23 research outputs found
Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts
A skew-symmetric graph is a directed graph with an
involution on the set of vertices and arcs. In this paper, we
introduce a separation problem, -Skew-Symmetric Multicut, where we are given
a skew-symmetric graph , a family of of -sized subsets of
vertices and an integer . The objective is to decide if there is a set
of arcs such that every set in the family has a vertex
such that and are in different connected components of
. In this paper, we give an algorithm for
this problem which runs in time , where is the
number of arcs in the graph, the number of vertices and the length
of the family given in the input.
Using our algorithm, we show that Almost 2-SAT has an algorithm with running
time and we obtain algorithms for {\sc Odd Cycle Transversal}
and {\sc Edge Bipartization} which run in time and
respectively. This resolves an open problem posed by Reed,
Smith and Vetta [Operations Research Letters, 2003] and improves upon the
earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010].
We also show that Deletion q-Horn Backdoor Set Detection is a special case of
3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor
Set Detection which runs in time . This gives the first
fixed-parameter tractable algorithm for this problem answering a question posed
in a paper by a superset of the authors [STACS, 2013]. Using this result, we
get an algorithm for Satisfiability which runs in time where
is the size of the smallest q-Horn deletion backdoor set, with being
the length of the input formula
Parameterized Approximation Schemes for Independent Set of Rectangles and Geometric Knapsack
The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+epsilon)-approximations in f(k,epsilon)n^O(1) time where k is some parameter of the input. The goal is to overcome lower bounds from either of the areas. We obtain the following results on parameterized approximability:
- In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA\u2705]. The best-known polynomial-time approximation factor is O(log log n) [Chalermsook and Chuzhoy, SODA\u2709] and it admits a QPTAS [Adamaszek and Wiese, FOCS\u2713; Chuzhoy and Ene, FOCS\u2716]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant epsilon>0 and integer k>0, in time f(k,epsilon)n^g(epsilon), either outputs a solution of size at least k/(1+epsilon), or declares that the optimum solution has size less than k.
- In the (2-dimensional) geometric knapsack problem (2DK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of 2DK with rotations (2DKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factor is 2+epsilon [Jansen and Zhang, SODA\u2704]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA\u2715]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for 2DKR.
For all considered problems, getting time f(k,epsilon)n^O(1), rather than f(k,epsilon)n^g(epsilon), would give FPT time f\u27(k)n^O(1) exact algorithms by setting epsilon=1/(k+1), contradicting W[1]-hardness. Instead, for each fixed epsilon>0, our PASs give (1+epsilon)-approximate solutions in FPT time.
For both MISR and 2DKR our techniques also give rise to preprocessing algorithms that take n^g(epsilon) time and return a subset of at most k^g(epsilon) rectangles/items that contains a solution of size at least k/(1+epsilon) if a solution of size k exists. This is a special case of the recently introduced notion of a polynomial-size approximate kernelization scheme [Lokshtanov et al., STOC\u2717]
FPT-approximation for FPT problems
Over the past decade, many results have focused on the design of parameterized approximation algorithms for W[1]-hard problems. However, there are fundamental problems within the class FPT for which the best known algorithms have seen no progress over the course of the decade. In this paper, we expand the study of FPT-approximation and initiate a systematic study of FPT-approximation for problems that are in FPT. We design FPT-approximation algorithms for problems that are in FPT, with running times that are significantly faster than the corresponding best known FPT-algorithm, and while achieving approximation ratios that are significantly better than what is possible in polynomial time
Half-integrality, LP-branching and FPT Algorithms
A recent trend in parameterized algorithms is the application of polytope
tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011;
Narayanaswamy et al., 2012). However, although interesting results have been
achieved, the methods require the underlying polytope to have very restrictive
properties (half-integrality and persistence), which are known only for few
problems (essentially Vertex Cover (Nemhauser and Trotter, 1975) and Node
Multiway Cut (Garg et al., 1994)). Taking a slightly different approach, we
view half-integrality as a \emph{discrete} relaxation of a problem, e.g., a
relaxation of the search space from to such that
the new problem admits a polynomial-time exact solution. Using tools from CSP
(in particular Thapper and \v{Z}ivn\'y, 2012) to study the existence of such
relaxations, we provide a much broader class of half-integral polytopes with
the required properties, unifying and extending previously known cases.
