112 research outputs found
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
Linear-Time FPT Algorithms via Network Flow
In the area of parameterized complexity, to cope with NP-Hard problems, we
introduce a parameter k besides the input size n, and we aim to design
algorithms (called FPT algorithms) that run in O(f(k)n^d) time for some
function f(k) and constant d. Though FPT algorithms have been successfully
designed for many problems, typically they are not sufficiently fast because of
huge f(k) and d. In this paper, we give FPT algorithms with small f(k) and d
for many important problems including Odd Cycle Transversal and Almost 2-SAT.
More specifically, we can choose f(k) as a single exponential (4^k) and d as
one, that is, linear in the input size. To the best of our knowledge, our
algorithms achieve linear time complexity for the first time for these
problems. To obtain our algorithms for these problems, we consider a large
class of integer programs, called BIP2. Then we show that, in linear time, we
can reduce BIP2 to Vertex Cover Above LP preserving the parameter k, and we can
compute an optimal LP solution for Vertex Cover Above LP using network flow.
Then, we perform an exhaustive search by fixing half-integral values in the
optimal LP solution for Vertex Cover Above LP. A bottleneck here is that we
need to recompute an LP optimal solution after branching. To address this
issue, we exploit network flow to update the optimal LP solution in linear
time.Comment: 20 page
A Linear Time Parameterized Algorithm for Node Unique Label Cover
The optimization version of the Unique Label Cover problem is at the heart of
the Unique Games Conjecture which has played an important role in the proof of
several tight inapproximability results. In recent years, this problem has been
also studied extensively from the point of view of parameterized complexity.
Cygan et al. [FOCS 2012] proved that this problem is fixed-parameter tractable
(FPT) and Wahlstr\"om [SODA 2014] gave an FPT algorithm with an improved
parameter dependence. Subsequently, Iwata, Wahlstr\"om and Yoshida [2014]
proved that the edge version of Unique Label Cover can be solved in linear
FPT-time. That is, there is an FPT algorithm whose dependence on the input-size
is linear. However, such an algorithm for the node version of the problem was
left as an open problem. In this paper, we resolve this question by presenting
the first linear-time FPT algorithm for Node Unique Label Cover
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
FPT algorithms for path-transversal and cycle-transversal problems
AbstractWe study the parameterized complexity of several vertex- and edge-deletion problems on graphs, parameterized by the number p of deletions. The first kind of problems are separation problems on undirected graphs, where we aim at separating distinguished vertices in a graph. The second kind of problems are feedback set problems on group-labelled graphs, where we aim at breaking nonnull cycles in a graph having its arcs labelled by elements of a group. We obtain new FPT algorithms for these different problems, relying on a generic O∗(4p) algorithm for breaking paths of a homogeneous path system
Branch-and-Reduce Exponential/FPT Algorithms in Practice: A Case Study of Vertex Cover
We investigate the gap between theory and practice for exact branching
algorithms. In theory, branch-and-reduce algorithms currently have the best
time complexity for numerous important problems. On the other hand, in
practice, state-of-the-art methods are based on different approaches, and the
empirical efficiency of such theoretical algorithms have seldom been
investigated probably because they are seemingly inefficient because of the
plethora of complex reduction rules. In this paper, we design a
branch-and-reduce algorithm for the vertex cover problem using the techniques
developed for theoretical algorithms and compare its practical performance with
other state-of-the-art empirical methods. The results indicate that
branch-and-reduce algorithms are actually quite practical and competitive with
other state-of-the-art approaches for several kinds of instances, thus showing
the practical impact of theoretical research on branching algorithms.Comment: To appear in ALENEX 201
Improved Analysis of Highest-Degree Branching for Feedback Vertex Set
Recent empirical evaluations of exact algorithms for Feedback Vertex Set have demonstrated the efficiency of a highest-degree branching algorithm with a degree-based pruning heuristic. In this paper, we prove that this empirically fast algorithm runs in O(3.460^k n) time, where k is the solution size. This improves the previous best O(3.619^k n)-time deterministic algorithm obtained by Kociumaka and Pilipczuk (Inf. Process. Lett., 2014)
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