Cycle Packing Research Articles

Abstract 4252: Establishment of a multicellular 3D cell culture model for tumor - endothelial cell interaction

Abstract Introduction: Due to the increasing understanding of the mechanisms relevant to the genesis of cancer, we are experiencing a transition from disease to target-oriented therapy. One major hurdle for the development of these targeted therapeutic regimens, however, is the limited availability of predictive in vitro models. The critical challenge is to develop culture models better reflecting in vivo conditions. We present data that highlights the differences of RNA expression of in vivo like 3D microtissues consisting of tumour cells, fibroblasts and two different endothelial cell lines compared to normal 2D cell culture conditions. Methods: 96-well hanging drop microtiter plates (InSphero AG, Zürich, Switzerland) were applied for the production of 3D mono-, co- and tri-cultures including the human lung cancer cell lines A549 or Colo699 alone or in combination with a human lung fibroblast cell line (SV-80) and either a human umbilical vein endothelial cell line (HUVEC) or the primary human lung microvascular endothelial cell line (HMVEC-L). In addition, to conventional histology (H&E), tumour endothelial spheroid aggregation was displayed immunohistochemically (IHC) by protein expression of e-cadherin, CD31, von Willebrand factor (vWF) and α-muscle actin (α-SMA). RNA expression profiling by Affymetrix chip analysis was performed for multicellular 3D microtissues and 2D cultured cell lines. Results: Endothelial cells aggregated in coherent tube like structures preferentially in the fibroblast consisting core of all microtissues. Furthermore, endothelial cells expressed α-SMA only in microtissues that consisted both of fibroblasts and tumour cells indicating an interaction between these two cell types. RNA expression profiles revealed a high number of regulated genes in tri-cultures when compared to microtissues only consisting of mono- or co-cultures or to traditional 2D cultivated cells. Regulated genes played an important role either in cell cycle, organelle fission, wound healing and DNA packing. Interestingly, no difference in the RNA expression was displayed in microtissues containing either immortalized or primary endothelial cells, Finally, a relation of RNA expression between our cell culture model and patient data was identified. Conclusion: We demonstrate that cultivation of cells as multicellular microtissues in a 3D environment led not only to a difference in RNA but also in protein expression due to cell - cell interactions. Our data support the importance of performing complex co-culture for investigating tumour stroma interactions. Citation Format: Arno Amann, Marit Zwierzina, Johann Kern, Gabriele Gamerith, Stefan Koeck, Edith Lorenz, Johannes Rainer, Heinz Zwierzina. Establishment of a multicellular 3D cell culture model for tumor - endothelial cell interaction. [abstract]. In: Proceedings of the 107th Annual Meeting of the American Association for Cancer Research; 2016 Apr 16-20; New Orleans, LA. Philadelphia (PA): AACR; Cancer Res 2016;76(14 Suppl):Abstract nr 4252.

Edge-disjoint packing of stars and cycles

We study the parameterized complexity of two graph packing problems, Edge-Disjointk-Packing ofs-Stars and Edge-Disjointk-Packing ofs-Cycles. With respect to the choice of parameters, we show that although the two problems are FPT with both k and s as parameters, they are unlikely to be fixed-parameter tractable when parameterized by only k or only s. In terms of kernelization complexity, we show that Edge-Disjointk-Packing ofs-Stars has a kernel with size polynomial in both k and s, but in contrast, unless NP ⊆ coNP/poly, Edge-Disjointk-Packing ofs-Cycles does not have a kernel with size polynomial in both k and s, and moreover does not have a kernel with size polynomial in s for any fixed k. We also show that Edge-Disjointk-Packing of 4-Cycles admits a 96k2 kernel in general graphs and a 96k kernel in planar graphs.

