Given an undirected graph G=(V,E) and subset of terminals T \subseteq V, the element-connectivity of two terminals u,v \in T is the maximum number of u-v paths that are pairwise disjoint in both edges and non-terminals V \setminus T (the paths need not be disjoint in terminals). Element-connectivity is more general than edge-connectivity and less general than vertex-connectivity. Hind and Oellermann gave a graph reduction step that preserves the global element-connectivity of the graph. We show that this step also preserves local connectivity, that is, all the pairwise element-connectivities of the terminals. We give two applications of this reduction step to connectivity and network design problems:
1. Given a graph G and disjoint terminal sets T_1, T_2, ..., T_m, we seek a maximum number of element-disjoint Steiner forests where each forest connects each T_i. We prove that if each T_i is k-element-connected then there exist Ω(\frac{k}{\log h \log m}) element-disjoint Steiner forests, where h = |\bigcup_i T_i|. If G is planar (or more generally, has fixed genus), we show that there exist Ω(k) Steiner forests. Our proofs are constructive, giving poly-time algorithms to find these forests; these are the first non-trivial algorithms for packing element-disjoint Steiner Forests.
2. We give a very short and intuitive proof of a spider-decomposition theorem of Chuzhoy and Khanna in the context of the single-sink k-vertex-connectivity problem; this yields a simple and alternative analysis of an O(k \log n) approximation.
Our results highlight the effectiveness of the element-connectivity reduction step; we believe it will find more applications in the future.
We study several problems related to graph modification problems under connectivity constraints from the perspective of parameterized complexity: {\sc (Weighted) Biconnectivity Deletion}, where we are tasked with deleting~$k$ edges while preserving biconnectivity in an undirected graph, {\sc Vertex-deletion Preserving Strong Connectivity}, where we want to maintain strong connectivity of a digraph while deleting exactly~$k$ vertices, and {\sc Path-contraction Preserving Strong Connectivity}, in which the operation of path contraction on arcs is used instead. The parameterized tractability of t…
We consider a discrete system of $n$ devices lying on a 2-dimensional square grid and forming an initial connected shape $S_I$. Each device is equipped with a linear-strength mechanism which enables it to move a whole line of consecutive devices in a single time-step. We study the problem of transforming $S_I$ into a given connected target shape $S_F$ of the same number of devices, via a finite sequence of \emph{line moves}. Our focus is on designing \emph{centralised} transformations aiming at \emph{minimising the total number of moves} subject to the constraint of \emph{preserving connectivi…
The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The Connectivity Augmentation Problem (CAP) is arguably one of the most basic problems in this area: given a $k$(-edge)-connected graph $G$ and a set of extra edges (links), select a minimum cardinality subset $A$ of links such that adding $A$ to $G$ increases its edge connectivity to $k+1$. Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation fact…
The vertex connectivity of an $m$-edge $n$-vertex undirected graph is the smallest number of vertices whose removal disconnects the graph, or leaves only a singleton vertex. In this paper, we give a reduction from the vertex connectivity problem to a set of maxflow instances. Using this reduction, we can solve vertex connectivity in $\tilde O(m^α)$ time for any $α\ge 1$, if there is a $m^α$-time maxflow algorithm. Using the current best maxflow algorithm that runs in $m^{4/3+o(1)}$ time (Kathuria, Liu and Sidford, FOCS 2020), this yields a $m^{4/3+o(1)}$-time vertex connectivity algorithm. Thi…
An important open question in the area of vertex sparsification is whether $(1+ε)$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. The work Chalermsook et al. (SODA 2021) introduced a relaxation called connectivity-$c$ mimicking networks, which asks to construct a vertex sparsifier which preserves connectivity among $k$ terminals exactly up to the value of $c$, and showed applications to dynamic connectivity data structures and survivable network design. We show that connectivity-$c$ mimicking networks with $\widetilde{O}(kc^3)$ edges exist and c…
We consider network design problems in which we are given a graph and seek a min-size $2$-connected subgraph that satisfies a prescribed property. $\bullet$ In the 1-Connectivity Augmentation problem the goal is to augment a connected graph by a min-size edge subset of a specified edge set such that the augmented graph is $2$-connected. We breach the natural ratio of $2$ for this problem and also for the more general Crossing Family Cover problem. $\bullet$ In the $2$-Connected Dominating Set problem we seek a minimum size $2$-connected subgraph that dominates all nodes. We give the first non-…
We present a $2$-approximation algorithm for the Flexible Graph Connectivity problem [AHM20] via a reduction to the minimum cost $r$-out $2$-arborescence problem.
