Approximating subset kk-connectivity problems

A subset $T \subseteq V$ of terminals is $k$-connected to a root $s$ in a directed/undirected graph $J$ if $J$ has $k$ internally-disjoint $vs$-paths for every $v \in T$; $T$ is $k$-connected in $J$ if $T$ is $k$-connected to every $s \in T$. We consider the {\sf Subset $k$-Connectivity Augmentation} problem: given a graph $G=(V,E)$ with edge/node-costs, node subset $T \subseteq V$, and a subgraph $J=(V,E_J)$ of $G$ such that $T$ is $k$-connected in $J$, find a minimum-cost augmenting edge-set $F \subseteq E \setminus E_J$ such that $T$ is $(k+1)$-connected in $J \cup F$. The problem admits trivial ratio $O(|T|^2)$. We consider the case $|T|>k$ and prove that for directed/undirected graphs and edge/node-costs, a $\rho$-approximation for {\sf Rooted Subset $k$-Connectivity Augmentation} implies the following ratios for {\sf Subset $k$-Connectivity Augmentation}: (i) $b(\rho+k) + {(\frac{3|T|}{|T|-k})}^2 H(\frac{3|T|}{|T|-k})$; (ii) $\rho \cdot O(\frac{|T|}{|T|-k} \log k)$, where b=1 for undirected graphs and b=2 for directed graphs, and $H(k)$ is the $k$th harmonic number. The best known values of $\rho$ on undirected graphs are $\min\{|T|,O(k)\}$ for edge-costs and $\min\{|T|,O(k \log |T|)\}$ for node-costs; for directed graphs $\rho=|T|$ for both versions. Our results imply that unless $k=|T|-o(|T|)$, {\sf Subset $k$-Connectivity Augmentation} admits the same ratios as the best known ones for the rooted version. This improves the ratios in \cite{N-focs,L}

Spider covers for prize-collecting network activation problem

In the network activation problem, each edge in a graph is associated with an activation function, that decides whether the edge is activated from node-weights assigned to its end-nodes. The feasible solutions of the problem are the node-weights such that the activated edges form graphs of required connectivity, and the objective is to find a feasible solution minimizing its total weight. In this paper, we consider a prize-collecting version of the network activation problem, and present first non- trivial approximation algorithms. Our algorithms are based on a new LP relaxation of the problem. They round optimal solutions for the relaxation by repeatedly computing node-weights activating subgraphs called spiders, which are known to be useful for approximating the network activation problem

How to Secure Matchings Against Edge Failures

Suppose we are given a bipartite graph that admits a perfect matching and an adversary may delete any edge from the graph with the intention of destroying all perfect matchings. We consider the task of adding a minimum cost edge-set to the graph, such that the adversary never wins. We show that this problem is equivalent to covering a digraph with non-trivial strongly connected components at minimal cost. We provide efficient exact and approximation algorithms for this task. In particular, for the unit-cost problem, we give a log_2 n-factor approximation algorithm and a polynomial-time algorithm for chordal-bipartite graphs. Furthermore, we give a fixed parameter algorithm for the problem parameterized by the treewidth of the input graph. For general non-negative weights we give tight upper and lower approximation bounds relative to the Directed Steiner Forest problem. Additionally we prove a dichotomy theorem characterizing minor-closed graph classes which allow for a polynomial-time algorithm. To obtain our results, we exploit a close relation to the classical Strong Connectivity Augmentation problem as well as directed Steiner problems

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

Algorithms for Graph Connectivity and Cut Problems - Connectivity Augmentation, All-Pairs Minimum Cut, and Cut-Based Clustering

We address a collection of related connectivity and cut problems in simple graphs that reach from the augmentation of planar graphs to be k-regular and c-connected to new data structures representing minimum separating cuts and algorithms that smoothly maintain Gomory-Hu trees in evolving graphs, and finally to an analysis of the cut-based clustering approach of Flake et al. and its adaption to dynamic scenarios

Covering symmetric skew-supermodular functions with hyperedges

In this paper we give results related to a theorem of Szigeti that concerns the covering of symmetric skew-supermodular set functions with hyperedges of minimum total size. In particular, we show the following generalization using a variation of Schrijver’s supermodular colouring theorem: if p1 and p2 are skewsupermodular functions whose maximum value is the same, then it is possible to find in polynomial time a hypergraph of minimum total size that covers both of them. Note that without the assumption on the maximum values this problem is NP-hard. The result has applications concerning the local edge-connectivity augmentation problem of hypergraphs and the global edge-connectivity augmentation problem of mixed hypergraphs. We also present some results on the case when the hypergraph must be obtained by merging given hyperedges

Minimizing Hitting Time between Disparate Groups with Shortcut Edges

Structural bias or segregation of networks refers to situations where two or more disparate groups are present in the network, so that the groups are highly connected internally, but loosely connected to each other. In many cases it is of interest to increase the connectivity of disparate groups so as to, e.g., minimize social friction, or expose individuals to diverse viewpoints. A commonly-used mechanism for increasing the network connectivity is to add edge shortcuts between pairs of nodes. In many applications of interest, edge shortcuts typically translate to recommendations, e.g., what video to watch, or what news article to read next. The problem of reducing structural bias or segregation via edge shortcuts has recently been studied in the literature, and random walks have been an essential tool for modeling navigation and connectivity in the underlying networks. Existing methods, however, either do not offer approximation guarantees, or engineer the objective so that it satisfies certain desirable properties that simplify the optimization~task. In this paper we address the problem of adding a given number of shortcut edges in the network so as to directly minimize the average hitting time and the maximum hitting time between two disparate groups. Our algorithm for minimizing average hitting time is a greedy bicriteria that relies on supermodularity. In contrast, maximum hitting time is not supermodular. Despite, we develop an approximation algorithm for that objective as well, by leveraging connections with average hitting time and the asymmetric k-center problem.Comment: To appear in KDD 202