Edge-disjoint odd cycles in 4-edge-connected graphs

Finding edge-disjoint odd cycles is one of the most important problems in graph theory, graph algorithms and combinatorial optimization. One of the difficulties of this problem is that the Erdős–Pósa property does not hold for odd cycles in general. Motivated by this fact, we prove that for any positive integer k, there exists an integer f(k) satisfying the following. For any 4-edge-connected graph G=(V,E), either G has k edge-disjoint odd cycles or there exists an edge set F⊆E with |F|≤f(k) such that G−F is bipartite. We note that the 4-edge-connectivity is best possible in this statement.A similar approach can be applied to an algorithmic question. Suppose that the input graph G is a 4-edge-connected graph with n vertices. We show that, for any ε>0, if k=O((log⁡log⁡log⁡n)1/2−ε), then the k edge-disjoint odd cycle packing problem in G can be solved in polynomial time in n. This result implies the authors' algorithm for the k edge-disjoint paths problem in 4-edge-connected graphs [13].

A Dynamic Programming Approach for the Max-Min Cycle Packing Problem in Even Graphs

Let be an undirected graph. The maximum cycle packing problem in G then is to find a collection of edge-disjoint cycles Ciin G such that s is maximum. In general, the maximum cycle packing problem is NP-hard. In this paper, it is shown for even graphs that if such a collection satisfies the condition that it minimizes the quantityon the set of all edge-disjoint cycle collections, then it is a maximum cycle packing. The paper shows that the determination of such a packing can be solved by a dynamic programming approach. For its solution, an-shortest path procedure on an appropriate acyclic networkis presented. It uses a particular monotonous node potential.

Tight Co‐Degree Condition for Packing of Loose Cycles in 3‐Graphs

AbstractWe show that for every even integer there is n0 such that, if H is a 3‐uniform hypergraph on , vertices such that the minimum co‐degree of H is at least , then H can be tiled with copies of a loose cycle on s vertices. The co‐degree condition is tight.

Tree Derived Architectures with Decycling Number Equal to Cycle Packing Number

Abstract The decycling number of a graph G denoted by Δ ( G ), is the smallest number of vertices that can be deleted from G so that the resultant graph contains no cycles. The cycle packing number of a graph G denoted byc( G ), is the maximum number of vertex disjoint cycles in G . It is clear that c( G ) ≤ .Δ( G ) We find certain networks for which decycling number equals the cycle packing number.

ON CYCLE PACKINGS AND FEEDBACK VERTEX SETS

For a graph $G$ , let $\mathbf{fvs}$ and $\mathbf{cp}$ denote the minimum size of a feedback vertex set in $G$ and the maximum size of a cycle packing in $G$ , respectively. Kloks , Lee, and Liu conjectured that $\mathbf{fvs}(G)\le 2\,\mathbf{cp}(G)$ if $G$ is planar. They proved a weaker inequality, replacing $2$ by $5$ . We improve this, replacing $5$ by $3$ , and verifying the resulting inequality for graphs embedded in surfaces of nonnegative Euler characteristic. We also generalize to arbitrary surfaces. We show that, if a graph $G$ embeds in a surface of Euler characteristic $c\le 0$ , then $\mathbf{fvs}(G)\le 3\,\mathbf{cp}(G) + 103(-c)$ . Lastly, we consider what the best possible bound on $\mathbf{fvs}$ might be, and give some open problems.

