Approximation algorithms for maximally balanced connected graph partition

Given a simple connected graph $G = (V, E)$, we seek to partition the vertex set $V$ into $k$ non-empty parts such that the subgraph induced by each part is connected, and the partition is maximally balanced in the way that the maximum cardinality of these $k$ parts is minimized. We refer this problem to as {\em min-max balanced connected graph partition} into $k$ parts and denote it as {\sc $k$-BGP}. The general vertex-weighted version of this problem on trees has been studied since about four decades ago, which admits a linear time exact algorithm; the vertex-weighted {\sc $2$-BGP} and {\sc $3$-BGP} admit a $5/4$-approximation and a $3/2$-approximation, respectively; but no approximability result exists for {\sc $k$-BGP} when $k \ge 4$, except a trivial $k$-approximation. In this paper, we present another $3/2$-approximation for our cardinality {\sc $3$-BGP} and then extend it to become a $k/2$-approximation for {\sc $k$-BGP}, for any constant $k \ge 3$. Furthermore, for {\sc $4$-BGP}, we propose an improved $24/13$-approximation. To these purposes, we have designed several local improvement operations, which could be useful for related graph partition problems.Comment: 23 pages, 7 figures, accepted for presentation at COCOA 2019 (Xiamen, China

Numerical simulation of water exit of an initially fully submerged buoyant spheroid in an axisymmetric flow

The free water exit of an initially fully submerged buoyant spheroid in an axisymmetric flow, which is driven by the difference between the vertical fluid force and gravity, is investigated. The fluid is assumed to be incompressible and inviscid, and the flow to be irrotational. The velocity potential theory is adopted together with fully nonlinear boundary conditions on the free surface. The surface tension is neglected and the pressure is taken as constant on the free surface. The acceleration of the body at each time step is obtained as part of the solution. Its nonlinear mutual dependence on the fluid force is decoupled through the auxiliary function method. The free-surface breakup by body penetration and water detachment from the body are treated through numerical conditions. The slender body theory based on the zero potential assumption on the undisturbed flat free surface is adopted, through which a condition for full water exit of a spheroid is obtained. Comparison is made between the results from the slender body theory and from the fully nonlinear theory through the boundary-element method, and good agreement is found when the spheroid is slender. Extensive case studies are undertaken to investigate the effects of body density, dimensions and the initial submergence

Effective Edge-Fault-Tolerant Single-Source Spanners via Best (or Good) Swap Edges

Computing \emph{all best swap edges} (ABSE) of a spanning tree $T$ of a given $n$-vertex and $m$-edge undirected and weighted graph $G$ means to select, for each edge $e$ of $T$, a corresponding non-tree edge $f$, in such a way that the tree obtained by replacing $e$ with $f$ enjoys some optimality criterion (which is naturally defined according to some objective function originally addressed by $T$). Solving efficiently an ABSE problem is by now a classic algorithmic issue, since it conveys a very successful way of coping with a (transient) \emph{edge failure} in tree-based communication networks: just replace the failing edge with its respective swap edge, so as that the connectivity is promptly reestablished by minimizing the rerouting and set-up costs. In this paper, we solve the ABSE problem for the case in which $T$ is a \emph{single-source shortest-path tree} of $G$, and our two selected swap criteria aim to minimize either the \emph{maximum} or the \emph{average stretch} in the swap tree of all the paths emanating from the source. Having these criteria in mind, the obtained structures can then be reviewed as \emph{edge-fault-tolerant single-source spanners}. For them, we propose two efficient algorithms running in $O(m n +n^2 \log n)$ and $O(m n \log \alpha(m,n))$ time, respectively, and we show that the guaranteed (either maximum or average, respectively) stretch factor is equal to 3, and this is tight. Moreover, for the maximum stretch, we also propose an almost linear $O(m \log \alpha(m,n))$ time algorithm computing a set of \emph{good} swap edges, each of which will guarantee a relative approximation factor on the maximum stretch of $3/2$ (tight) as opposed to that provided by the corresponding BSE. Surprisingly, no previous results were known for these two very natural swap problems.Comment: 15 pages, 4 figures, SIROCCO 201

