We consider the bilevel minimum spanning tree (BMST) problem where the leader and the follower choose a spanning tree together, according to different objective functions. By showing that this problem is NP-hard in general, we answer an open question stated in by Shi et al. We prove that BMST remains hard even in the special case where the follower only controls a matching. Moreover, by a polynomial reduction from the vertex-disjoint Steiner trees problem, we give some evidence that BMST might even remain hard in case the follower controls only few edges. On the positive side, we present a polynomial-time $(|V|-1)$-approximation algorithm for BMST, where $|V|$ is the number of vertices in the input graph. Moreover, considering the number of edges controlled by the follower as parameter, we show that 2-approximating BMST is fixed-parameter tractable and that, in case of uniform costs on leader's edges, even solving BMST exactly is fixed-parameter tractable. We finally consider bottleneck variants of BMST and settle the complexity landscape of all combinations of sum or bottleneck objective functions for the leader and follower, for the optimistic as well as the pessimistic setting.
This article studies the Minimum Spanning Tree Problem under Explorable Uncertainty as well as a related vertex uncertainty version of the problem. We particularly consider special instance types, including cactus graphs, for which we provide randomized algorithms. We introduce the problem of finding a minimum weight spanning star under uncertainty for which we show that no algorithm can achieve constant competitive ratio.
We introduce a novel Quadratic Unconstrained Binary Optimization (QUBO) formulation method for spanning tree problems. Instead of encoding the presence of edges in the tree individually, we opt to encode spanning trees as a permutation problem. We apply our method to four NP-hard spanning tree variants, namely the k-minimum spanning tree, degree-constrained minimum spanning tree, minimum leaf spanning tree, and maximum leaf spanning tree. Our main result is a formulation with $\mathcal{O}(|V|k)$ variables for the k-minimum spanning tree problem, beating related strategies that need $\mathcal{O…
We introduce the minimum labelling spanning bi-connected subgraph problem (MLSBP) replacing connectivity by bi-connectivity in the well known minimum labelling spanning tree problem (MLSTP). A graph is bi-connected if, for every two vertices, there are, at least, two vertex-disjoint paths joining them. The problem consists in finding the spanning bi-connected subgraph or block with minimum set of labels. We adapt the exact method of the MLSTP to solve the MLSTB and the basic greedy constructive heuristic, the maximum vertex covering algorithm (MVCA). This proce- dure is a basic component in th…
This paper give a simple linear-time algorithm that, given a weighted digraph, finds a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the two trees and epsilon > 0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+epsilon times the shortest-path distance, and yet the total weight of the tree is at most 1+2/epsilon times the weight of a minimum spanning tree. This is the best tradeoff possible. The paper also des…
We investigate special cases of the quadratic minimum spanning tree problem (QMSTP) on a graph $G=(V,E)$ that can be solved as a linear minimum spanning tree problem. Characterization of such problems on graphs with special properties are given. This include complete graphs, complete bipartite graphs, cactuses among others. Our characterization can be verified in $O(|E|^2)$ time. In the case of complete graphs and when the cost matrix is given in factored form, we show that our characterization can be verified in $O(|E|)$ time. Related open problems are also indicated.
We present the first polynomial-time approximation schemes, i.e., (1 + ε)-approximation algorithm for any constant ε > 0, for the minimum three-edge connected spanning subgraph problem and the minimum three-vertex connected spanning subgraph problem in undirected planar graphs. Both the approximation schemes run in linear time.
The quadratic minimum spanning tree problem and its variations such as the quadratic bottleneck spanning tree problem, the minimum spanning tree problem with conflict pair constraints, and the bottleneck spanning tree problem with conflict pair constraints are useful in modeling various real life applications. All these problems are known to be NP-hard. In this paper, we investigate these problems to obtain additional insights into the structure of the problems and to identify possible demarcation between easy and hard special cases. New polynomially solvable cases have been identified, as wel…
A spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges. We consider the problem of computing a minimum average stretch spanning tree in polygonal 2-trees, a super class of 2-connected outerplanar graphs. For a polygonal 2-tree on $n$ vertices, we present an algorithm to compute a minimum average stretch spanning tree in $O(n \log n)$ time. This algorithm also finds a minimum fundamental cycle basis in polygonal 2-trees.
Let $G=(V,E)$ be a strongly biconnected directed graph. In this paper we consider the problem of computing an edge subset $H \subseteq E$ of minimum size such that the directed subgraph $(V,H)$ is strongly biconnected.