The role of planarity in connectivity problems parameterized by treewidth

For some years it was believed that for “connectivity” problems such as Hamiltonian Cycle, algorithms running in time 2O(tw)⋅nO(1) – called single-exponential – existed only on planar and other topologically constrained graph classes, where tw stands for the treewidth of the n-vertex input graph. This was recently disproved by Cygan et al. [3, FOCS 2011], Bodlaender et al. [1, ICALP 2013], and Fomin et al. [11, SODA 2014], who provided single-exponential algorithms on general graphs for most connectivity problems that were known to be solvable in single-exponential time on topologically constrained graphs. In this article we further investigate the role of planarity in connectivity problems parameterized by treewidth, and convey that several problems can indeed be distinguished according to their behavior on planar graphs. Known results from the literature imply that there exist problems, like Cycle Packing, that cannot be solved in time 2o(twlog⁡tw)⋅nO(1) on general graphs but that can be solved in time 2O(tw)⋅nO(1) when restricted to planar graphs. Our main contribution is to show that there exist natural problems that can be solved in time 2O(twlog⁡tw)⋅nO(1) on general graphs but that cannot be solved in time 2o(twlog⁡tw)⋅nO(1) even when restricted to planar graphs. Furthermore, we prove that Planar Cycle Packing and Planar Disjoint Pathscannot be solved in time 2o(tw)⋅nO(1). The mentioned negative results hold unless the ETH fails. We feel that our results constitute a first step in a subject that can be further exploited.

Cycle packing

AbstractIn the 1960s, Erdős and Gallai conjectured that the edge set of every graph on n vertices can be partitioned into O(n) cycles and edges. They observed that one can easily get an O(nlogn) upper bound by repeatedly removing the edges of the longest cycle. We make the first progress on this problem, showing that O(nloglogn) cycles and edges suffice. We also prove the Erdős‐Gallai conjecture for random graphs and for graphs with linear minimum degree. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 608–626, 2014

English

This paper addresses upper and lower bounds for the cardinality of a maximum vertex-/edge-disjoint cycle packing in a polyhedral graph G. Bounds on the cardinality of such packings are provided, that depend on the size, the order or the number of faces of G, respectively. Polyhedral graphs are constructed, that attain these bounds.

Hamilton decompositions of regular expanders: Applications

In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into (n−1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large. This verified a conjecture of Kelly from 1968. In this paper, we derive a number of further consequences of our result on robust outexpanders, the main ones are the following:(i)an undirected analogue of our result on robust outexpanders;(ii)best possible bounds on the size of an optimal packing of edge-disjoint Hamilton cycles in a graph of minimum degree δ for a large range of values for δ.(iii)a similar result for digraphs of given minimum semidegree;(iv)an approximate version of a conjecture of Nash-Williams on Hamilton decompositions of dense regular graphs;(v)a verification of the ‘very dense’ case of a conjecture of Frieze and Krivelevich on packing edge-disjoint Hamilton cycles in random graphs;(vi)a proof of a conjecture of Erdős on the size of an optimal packing of edge-disjoint Hamilton cycles in a random tournament.

Half-integral packing of odd cycles through prescribed vertices

The well-known theorem of Erd?s-Posa says that a graph G has either k disjoint cycles or a vertex set X of order at most f(k) for some function f such that G\X is a forest. Starting with this result, there are many results concerning packing and covering cycles in graph theory and combinatorial optimization. In this paper, we discuss packing disjoint S-cycles, i.e., cycles that are required to go through a set S of vertices. For this problem, Kakimura-Kawarabayashi-Marx (2011) and Pontecorvi-Wollan (2010) recently showed the Erd?s-Posa-type result holds. We further try to generalize this result to packing S-cycles of odd length. In contrast to packing S-cycles, the Erd?s-Posa-type result does not hold for packing odd S-cycles. We then relax packing odd S-cycles to half-integral packing, and show the Erd?s-Posa-type result for the half-integral packing of odd S-cycles, which is a generalization of Reed (1999) when S=V. That is, we show that given an integer k and a vertex set S, a graph G has either 2k odd S-cycles so that each vertex is in at most two of these cycles, or a vertex set X of order at most f(k) (for some function f) such that G\X has no odd S-cycle.