In addition to the insight into problems with half-integral relaxations, our
results yield a range of new and improved FPT algorithms, including an
-time algorithm for node-deletion Unique Label Cover with
label set and an -time algorithm for Group Feedback Vertex
Set, including the setting where the group is only given by oracle access. All
these significantly improve on previous results. The latter result also implies
the first single-exponential time FPT algorithm for Subset Feedback Vertex Set,
answering an open question of Cygan et al. (2012).
Additionally, we propose a network flow-based approach to solve some cases of
the relaxation problem. This gives the first linear-time FPT algorithm to
edge-deletion Unique Label Cover.Comment: Added results on linear-time FPT algorithms (not present in SODA
paper
Edge Bipartization Faster Than 2^k
In the Edge Bipartization problem one is given an undirected graph and an
integer , and the question is whether edges can be deleted from so
that it becomes bipartite. In 2006, Guo et al. [J. Comput. Syst. Sci.,
72(8):1386-1396, 2006] proposed an algorithm solving this problem in time
; today, this algorithm is a textbook example of an application of
the iterative compression technique. Despite extensive progress in the
understanding of the parameterized complexity of graph separation problems in
the recent years, no significant improvement upon this result has been yet
reported.
We present an algorithm for Edge Bipartization that works in time , which is the first algorithm with the running time dependence on the
parameter better than . To this end, we combine the general iterative
compression strategy of Guo et al. [J. Comput. Syst. Sci., 72(8):1386-1396,
2006], the technique proposed by Wahlstrom [SODA 2014, 1762-1781] of using a
polynomial-time solvable relaxation in the form of a Valued Constraint
Satisfaction Problem to guide a bounded-depth branching algorithm, and an
involved Measure & Conquer analysis of the recursion tree
Combinatorial Optimization
Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Measure-Driven Algorithm Design and Analysis: A New Approach for Solving NP-hard Problems
NP-hard problems have numerous applications in various fields such as networks,
computer systems, circuit design, etc. However, no efficient algorithms have
been found for NP-hard problems. It has been commonly believed that no efficient algorithms
for NP-hard problems exist, i.e., that P6=NP. Recently, it has been observed
that there are parameters much smaller than input sizes in many instances of NP-hard
problems in the real world. In the last twenty years, researchers have been interested
in developing efficient algorithms, i.e., fixed-parameter tractable algorithms, for those
instances with small parameters. Fixed-parameter tractable algorithms can practically
find exact solutions to problem instances with small parameters, though those
problems are considered intractable in traditional computational theory.
In this dissertation, we propose a new approach of algorithm design and analysis:
discovering better measures for problems. In particular we use two measures instead of
the traditional single measure?input size to design algorithms and analyze their time
complexity. For several classical NP-hard problems, we present improved algorithms
designed and analyzed with this new approach,
First we show that the new approach is extremely powerful for designing fixedparameter
tractable algorithms by presenting improved fixed-parameter tractable algorithms
for the 3D-matching and 3D-packing problems, the multiway cut problem, the feedback vertex set problems on both directed and undirected
graph and the max-leaf problems on both directed and undirected graphs. Most of
our algorithms are practical for problem instances with small parameters.
Moreover, we show that this new approach is also good for designing exact algorithms
(with no parameters) for NP-hard problems by presenting an improved exact
algorithm for the well-known satisfiability problem.
Our results demonstrate the power of this new approach to algorithm design and
analysis for NP-hard problems. In the end, we discuss possible future directions on
this new approach and other approaches to algorithm design and analysis