Given a $2$-vertex-twinless connected directed graph $G=(V,E)$, the minimum $2$-vertex-twinless connected spanning subgraph problem is to find a minimum cardinality edge subset $E^{t} \subseteq E$ such that the subgraph $(V,E^{t})$ is $2$-vertex-twinless connected. Let $G^{1}$ be a minimal $2$-vertex-connected subgraph of $G$. In this paper we present a $(2+a_{t}/2)$-approximation algorithm for the minimum $2$-vertex-twinless connected spanning subgraph problem, where $a_{t}$ is the number of twinless articulation points in $G^{1}$.
We give a new data structure for the fully-dynamic minimum spanning forest problem in simple graphs. Edge updates are supported in $O(\log^4n/\log\log n)$ amortized time per operation, improving the $O(\log^4n)$ amortized bound of Holm et al. (STOC'98, JACM'01). We assume the Word-RAM model with standard instructions.
In the weighted minimum strongly connected spanning subgraph (WMSCSS) problem we must purchase a minimum-cost strongly connected spanning subgraph of a digraph. We show that half-integral linear program (LP) solutions for WMSCSS can be efficiently rounded to integral solutions at a multiplicative $1.5$ cost. This rounding matches a known $1.5$ integrality gap lower bound for a half-integral instance. More generally, we show that LP solutions whose non-zero entries are at least a value $f > 0$ can be rounded at a multiplicative cost of $2 - f$.
We obtain a polynomial-time 17/12-approximation algorithm for the minimum-cost 2-vertex-connected spanning subgraph problem, restricted to graphs of minimum degree at least 3. Our algorithm uses the framework of ear-decompositions for approximating connectivity problems, which was previously used in algorithms for finding the smallest 2-edge-connected spanning subgraph by Cheriyan, Sebő and Szigeti (SIAM J.Discrete Math. 2001) who gave a 17/12-approximation algorithm for this problem, and by Sebő and Vygen (Combinatorica 2014), who improved the approximation ratio to 4/3.
Previous studies has shown that for a weighted undirected graph having $n$ vertices and $m$ edges, a minimal weight spanning tree can be found with $O^*(\sqrt{mn})$ calls to the weight oracle. The present note shows that a given spanning tree can be verified to be a minimal weight spanning tree with only $O(n\bigr)$ calls to the weight oracle and $O(n+\sqrt{m}\log n)$ total work.
We are interested in the consequences of imposing edges in $T$ a minimum spanning tree. We prove that the sum of the replacement costs in $T$ of the imposed edges is a lower bounds of the additional costs. More precisely if r-cost$(T,e)$ is the replacement cost of the edge $e$, we prove that if we impose a set $I$ of nontree edges of $T$ then $\sum_{e \in I} $ r-cost$(T,e) \leq$ cost$(T_{e \in I})$, where $I$ is the set of imposed edges and $T_{e \in I}$ a minimum spanning tree containing all the edges of $I$.
Given a spatio-temporal network (ST network) where edge properties vary with time, a time-sub-interval minimum spanning tree (TSMST) is a collection of minimum spanning trees of the ST network, where each tree is associated with a time interval. During this time interval, the total cost of tree is least among all the spanning trees. The TSMST problem aims to identify a collection of distinct minimum spanning trees and their respective time-sub-intervals under the constraint that the edge weight functions are piecewise linear. This is an important problem in ST network application domains such …
We analyze a new general representation for the Minimum Weight Steiner Tree (MST) problem which translates the topological connectivity constraint into a set of local conditions which can be analyzed by the so called cavity equations techniques. For the limit case of the Spanning tree we prove that the fixed point of the algorithm arising from the cavity equations leads to the global optimum.