Fixed-parameter tractability for the subset feedback set problem and the S-cycle packing problem

We investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our problem setting is that we are given a graph and a vertex set S called “terminals”. Our purpose here is to consider the following problems:1.The feedback set problem with respect to the terminals S. We call it the subset feedback set problem.2.The cycle packing problem with respect to the terminals S, i.e., each cycle has to contain a vertex in S (such a cycle is called an S-cycle). We call it the S-cycle packing problem. We give the first fixed parameter algorithms for the two problems. Namely;1.For fixed k, we can either find a vertex set X of size k such that G−X has no S-cycle, or conclude that such a vertex set does not exist in O(n2m) time, where n is the number of vertices of the input graph and m is the number of edges of the input graph.2.For fixed k, we can either find k vertex-disjoint S-cycles or conclude that such k disjoint cycles do not exist in O(n3) time.

Induced packing of odd cycles in planar graphs

An induced packing of odd cycles in a graph is a packing such that there is no edge in the graph between any two odd cycles in the packing. We prove that an induced packing of k odd cycles in an n -vertex graph can be found (if it exists) in time 2 O ( k 3 / 2 ) ⋅ n 2 + ϵ (for any constant ϵ > 0 ) when the input graph is planar. We also show that deciding if a graph has an induced packing of two odd induced cycles is NP-complete.

Network Coding in Node-Constrained Line and Star Networks

Line and star networks with both node and edge constraints are studied in the network coding framework. For line networks, the capacity region of the general multiple multicast problem is established. The coding theorem is based on a binary linear coding scheme, while the converse requires new upper bounds that improve on standard cut-based bounds. For star networks, the multiple unicast problem is examined. Capacity upper bounds are derived and a simple linear coding scheme is proposed which is based on the combinatorial optimization problem of cycle packing in directed graphs. The optimality of this scheme is established for a broad class of demands. The connection of node-constrained network coding in star networks, and index coding with side information is discussed and used to partially characterize the optimal linear code for general rates.

Approximation algorithms for grooming in optical network design

We study traffic grooming in optical network design, where the goal is to aggregate low-bandwidth traffic streams to utilize efficiently high-bandwidth media such as wavelength channels. More precisely, given traffic demands to be routed in a network, the design problem is to define a collection of light paths such that each demand can follow a sequence of consecutive light paths. Each light path has a unit-wavelength bandwidth, and multiple sub-wavelength demands may share a common light path. Traffic must enter and depart from a light path at its two endpoints only. Most previous work on grooming focused on the ring topology and typically involved only uniform bandwidth demands, whereas we deal with more general settings. Two objectives are considered. One is to minimize the total cost of equipment necessary to support the light paths; the other is simply to minimize the number of light paths. Even for the extremely restricted special case of a line topology and traffic demands that request half wavelength bandwidth, we show that both objectives are APX -hard to optimize, which means we cannot approximate the optimum arbitrarily closely. On the other extreme of generality, for arbitrary network topologies and traffic demands that request arbitrary amounts of bandwidth, we show a logarithmic approximation for cost minimization and a 2-approximation for minimizing light path counts. Furthermore, we discover that the special case of half-wavelength demands has rich combinatorial properties, closely related to graph problems such as cycle packing and pattern matching problems such as interchange distance in strings. We show how to approximate both objectives up to small constant factors in this case, while similarly improving the approximation and hardness of the interchange distance problem as well.

On the Resilience of Hamiltonicity and Optimal Packing of Hamilton Cycles in Random Graphs

