25 research outputs found

    Transversal Problems In Sparse Graphs

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    Graph transversals are a classical branch of graph algorithms. In such a problem, one seeks a minimum-weight subset of nodes in a node-weighted graph GG which intersects all copies of subgraphs~FF from a fixed family F\mathcal F. In the first portion of this thesis we show two results related to even cycle transversal. %%Note rephrase this later. In Chapter \ref{ECTChapter}, we present our 47/7-approximation for even cycle transversal. To do this, we first apply a graph ``compression" method of Fiorini et al. % \cite{FioriniJP2010} which we describe in Chapter \ref{PreliminariesChapter}. For the analysis we repurpose the theory behind the 18/7-approximation for ``uncrossable" feedback vertex set problems of Berman and Yaroslavtsev %% \cite{BermanY2012} noting that we do not need the larger ``witness" cycles to be a cycle that we need to hit. This we do in Chapter \ref{BermanYaroChapter}. In Chapter \ref{ErdosPosaChapter} we present a simple proof of an Erdos Posa result, that for any natural number kk a planar graph GG either contains kk vertex disjoint even cycles, or a set XX of at most 9k9k such that G\XG \backslash X contains no even cycle. In the rest of this thesis, we show a result for dominating set. A dominating set SS in a graph is a set of vertices such that each node is in SS or adjacent to SS. In Chapter 6 we present a primal-dual (a+1)(a+1)-approximation for minimum weight dominating set in graphs of arboricity aa. At the end, we propose five open problems for future research

    Deterministic Decremental Reachability, SCC, and Shortest Paths via Directed Expanders and Congestion Balancing

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    Let G=(V,E,w)G = (V,E,w) 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 ss and the goal is to maintain a data structure that can answer path-queries svs \rightarrowtail v for any vVv \in V. In the more general single-source shortest paths (SSSP) problem the goal is to return an approximate shortest path to vv, and in the SCC problem the goal is to maintain strongly connected components of GG 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 O(mn)O(mn). We present the first algorithms to break through this barrier: 1) deterministic decremental SSR/SCC with total update time mn2/3+o(1)mn^{2/3 + o(1)} 2) deterministic decremental SSSP with total update time n2+2/3+o(1)n^{2+2/3+o(1)}. 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

    Parameterization Above a Multiplicative Guarantee

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    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

    A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

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    Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

    A Polynomial time Algorithm for Hamilton Cycle with maximum Degree 3

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    Based on the famous Rotation-Extension technique, by creating the new concepts and methods: broad cycle, main segment, useful cut and insert, destroying edges for a main segment, main goal Hamilton cycle, depth-first search tree, we develop a polynomial time algorithm for a famous NPC: the Hamilton cycle problem. Thus we proved that NP=P. The key points of this paper are: 1) there are two ways to get a Hamilton cycle in exponential time: a full permutation of n vertices; or, chose n edges from all k edges, and check all possible combinations. The main problem is: how to avoid checking all combinations of n edges from all edges. My algorithm can avoid this. Lemma 1 and lemma 2 are very important. They are the foundation that we always can get a good branch in the depth-first search tree and can get a series of destroying edges (all are bad edges) for this good branch in polynomial time. The extraordinary insights are: destroying edges, a tree contains each main segment at most one time at the same time, and dynamic combinations. The difficult part is to understand how to construct a main segment's series of destroying edges by dynamic combinations (see the proof of lemma 4). The proof logic is: if there is at least on Hamilton cycle in the graph, we always can do useful cut and inserts until a Hamilton cycle is got. The times of useful cut and inserts are polynomial. So if at any step we cannot have a useful cut and insert, this means that there are no Hamilton cycles in the graph.Comment: 49 pages. This time, I add a detailed polynomial time algorithm and proof for 3S

    An O(N) Time Algorithm for Finding Hamilton Cycles with High Probability

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    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

    Packing cycles faster than Erdos-Posa

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    The Cycle Packing problem asks whether a given undirected graph G=(V,E)G=(V,E) contains kk 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 2O(k2)V2^{\mathcal{O}(k^2)}\cdot |V| 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 2O(klog2k)V2^{\mathcal{O}(k\log^2k)}\cdot |V|-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 2o(klog2k)VO(1)2^{o(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}, beating the bound 2O(klog2k)VO(1)2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}, has been found. In light of this, it seems natural to ask whetherthe 2O(klog2k)VO(1)2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)} bound is essentially optimal. In this paper, we answer this question negatively by developing a 2O(klog2kloglogk)V2^{\mathcal{O}(\frac{k\log^2k}{\log\log k})}\cdot |V|-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 2O(klog2k)VO(1)2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}, our algorithm runs in time linear in V|V|, and its space complexity is polynomial in the input size.publishedVersio

    Graph Theory

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    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

    35th Symposium on Theoretical Aspects of Computer Science: STACS 2018, February 28-March 3, 2018, Caen, France

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