Edge-connectivity augmentation of graphs over symmetric parity families

AbstractIn this note we solve the edge-connectivity augmentation problem over symmetric parity families. It provides a solution for the minimum T-cut augmentation problem. We also extend a recent result of Zhang [C.Q. Zhang, Circular flows of nearly eulerian graphs and vertex splitting, J. Graph Theory 40 (2002) 147–161]

Optimization methods for graphical degree sequence problems and their extensions

制度:新 ; 文部省報告番号:甲2374号 ; 学位の種類:博士(工学) ; 授与年月日:2007/3/15 ; 早大学位記番号:新446

インターネット問題のモデル化法と効率的算法の研究

平成16-17度科学研究費補助金(基盤研究(C))研究成果報告書　課題番号：16500010 研究代表者：伊藤大雄 (京都大学大学院情報学研究科

The (2,k)(2,k)-connectivity augmentation problem: Algorithmic aspects

Durand de Gevigney and Szigeti \cite{DgGSz} have recently given a min-max theorem for the $(2,k)$-connectivity augmentation problem. This article provides an $O(n^3(m+ n \textrm{ }log\textrm{ }n))$ algorithm to find an optimal solution for this problem

Posimodular Function Optimization

Given a posimodular function $f: 2^V \to \mathbb{R}$ on a finite set $V$, we consider the problem of finding a nonempty subset $X$ of $V$ that minimizes $f(X)$. Posimodular functions often arise in combinatorial optimization such as undirected cut functions. In this paper, we show that any algorithm for the problem requires $\Omega(2^{\frac{n}{7.54}})$ oracle calls to $f$, where $n=|V|$. It contrasts to the fact that the submodular function minimization, which is another generalization of cut functions, is polynomially solvable. When the range of a given posimodular function is restricted to be $D=\{0,1,...,d\}$ for some nonnegative integer $d$, we show that $\Omega(2^{\frac{d}{15.08}})$ oracle calls are necessary, while we propose an $O(n^dT_f+n^{2d+1})$-time algorithm for the problem. Here, $T_f$ denotes the time needed to evaluate the function value $f(X)$ for a given $X \subseteq V$. We also consider the problem of maximizing a given posimodular function. We show that $\Omega(2^{n-1})$ oracle calls are necessary for solving the problem, and that the problem has time complexity $\Theta(n^{d-1}T_f)$ when $D=\{0,1,..., d\}$ is the range of $f$ for some constant $d$.Comment: 18 page

Edge Augmentation on Disconnected Graphs via Eigenvalue Elevation

The graph-theoretical task of determining most likely inter-community edges based on disconnected subgraphs' intra-community connectivity is proposed. An algorithm is developed for this edge augmentation task, based on elevating the zero eigenvalues of graph's spectrum. Upper bounds for eigenvalue elevation amplitude and for the corresponding augmented edge density are derived and are authenticated with simulation on random graphs. The algorithm works consistently across synthetic and real networks, yielding desirable performance at connecting graph components. Edge augmentation reverse-engineers graph partition under different community detection methods (Girvan-Newman method, greedy modularity maximization, label propagation, Louvain method, and fluid community), in most cases producing inter-community edges at >50% frequency.Comment: 6 pages, 3 figure

Approximating Minimum Cost Connectivity Orientation and Augmentation

We investigate problems addressing combined connectivity augmentation and orientations settings. We give a polynomial-time 6-approximation algorithm for finding a minimum cost subgraph of an undirected graph $G$ that admits an orientation covering a nonnegative crossing $G$-supermodular demand function, as defined by Frank. An important example is $(k,\ell)$-edge-connectivity, a common generalization of global and rooted edge-connectivity. Our algorithm is based on a non-standard application of the iterative rounding method. We observe that the standard linear program with cut constraints is not amenable and use an alternative linear program with partition and co-partition constraints instead. The proof requires a new type of uncrossing technique on partitions and co-partitions. We also consider the problem setting when the cost of an edge can be different for the two possible orientations. The problem becomes substantially more difficult already for the simpler requirement of $k$-edge-connectivity. Khanna, Naor, and Shepherd showed that the integrality gap of the natural linear program is at most $4$ when $k=1$ and conjectured that it is constant for all fixed $k$. We disprove this conjecture by showing an $\Omega(|V|)$ integrality gap even when $k=2$