331 research outputs found
Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder
Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat).
The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem
Counting Problems in Parameterized Complexity
This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity. After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large graphs, and counting perfect matchings and Hamiltonian cycles in well-structured graphs.
While this survey presupposes familiarity with parameterized algorithms and complexity, we aim at explaining all relevant notions from counting complexity in a self-contained way
Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths
We systematically investigate the complexity of counting subgraph patterns
modulo fixed integers. For example, it is known that the parity of the number
of -matchings can be determined in polynomial time by a simple reduction to
the determinant. We generalize this to an -time algorithm to
compute modulo the number of subgraph occurrences of patterns that are
vertices away from being matchings. This shows that the known
polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry
over into the setting of counting modulo .
Complementing our algorithm, we also give a simple and self-contained proof
that counting -matchings modulo odd integers is Mod_q-W[1]-complete and
prove that counting -paths modulo is Parity-W[1]-complete, answering an
open question by Bj\"orklund, Dell, and Husfeldt (ICALP 2015).Comment: 23 pages, to appear at ESA 202
Extensor-coding
We devise an algorithm that approximately computes the number of paths of
length in a given directed graph with vertices up to a multiplicative
error of . Our algorithm runs in time . The algorithm is based on associating with
each vertex an element in the exterior (or, Grassmann) algebra, called an
extensor, and then performing computations in this algebra. This connection to
exterior algebra generalizes a number of previous approaches for the longest
path problem and is of independent conceptual interest. Using this approach, we
also obtain a deterministic time algorithm
to find a -path in a given directed graph that is promised to have few of
them. Our results and techniques generalize to the subgraph isomorphism problem
when the subgraphs we are looking for have bounded pathwidth. Finally, we also
obtain a randomized algorithm to detect -multilinear terms in a multivariate
polynomial given as a general algebraic circuit. To the best of our knowledge,
this was previously only known for algebraic circuits not involving negative
constants.Comment: To appear at STOC 2018: Symposium on Theory of Computing, June 23-27,
2018, Los Angeles, CA, US
Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large
We study the parameterized complexity of two classic problems on directed graphs: Hamiltonian Cycle and its generalization Longest Cycle. Since 2008, it is known that Hamiltonian Cycle is W[1]-hard when parameterized by directed treewidth [Lampis et al., ISSAC\u2708]. By now, the question of whether it is FPT parameterized by the directed feedback vertex set (DFVS) number has become a longstanding open problem. In particular, the DFVS number is the largest natural directed width measure studied in the literature. In this paper, we provide a negative answer to the question, showing that even for the DFVS number, the problem remains W[1]-hard. As a consequence, we also obtain that Longest Cycle is W[1]-hard on directed graphs when parameterized multiplicatively above girth, in contrast to the undirected case. This resolves an open question posed by Fomin et al. [ACM ToCT\u2721] and Gutin and Mnich [arXiv:2207.12278]. Our hardness results apply to the path versions of the problems as well. On the positive side, we show that Longest Path parameterized multiplicatively above girth belongs to the class XP
Recent Advances in Fully Dynamic Graph Algorithms
In recent years, significant advances have been made in the design and
analysis of fully dynamic algorithms. However, these theoretical results have
received very little attention from the practical perspective. Few of the
algorithms are implemented and tested on real datasets, and their practical
potential is far from understood. Here, we present a quick reference guide to
recent engineering and theory results in the area of fully dynamic graph
algorithms
Detecting Feedback Vertex Sets of Size in Time
In the Feedback Vertex Set problem, one is given an undirected graph and
an integer , and one needs to determine whether there exists a set of
vertices that intersects all cycles of (a so-called feedback vertex set).
Feedback Vertex Set is one of the most central problems in parameterized
complexity: It served as an excellent test bed for many important algorithmic
techniques in the field such as Iterative Compression~[Guo et al. (JCSS'06)],
Randomized Branching~[Becker et al. (J. Artif. Intell. Res'00)] and
Cut\&Count~[Cygan et al. (FOCS'11)]. In particular, there has been a long race
for the smallest dependence in run times of the type ,
where the notation omits factors polynomial in . This race seemed
to be run in 2011, when a randomized algorithm time algorithm
based on Cut\&Count was introduced.
In this work, we show the contrary and give a time
randomized algorithm. Our algorithm combines all mentioned techniques with
substantial new ideas: First, we show that, given a feedback vertex set of size
of bounded average degree, a tree decomposition of width
can be found in polynomial time. Second, we give a randomized branching
strategy inspired by the one from~[Becker et al. (J. Artif. Intell. Res'00)] to
reduce to the aforementioned bounded average degree setting. Third, we obtain
significant run time improvements by employing fast matrix multiplication.Comment: SODA 2020, 22 page
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