping (where each token has a
set of possible destinations). To complement the hardness result, we prove that
even the most general variant, Subset Token Swapping, is FPT in nowhere-dense
graph classes.
Finally, we consider the complexities of all three problems in very
restricted classes of graphs: graphs of bounded treewidth and diameter, stars,
cliques, and paths, trying to identify the borderlines between polynomial and
NP-hard cases.Comment: 23 pages, 7 Figure
Counting Houses of Pareto Optimal Matchings in the House Allocation Problem
Let with and be two sets. We assume that every
element has a reference list over all elements from . We call an
injective mapping from to a matching. A blocking coalition of
is a subset of such that there exists a matching that
differs from only on elements of , and every element of
improves in , compared to according to its preference list. If
there exists no blocking coalition, we call the matching an exchange
stable matching (ESM). An element is reachable if there exists an
exchange stable matching using . The set of all reachable elements is
denoted by . We show This is
asymptotically tight. A set is reachable (respectively exactly
reachable) if there exists an exchange stable matching whose image
contains as a subset (respectively equals ). We give bounds for the
number of exactly reachable sets. We find that our results hold in the more
general setting of multi-matchings, when each element of is matched
with elements of instead of just one. Further, we give complexity
results and algorithms for corresponding algorithmic questions. Finally, we
characterize unavoidable elements, i.e., elements of that are used by all
ESM's. This yields efficient algorithms to determine all unavoidable elements.Comment: 24 pages 2 Figures revise
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