9 research outputs found
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
The Odd Cycle Transversal problem (OCT) asks whether a given graph can be
made bipartite by deleting at most of its vertices. In a breakthrough
result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a
\BigOh(4^kkmn) time algorithm for it, the first algorithm with polynomial
runtime of uniform degree for every fixed . It is known that this implies a
polynomial-time compression algorithm that turns OCT instances into equivalent
instances of size at most \BigOh(4^k), a so-called kernelization. Since then
the existence of a polynomial kernel for OCT, i.e., a kernelization with size
bounded polynomially in , has turned into one of the main open questions in
the study of kernelization.
This work provides the first (randomized) polynomial kernelization for OCT.
We introduce a novel kernelization approach based on matroid theory, where we
encode all relevant information about a problem instance into a matroid with a
representation of size polynomial in . For OCT, the matroid is built to
allow us to simulate the computation of the iterative compression step of the
algorithm of Reed, Smith, and Vetta, applied (for only one round) to an
approximate odd cycle transversal which it is aiming to shrink to size . The
process is randomized with one-sided error exponentially small in , where
the result can contain false positives but no false negatives, and the size
guarantee is cubic in the size of the approximate solution. Combined with an
\BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a
reduction of the instance to size \BigOh(k^{4.5}), implying a randomized
polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape
Brazilian Congress structural balance analysis
In this work, we study the behavior of Brazilian politicians and political
parties with the help of clustering algorithms for signed social networks. For
this purpose, we extract and analyze a collection of signed networks
representing voting sessions of the lower house of Brazilian National Congress.
We process all available voting data for the period between 2011 and 2016, by
considering voting similarities between members of the Congress to define
weighted signed links. The solutions obtained by solving Correlation Clustering
(CC) problems are the basis for investigating deputies voting networks as well
as questions about loyalty, leadership, coalitions, political crisis, and
social phenomena such as mediation and polarization.Comment: 27 pages, 15 tables, 6 figures; entire article was revised, new
references added (including international press); correcting typing error
Parameterized complexity of edge-coloured and signed graph homomorphism problems
We study the complexity of graph modification problems for homomorphism-based
properties of edge-coloured graphs. A homomorphism from an edge-coloured graph
to an edge-coloured graph is a vertex-mapping from to that
preserves adjacencies and edge-colours. We consider the property of having a
homomorphism to a fixed edge-coloured graph . Given an edge-coloured graph
, can we perform graph operations so that the resulting graph has a
homomorphism to ? The operations we consider are vertex-deletion,
edge-deletion and switching (an operation that permutes the colours of the
edges incident to a given vertex). Switching plays an important role in the
theory of signed graphs, that are -edge-coloured graphs whose colours are
and . We denote the corresponding problems (parameterized by ) by
VERTEX DELETION -COLOURING, EDGE DELETION -COLOURING and SWITCHING
-COLOURING. These generalise -COLOURING (where one has to decide if an
input graph admits a homomorphism to ). Our main focus is when has order
at most , a case that includes standard problems such as VERTEX COVER, ODD
CYCLE TRANSVERSAL and EDGE BIPARTIZATION. For such a graph , we give a
P/NP-complete complexity dichotomy for all three studied problems. Then, we
address their parameterized complexity. We show that all VERTEX DELETION
-COLOURING and EDGE DELETION -COLOURING problems for such are FPT.
This is in contrast with the fact that already for some of order~,
unless P=NP, none of the three considered problems is in XP. We show that the
situation is different for SWITCHING -COLOURING: there are three
-edge-coloured graphs of order for which this is W-hard, and
assuming the ETH, admits no algorithm in time for inputs of size
. For the other cases, SWITCHING -COLOURING is FPT.Comment: 18 pages, 8 figures, 1 table. To appear in proceedings of IPEC 201
Fixed-parameter algorithms for some combinatorial problems in bioinformatics
Fixed-parameterized algorithmics has been developed in 1990s as an approach to solve NP-hard problem optimally in a guaranteed running time. It offers a new opportunity to solve NP-hard problems exactly even on large problem instances.
In this thesis, we apply fixed-parameter algorithms to cope with three NP-hard problems in bioinformatics:
Flip Consensus Tree Problem is a combinatorial problem arising in computational phylogenetics. Using the formulation of the Flip Consensus Tree Problem as a graph-modification problem, we present a set of data reduction rules and two fixed-parameter algorithms with respect to the number of modifications. Additionally, we discuss several heuristic improvements to accelerate the running time of our algorithms in practice. We also report computational results on phylogenetic data.
Weighted Cluster Editing Problem is a graph-modification problem, that arises in computational biology when clustering objects with respect to a given similarity or distance measure. We present one of our fixed-parameter algorithms with respect to the minimum modification cost and describe the idea of our fastest algorithm for this problem and its unweighted counterpart.
Bond Order Assignment Problem asks for a bond order assignment of a molecule graph that minimizes a penalty function. We prove several complexity results on this problem and give two exact fixed-parameter algorithms for the problem. Our algorithms base on the dynamic programming approach on a tree decomposition of the molecule graph. Our algorithms are fixed-parameter with respect to the treewidth of the molecule graph and the maximum atom valence. We implemented one of our algorithms with several heuristic improvements and evaluate our algorithm on a set of real molecule graphs. It turns out that our algorithm is very fast on this dataset and even outperforms a heuristic algorithm that is usually used in practice