We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in $O(m + n \operatorname{polylog} n)$ total time. Interspersed with the deletions, it can answer queries to whether any two given vertices currently belong to the same (2-edge-)connected component in constant time. Our result is based on a general Monte-Carlo randomized reduction from decremental $c$-edge-connectivity to a variant of fully-dynamic $c$-edge-connectivity…
We study a model of programmable matter systems consisting of $n$ devices lying on a 2-dimensional square grid, which are able to perform the minimal mechanical operation of rotating around each other. The goal is to transform an initial shape A into a target shape B. We are interested in characterising the class of shapes which can be transformed into each other in such a scenario, under the additional constraint of maintaining global connectivity at all times. This was one of the main problems left open by $[$Michail et al., JCSS'19$]$. Note that the considered question is about structural f…
Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $(1+ε)$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we study a thresholded version of the problem: for a given parameter $c$, find a smaller graph, which we call connectivity-$c$ mimicking network, which preserves connectivity among $k$ terminals exactly up to the value of $c$. We show that connectivity-$c$ mimicking networks with $O(kc^4)$ edges exist and can be fo…
We present quantum algorithms for various problems related to graph connectivity. We give simple and query-optimal algorithms for cycle detection and odd-length cycle detection (bipartiteness) using a reduction to st-connectivity. Furthermore, we show that our algorithm for cycle detection has improved performance under the promise of large circuit rank or a small number of edges. We also provide algorithms for detecting even-length cycles and for estimating the circuit rank of a graph. All of our algorithms have logarithmic space complexity.
We show the first dimension-preserving search-to-decision reductions for approximate SVP and CVP. In particular, for any $γ\leq 1 + O(\log n/n)$, we obtain an efficient dimension-preserving reduction from $γ^{O(n/\log n)}$-SVP to $γ$-GapSVP and an efficient dimension-preserving reduction from $γ^{O(n)}$-CVP to $γ$-GapCVP. These results generalize the known equivalences of the search and decision versions of these problems in the exact case when $γ= 1$. For SVP, we actually obtain something slightly stronger than a search-to-decision reduction---we reduce $γ^{O(n/\log n)}$-SVP to $γ$-unique SVP…
This note concerns the trade-off between the degree of the constraint graph and the gap in hardness of approximating the Min-Rep variant of Label Cover (aka Projection Game). We make a very simple observation that, for NP-hardness with gap $g$, the degree can be made as small as $O(g \log g)$, which improves upon the previous $\tilde{O}(g^{1/2})$ bound from a work of Laekhanukit (SODA'14). Note that our bound is optimal up to a logarithmic factor since there is a trivial $Δ$-approximation for Min-Rep where $Δ$ is the maximum degree of the constraint graph. Thanks to known reductions, this impr…
We study graph connectivity problem in MPC model. On an undirected graph with $n$ nodes and $m$ edges, $O(\log n)$ round connectivity algorithms have been known for over 35 years. However, no algorithms with better complexity bounds were known. In this work, we give fully scalable, faster algorithms for the connectivity problem, by parameterizing the time complexity as a function of the diameter of the graph. Our main result is a $O(\log D \log\log_{m/n} n)$ time connectivity algorithm for diameter-$D$ graphs, using $Θ(m)$ total memory. If our algorithm can use more memory, it can terminate in…
The concept of nimbers--a.k.a. Grundy-values or nim-values--is fundamental to combinatorial game theory. Nimbers provide a complete characterization of strategic interactions among impartial games in their disjunctive sums as well as the winnability. In this paper, we initiate a study of nimber-preserving reductions among impartial games. These reductions enhance the winnability-preserving reductions in traditional computational characterizations of combinatorial games. We prove that Generalized Geography is complete for the natural class, $\cal{I}^P$ , of polynomially-short impartial rulesets…
The basic goal of survivable network design is to construct low-cost networks which preserve a sufficient level of connectivity despite the failure or removal of a few nodes or edges. One of the most basic problems in this area is the $2$-Edge-Connected Spanning Subgraph problem (2-ECSS): given an undirected graph $G$, find a $2$-edge-connected spanning subgraph $H$ of $G$ with the minimum number of edges (in particular, $H$ remains connected after the removal of one arbitrary edge). 2-ECSS is NP-hard and the best-known (polynomial-time) approximation factor for this problem is $4/3$. Interest…