The minimum degree spanning tree (MDST) problem requires the construction of a spanning tree $T$ for graph $G=(V,E)$ with $n$ vertices, such that the maximum degree $d$ of $T$ is the smallest among all spanning trees of $G$. In this paper, we present two new distributed approximation algorithms for the MDST problem. Our first result is a randomized distributed algorithm that constructs a spanning tree of maximum degree $\hat d = O(d\log{n})$. It requires $O((D + \sqrt{n}) \log^2 n)$ rounds (w.h.p.), where $D$ is the graph diameter, which matches (within log factors) the optimal round complexit…
We study approaches for the exact solution of the \NP--hard minimum spanning tree problem under conflict constraints. Given a graph $G(V,E)$ and a set $C \subset E \times E$ of conflicting edge pairs, the problem consists of finding a conflict-free minimum spanning tree, i.e. feasible solutions are allowed to include at most one of the edges from each pair in $C$. The problem was introduced recently in the literature, with several results on its complexity and approximability. Some formulations and both exact and heuristic algorithms were also discussed, but computational results indicate cons…
In this paper, we present a fully-dynamic distributed algorithm for maintaining a minimum spanning tree on general graphs with positive real edge weights. The goal of a dynamic MST algorithm is to update efficiently the minimum spanning tree after dynamic changes like edge weight changes, rather than having to recompute it from scatch each time. The first part of the paper surveys various algorithms available today both in sequential and distributed environments to solve static MST problem. We also present some of the efficient sequential algorithms for computing dynamic MST like the Frederick…
Given a graph $G = (V, E)$, we wish to compute a spanning tree whose maximum vertex degree, i.e. tree degree, is as small as possible. Computing the exact optimal solution is known to be NP-hard, since it generalizes the Hamiltonian path problem. For the approximation version of this problem, a $\tilde{O}(mn)$ time algorithm that computes a spanning tree of degree at most $Δ^* +1$ is previously known [Fürer \& Raghavachari 1994]; here $Δ^*$ denotes the minimum tree degree of all the spanning trees. In this paper we give the first near-linear time approximation algorithm for this problem. Speci…
We study three representations of hierarchies of partitions: dendrograms (direct representations), saliency maps, and minimum spanning trees. We provide a new bijection between saliency maps and hierarchies based on quasi-flat zones as used in image processing and characterize saliency maps and minimum spanning trees as solutions to constrained minimization problems where the constraint is quasi-flat zones preservation. In practice, these results form a toolkit for new hierarchical methods where one can choose the most convenient representation. They also invite us to process non-image data wi…
Given a set $P$ of $n$ points that are moving in the plane, we consider the problem of computing a spanning tree for these moving points that does not change its combinatorial structure during the point movement. The objective is to minimize the bottleneck weight of the spanning tree (i.e., the largest Euclidean length of all edges) during the whole movement. The problem was solved in $O(n^2)$ time previously [Akitaya, Biniaz, Bose, De Carufel, Maheshwari, Silveira, and Smid, WADS 2021]. In this paper, we present a new algorithm of $O(n^{4/3} \log^3 n)$ time.
We study a geometric facility location problem under imprecision. Given $n$ unit intervals in the real line, each with one of $k$ colors, the goal is to place one point in each interval such that the resulting \emph{minimum color-spanning interval} is as large as possible. A minimum color-spanning interval is an interval of minimum size that contains at least one point from a given interval of each color. We prove that if the input intervals are pairwise disjoint, the problem can be solved in $O(n)$ time, even for intervals of arbitrary length. For overlapping intervals, the problem becomes mu…
This paper makes two main contributions: The first is the construction of a near-minimum spanning tree with constant average distortion. The second is a general equivalence theorem relating two refined notions of distortion: scaling distortion and prioritized distortion. Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning…
A directed graph $G=(V,E)$ is strongly biconnected if $G$ is strongly connected and its underlying graph is biconnected. A strongly biconnected directed graph $G=(V,E)$ is called $2$-vertex-strongly biconnected if $|V|\geq 3$ and the induced subgraph on $V\setminus\left\lbrace w\right\rbrace $ is strongly biconnected for every vertex $w\in V$. In this paper we study the following problem. Given a $2$-vertex-strongly biconnected directed graph $G=(V,E)$, compute an edge subset $E^{2sb} \subseteq E$ of minimum size such that the subgraph $(V,E^{2sb})$ is $2$-vertex-strongly biconnected.
In this lecture we will consider the minimum weight spanning tree (MST) problem, i.e., one of the simplest and most vital combinatorial optimization problems. We will discuss a particular greedy algorithm that allows to compute a MST for undirected weighted graphs, namely Kruskal's algorithm, and we will study the structure of MSTs obtained for weighted scale free random graphs. This is meant to clarify whether the structure of MSTs is sensitive to correlations between edge weights and topology of the underlying scale free graphs.
Formulations of graph algorithms using sparse linear algebra have yielded highly scalable distributed algorithms for problems such as connectivity and shortest path computation. We develop the first formulation of the Awerbuch-Shiloach parallel minimum spanning forest (MSF) algorithm using linear algebra primitives. We introduce a multilinear kernel that operates on an adjacency matrix and two vectors. This kernel updates graph vertices by simultaneously using information from both adjacent edges and vertices. In addition, we explore optimizations to accelerate the shortcutting step in the Awe…
This paper gives a new deterministic algorithm for the dynamic Minimum Spanning Forest (MSF) problem in the EREW PRAM model, where the goal is to maintain a MSF of a weighted graph with $n$ vertices and $m$ edges while supporting edge insertions and deletions. We show that one can solve the dynamic MSF problem using $O(\sqrt n)$ processors and $O(\log n)$ worst-case update time, for a total of $O(\sqrt n \log n)$ work. This improves on the work of Ferragina [IPPS 1995] which costs $O(\log n)$ worst-case update time and $O(n^{2/3} \log{\frac{m}{n}})$ work.
