25 research outputs found
Packing cycles faster than Erdos-Posa
The Cycle Packing problem asks whether a given undirected graph contains vertex-disjoint cycles. Since the publication of the classic Erdös--Pósa theorem in 1965, this problem received significant attention in the fields of graph theory and algorithm design. In particular, this problem is one of the first problems studied in the framework of parameterized complexity. The nonuniform fixed-parameter tractability of Cycle Packing follows from the Robertson--Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle Packing can be solved in time using exponential space. In the case a solution exists, Bodlaender's algorithm also outputs a solution (in the same time). It has later become common knowledge that Cycle Packing admits a -time (deterministic) algorithm using exponential space, which is a consequence of the Erdös--Pósa theorem. Nowadays, the design of this algorithm is given as an exercise in textbooks on parameterized complexity. Yet, no algorithm that runs in time , beating the bound , has been found. In light of this, it seems natural to ask whetherthe bound is essentially optimal. In this paper, we answer this question negatively by developing a -time (deterministic) algorithm for Cycle Packing. In the case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the bound , our algorithm runs in time linear in , and its space complexity is polynomial in the input size.publishedVersio
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs
We study the problem of routing symmetric demand pairs in planar digraphs. The input consists of a directed planar graph G = (V, E) and a collection of k source-destination pairs M = {s_1t_1, ..., s_kt_k}. The goal is to maximize the number of pairs that are routed along disjoint paths. A pair s_it_i is routed in the symmetric setting if there is a directed path connecting s_i to t_i and a directed path connecting t_i to s_i. In this paper we obtain a randomized poly-logarithmic approximation with constant congestion for this problem in planar digraphs. The main technical contribution is to show that a planar digraph with directed treewidth h contains a constant congestion crossbar of size Omega(h/polylog(h))
Parameterization Above a Multiplicative Guarantee
Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (at most) k+g(I). Here, g(I) is usually a lower bound (resp. upper bound) on the maximum (resp. minimum) size of a solution. Since its introduction in 1999 for Max SAT and Max Cut (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research.
We highlight a multiplicative form of parameterization above a guarantee: Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (resp. at most) k ? g(I). In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, and provide a parameterized algorithm for this problem. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ?>0, multiplicative parameterization above g(I)^(1+?) of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of algorithms for other problems parameterized multiplicatively above girth
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
An O(N) Time Algorithm for Finding Hamilton Cycles with High Probability
We design a randomized algorithm that finds a Hamilton cycle in ?(n) time with high probability in a random graph G_{n,p} with edge probability p ? C log n / n. This closes a gap left open in a seminal paper by Angluin and Valiant from 1979
Graph Theory
This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
Deterministic Decremental Reachability, SCC, and Shortest Paths via Directed Expanders and Congestion Balancing
Let be a weighted, digraph subject to a sequence of adversarial
edge deletions. In the decremental single-source reachability problem (SSR), we
are given a fixed source and the goal is to maintain a data structure that
can answer path-queries for any . In the more
general single-source shortest paths (SSSP) problem the goal is to return an
approximate shortest path to , and in the SCC problem the goal is to
maintain strongly connected components of and to answer path queries within
each component. All of these problems have been very actively studied over the
past two decades, but all the fast algorithms are randomized and, more
significantly, they can only answer path queries if they assume a weaker model:
they assume an oblivious adversary which is not adaptive and must fix the
update sequence in advance. This assumption significantly limits the use of
these data structures, most notably preventing them from being used as
subroutines in static algorithms. All the above problems are notoriously
difficult in the adaptive setting. In fact, the state-of-the-art is still the
Even and Shiloach tree, which dates back all the way to 1981 and achieves total
update time . We present the first algorithms to break through this
barrier:
1) deterministic decremental SSR/SCC with total update time
2) deterministic decremental SSSP with total update time .
To achieve these results, we develop two general techniques of broader
interest for working with dynamic graphs: 1) a generalization of expander-based
tools to dynamic directed graphs, and 2) a technique that we call congestion
balancing and which provides a new method for maintaining flow under
adversarial deletions. Using the second technique, we provide the first
near-optimal algorithm for decremental bipartite matching.Comment: Reuploaded with some generalizations of previous theorem