Let $\bk=(k_1,...,k_n)$ be a sequence of $n$ integers. For an increasing monotone graph property $\mP$ we say that a base graph $G=([n],E)$ is \emph{$\bk$-resilient} with respect to $\mP$ if for every subgraph $H\subseteq G$ such that $d_H(i)\leq k_i$ for every $1\leq i\leq n$ the graph $G-H$ possesses $\mP$. This notion naturally extends the idea of the \emph{local resilience} of graphs recently initiated by Sudakov and Vu. In this paper we study the $\bk$-resilience of a typical graph from $\GNP$ with respect to the Hamiltonicity property where we let $p$ range over all values for which the base graph is expected to be Hamiltonian. In particular, we prove that for every $\epsilon>0$ and $p\geq\frac{\ln n+\ln\ln n +\omega(1)}{n}$ if a graph is sampled from $\GNP$ then with high probability removing from each vertex of "small" degree all incident edges but two and from any other vertex at most a $(\frac{1}{3}-\epsilon)$-fraction of the incident edges will result in a Hamiltonian graph. Considering this generalized approach to the notion of resilience allows to establish several corollaries which improve on the best known bounds of Hamiltonicity related questions. It implies that for every positive $\epsilon>0$ and large enough values of $K$, if $p>\frac{K\ln n}{n}$ then with high probability the local resilience of $\GNP$ with respect to being Hamiltonian is at least $(1-\epsilon)np/3$, improving on the previous bound for this range of $p$. Another implication is a result on optimal packing of edge disjoint Hamilton cycles in a random graph. We prove that if $p\leq\frac{1.02\ln n}{n}$ then with high probability a graph $G$ sampled from $\GNP$ contains $\lfloor\frac{\delta(G)}{2}\rfloor$ edge disjoint Hamilton cycles, extending the previous range of $p$ for which this was known to hold.

MAXIMUM WEIGHT CYCLE PACKING IN DIRECTED GRAPHS, WITH APPLICATION TO KIDNEY EXCHANGE PROGRAMS

Centralized matching programs have been established in several countries to organize kidney exchanges between incompatible patient-donor pairs. At the heart of these programs are algorithms to solve kidney exchange problems, which can be modelled as cycle packing problems in a directed graph, involving cycles of length 2, 3, or even longer. Usually, the goal is to maximize the number of transplants, but sometimes the total benefit is maximized by considering the differences between suitable kidneys. These problems correspond to computing cycle packings of maximum size or maximum weight in directed graphs. Here we prove the APX-completeness of the problem of finding a maximum size exchange involving only 2-cycles and 3-cycles. We also present an approximation algorithm and an exact algorithm for the problem of finding a maximum weight exchange involving cycles of bounded length. The exact algorithm has been used to provide optimal solutions to real kidney exchange problems arising from the National Matching Scheme for Paired Donation run by NHS Blood and Transplant, and we describe practical experience based on this collaboration.

Approximability of Packing Disjoint Cycles

Given a graph G, the edge-disjoint cycle packing problem is to find the largest set of cycles of which no two share an edge. For undirected graphs, the best known approximation algorithm has ratio $O(\sqrt{\log n})$ (Krivelevich et al. in ACM Trans. Algorithms, 2009, to appear). In fact, they proved the same upper bound for the integrality gap of this problem by presenting a simple greedy algorithm. Here we show that this is almost best possible. By modifying integrality gap and hardness results for the edge-disjoint paths problem (Andrews et al. in Proc. of 46th IEEE FOCS, pp. 226–244, 2005; Chuzhoy and Khanna in New hardness results for undirected edge disjoint paths. Manuscript, 2005), we show that the undirected edge-disjoint cycle packing problem is quasi-NP-hard to approximate within ratio of $O(\log^{\frac{1}{2}-\epsilon}n)$ for any constant e>0. The same result holds for the problem of packing vertex-disjoint cycles.

On packing shortest cycles in graphs

We study the problems to find a maximum packing of shortest edge-disjoint cycles in a graph of given girth g ( g-ESCP) and its vertex-disjoint analogue g-VSCP. In the case g = 3 , Caprara and Rizzi (2001) have shown that g-ESCP can be solved in polynomial time for graphs with maximum degree 4, but is APX-hard for graphs with maximum degree 5, while g-VSCP can be solved in polynomial time for graphs with maximum degree 3, but is APX-hard for graphs with maximum degree 4. For g ∈ { 4 , 5 } , we show that both problems allow polynomial time algorithms for instances with maximum degree 3, but are APX-hard for instances with maximum degree 4. For each g ⩾ 6 , both problems are APX-hard already for graphs with maximum degree 3.