Efficient GRASP+VND and GRASP+VNS metaheuristics for the traveling repairman problem

The traveling repairman problem is a customer-centric routing problem, in which the total waiting time of the customers is minimized, rather than the total travel time of a vehicle. To date, research on this problem has focused on exact algorithms and approximation methods. This paper presents the first metaheuristic approach for the traveling repairman problem

On the complexity of color-avoiding site and bond percolation

The mathematical analysis of robustness and error-tolerance of complex networks has been in the center of research interest. On the other hand, little work has been done when the attack-tolerance of the vertices or edges are not independent but certain classes of vertices or edges share a mutual vulnerability. In this study, we consider a graph and we assign colors to the vertices or edges, where the color-classes correspond to the shared vulnerabilities. An important problem is to find robustly connected vertex sets: nodes that remain connected to each other by paths providing any type of error (i.e. erasing any vertices or edges of the given color). This is also known as color-avoiding percolation. In this paper, we study various possible modeling approaches of shared vulnerabilities, we analyze the computational complexity of finding the robustly (color-avoiding) connected components. We find that the presented approaches differ significantly regarding their complexity.Comment: 14 page

Kernelization and Parameterized Algorithms for 3-Path Vertex Cover

A 3-path vertex cover in a graph is a vertex subset $C$ such that every path of three vertices contains at least one vertex from $C$. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most $k$. In this paper, we give a kernel of $5k$ vertices and an $O^*(1.7485^k)$-time and polynomial-space algorithm for this problem, both new results improve previous known bounds.Comment: in TAMC 2016, LNCS 9796, 201

Algorithms for the minimum non-separating path and the balanced connected bipartition problems on grid graphs (With erratum)

For given a pair of nodes in a graph, the minimum non-separating path problem looks for a minimum weight path between the two nodes such that the remaining graph after removing the path is still connected. The balanced connected bipartition (BCP$_2$) problem looks for a way to bipartition a graph into two connected subgraphs with their weights as equal as possible. In this paper we present an algorithm in time $O(N\log N)$ for finding a minimum weight non-separating path between two given nodes in a grid graph of $N$ nodes with positive weight. This result leads to a 5/4-approximation algorithm for the BCP$_2$ problem on grid graphs, which is the currently best ratio achieved in polynomial time. We also developed an exact algorithm for the BCP$_2$ problem on grid graphs. Based on the exact algorithm and a rounding technique, we show an approximation scheme, which is a fully polynomial time approximation scheme for fixed number of rows.Comment: With erratu

A Minimal Threshold of c-di-GMP Is Essential for Fruiting Body Formation and Sporulation in Myxococcus xanthus

Generally, the second messenger bis-(3’-5’)-cyclic dimeric GMP (c-di-GMP) regulates the switch between motile and sessile lifestyles in bacteria. Here, we show that c-di-GMP is an essential regulator of multicellular development in the social bacterium Myxococcus xanthus. In response to starvation, M. xanthus initiates a developmental program that culminates in formation of spore-filled fruiting bodies. We show that c-di-GMP accumulates at elevated levels during development and that this increase is essential for completion of development whereas excess c-di-GMP does not interfere with development. MXAN3735 (renamed DmxB) is identified as a diguanylate cyclase that only functions during development and is responsible for this increased c-di-GMP accumulation. DmxB synthesis is induced in response to starvation, thereby restricting DmxB activity to development. DmxB is essential for development and functions downstream of the Dif chemosensory system to stimulate exopolysaccharide accumulation by inducing transcription of a subset of the genes encoding proteins involved in exopolysaccharide synthesis. The developmental defects in the dmxB mutant are non-cell autonomous and rescued by co-development with a strain proficient in exopolysaccharide synthesis, suggesting reduced exopolysaccharide accumulation as the causative defect in this mutant. The NtrC-like transcriptional regulator EpsI/Nla24, which is required for exopolysaccharide accumulation, is identified as a c-diGMP receptor, and thus a putative target for DmxB generated c-di-GMP. Because DmxB can be—at least partially—functionally replaced by a heterologous diguanylate cyclase, these results altogether suggest a model in which a minimum threshold level of c-di-GMP is essential for the successful completion of multicellular development in M. xanthus