111 research outputs found

### Linear-Time Kernelization for Feedback Vertex Set

In this paper, we give an algorithm that, given an undirected graph G of m edges and an integer k, computes a graph G\u27 and an integer k\u27 in O(k^4 m) time such that (1) the size of the graph G\u27 is O(k^2), (2) k\u27 leq k, and (3) G has a feedback vertex set of size at most k if and only if G\u27 has a feedback vertex set of size at most k\u27. This is the first linear-time polynomial-size kernel for Feedback Vertex Set. The size of our kernel is 2k^2+k vertices and 4k^2 edges, which is smaller than the previous best of 4k^2 vertices and 8k^2 edges. Thus, we improve the size and the running time simultaneously. We note that under the assumption of NP notsubseteq coNP/poly, Feedback Vertex Set does not admit an O(k^{2-epsilon})-size kernel for any epsilon>0.
Our kernel exploits k-submodular relaxation, which is a recently developed technique for obtaining efficient FPT algorithms for various problems. The dual of k-submodular relaxation of Feedback Vertex Set can be seen as a half-integral variant of A-path packing, and to obtain the linear-time complexity, we give an efficient augmenting-path algorithm for this problem. We believe that this combinatorial algorithm is of independent interest.
A solver based on the proposed method won first place in the 1st Parameterized Algorithms and Computational Experiments (PACE) challenge

### 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

### 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 $\{0,1\}^V$ to $\{0,1/2,1\}^V$ 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
$O^*(|\Sigma|^{2k})$-time algorithm for node-deletion Unique Label Cover with
label set $\Sigma$ and an $O^*(4^k)$-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

### Fine-Grained Complexity of k-OPT in Bounded-Degree Graphs for Solving TSP

The Traveling Salesman Problem asks to find a minimum-weight Hamiltonian cycle in an edge-weighted complete graph. Local search is a widely-employed strategy for finding good solutions to TSP. A popular neighborhood operator for local search is k-opt, which turns a Hamiltonian cycle C into a new Hamiltonian cycle C\u27 by replacing k edges. We analyze the problem of determining whether the weight of a given cycle can be decreased by a k-opt move. Earlier work has shown that (i) assuming the Exponential Time Hypothesis, there is no algorithm that can detect whether or not a given Hamiltonian cycle C in an n-vertex input can be improved by a k-opt move in time f(k) n^o(k / log k) for any function f, while (ii) it is possible to improve on the brute-force running time of O(n^k) and save linear factors in the exponent. Modern TSP heuristics are very successful at identifying the most promising edges to be used in k-opt moves, and experiments show that very good global solutions can already be reached using only the top-O(1) most promising edges incident to each vertex. This leads to the following question: can improving k-opt moves be found efficiently in graphs of bounded degree? We answer this question in various regimes, presenting new algorithms and conditional lower bounds. We show that the aforementioned ETH lower bound also holds for graphs of maximum degree three, but that in bounded-degree graphs the best improving k-move can be found in time O(n^((23/135+epsilon_k)k)), where lim_{k -> infty} epsilon_k = 0. This improves upon the best-known bounds for general graphs. Due to its practical importance, we devote special attention to the range of k in which improving k-moves in bounded-degree graphs can be found in quasi-linear time. For k <= 7, we give quasi-linear time algorithms for general weights. For k=8 we obtain a quasi-linear time algorithm when the weights are bounded by O(polylog n). On the other hand, based on established fine-grained complexity hypotheses about the impossibility of detecting a triangle in edge-linear time, we prove that the k = 9 case does not admit quasi-linear time algorithms. Hence we fully characterize the values of k for which quasi-linear time algorithms exist for polylogarithmic weights on bounded-degree graphs

### 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)

### 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

### Cut Tree Construction from Massive Graphs

The construction of cut trees (also known as Gomory-Hu trees) for a given
graph enables the minimum-cut size of the original graph to be obtained for any
pair of vertices. Cut trees are a powerful back-end for graph management and
mining, as they support various procedures related to the minimum cut, maximum
flow, and connectivity. However, the crucial drawback with cut trees is the
computational cost of their construction. In theory, a cut tree is built by
applying a maximum flow algorithm for $n$ times, where $n$ is the number of
vertices. Therefore, naive implementations of this approach result in cubic
time complexity, which is obviously too slow for today's large-scale graphs. To
address this issue, in the present study, we propose a new cut-tree
construction algorithm tailored to real-world networks. Using a series of
experiments, we demonstrate that the proposed algorithm is several orders of
magnitude faster than previous algorithms and it can construct cut trees for
billion-scale graphs.Comment: Short version will appear at ICDM'1

### On the Power of Tree-Depth for Fully Polynomial FPT Algorithms

There are many classical problems in P whose time complexities have not been improved over the past decades.
Recent studies of "Hardness in P" have revealed that, for several of such problems, the current fastest algorithm is the best possible under some complexity assumptions.
To bypass this difficulty, the concept of "FPT inside P" has been introduced.
For a problem with the current best time complexity O(n^c), the goal is to design an algorithm running in k^{O(1)}n^{c\u27} time for a parameter k and a constant c\u27<c.
In this paper, we investigate the complexity of graph problems in P parameterized by tree-depth, a graph parameter related to tree-width.
We show that a simple divide-and-conquer method can solve many graph problems, including
Weighted Matching, Negative Cycle Detection, Minimum Weight Cycle, Replacement Paths, and 2-hop Cover,
in O(td m) time or O(td (m+nlog n)) time, where td is the tree-depth of the input graph.
Because any graph of tree-width tw has tree-depth at most (tw+1)log_2 n, our algorithms also run in O(tw mlog n) time or O(tw (m+nlog n)log n) time.
These results match or improve the previous best algorithms parameterized by tree-width.
Especially, we solve an open problem of fully polynomial FPT algorithm for Weighted Matching parameterized by tree-width posed by Fomin et al. (SODA 2017)

### Fast Exact Shortest-Path Distance Queries on Large Networks by Pruned Landmark Labeling

We propose a new exact method for shortest-path distance queries on
large-scale networks. Our method precomputes distance labels for vertices by
performing a breadth-first search from every vertex. Seemingly too obvious and
too inefficient at first glance, the key ingredient introduced here is pruning
during breadth-first searches. While we can still answer the correct distance
for any pair of vertices from the labels, it surprisingly reduces the search
space and sizes of labels. Moreover, we show that we can perform 32 or 64
breadth-first searches simultaneously exploiting bitwise operations. We
experimentally demonstrate that the combination of these two techniques is
efficient and robust on various kinds of large-scale real-world networks. In
particular, our method can handle social networks and web graphs with hundreds
of millions of edges, which are two orders of magnitude larger than the limits
of previous exact methods, with comparable query time to those of previous
methods.Comment: To appear in SIGMOD 201

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