We present a novel self-stabilizing algorithm for minimum spanning tree (MST) construction. The space complexity of our solution is $O(\log^2n)$ bits and it converges in $O(n^2)$ rounds. Thus, this algorithm improves the convergence time of all previously known self-stabilizing asynchronous MST algorithms by a multiplicative factor $Θ(n)$, to the price of increasing the best known space complexity by a factor $O(\log n)$. The main ingredient used in our algorithm is the design, for the first time in self-stabilizing settings, of a labeling scheme for computing the nearest common ancestor with …
A Surprisingly Simple Method for Distributed Euclidean-Minimum Spanning Tree / Single Linkage Dendrogram Construction from High Dimensional Embeddings via Distance Decomposition
We present a deterministic algorithm for computing the sensitivity of a minimum spanning tree (MST) or shortest path tree in $O(m\logα(m,n))$ time, where $α$ is the inverse-Ackermann function. This improves upon a long standing bound of $O(mα(m,n))$ established by Tarjan. Our algorithms are based on an efficient split-findmin data structure, which maintains a collection of sequences of weighted elements that may be split into smaller subsequences. As far as we are aware, our split-findmin algorithm is the first with superlinear but sub-inverse-Ackermann complexity. We also give a reduction fro…
Choi et. al (2011) introduced a minimum spanning tree (MST)-based method called CLGrouping, for constructing tree-structured probabilistic graphical models, a statistical framework that is commonly used for inferring phylogenetic trees. While CLGrouping works correctly if there is a unique MST, we observe an indeterminacy in the method in the case that there are multiple MSTs. In this work we remove this indeterminacy by introducing so-called vertex-ranked MSTs. We note that the effectiveness of CLGrouping is inversely related to the number of leaves in the MST. This motivates the problem of f…
We present a novel self-stabilizing algorithm for minimum spanning tree (MST) construction. The space complexity of our solution is $O(\log^2n)$ bits and it converges in $O(n^2)$ rounds. Thus, this algorithm improves the convergence time of previously known self-stabilizing asynchronous MST algorithms by a multiplicative factor $Θ(n)$, to the price of increasing the best known space complexity by a factor $O(\log n)$. The main ingredient used in our algorithm is the design, for the first time in self-stabilizing settings, of a labeling scheme for computing the nearest common ancestor with only…
In this paper I present general outlook on questions relevant to the basic graph algorithms; Finding the Shortest Path with Positive Weights and Minimum Spanning Tree. I will show so far known solution set of basic graph problems and present my own. My solutions to graph problems are characterized by their linear worst-case time complexity. It should be noticed that the algorithms which compute the Shortest Path and Minimum Spanning Tree problems not only analyze the weight of arcs (which is the main and often the only criterion of solution hitherto known algorithms) but also in case of identi…
Wu and Grumbach introduced the concept of strongly biconnected directed graphs. A directed graph $G=(V,E)$ is called strongly biconnected if the directed graph $G$ is strongly connected and the underlying undirected graph of $G$ is biconnected. A strongly biconnected directed graph $G=(V,E)$ is said to be $2$- edge strongly biconnected if it has at least three vertices and the directed subgraph $(V,E\setminus\left\lbrace e\right\rbrace )$ is strongly biconnected for all $e \in E$. Let $G=(V,E)$ be a $2$-edge-strongly biconnected directed graph. In this paper we study the problem of computing a…
In the minimum $k$-edge-connected spanning subgraph ($k$-ECSS) problem the goal is to find the minimum weight subgraph resistant to up to $k-1$ edge failures. This is a central problem in network design, and a natural generalization of the minimum spanning tree (MST) problem. While the MST problem has been studied extensively by the distributed computing community, for $k \geq 2$ less is known in the distributed setting. In this paper, we present fast randomized distributed approximation algorithms for $k$-ECSS in the CONGEST model. Our first contribution is an $\widetilde{O}(D + \sqrt{n})$-ro…
Motivated by applications to sensor networks, as well as to many other areas, this paper studies the construction of minimum-degree spanning trees. We consider the classical node-register state model, with a weakly fair scheduler, and we present a space-optimal \emph{silent} self-stabilizing construction of minimum-degree spanning trees in this model. Computing a spanning tree with minimum degree is NP-hard. Therefore, we actually focus on constructing a spanning tree whose degree is within one from the optimal. Our algorithm uses registers on $O(\log n)$ bits, converges in a polynomial number…
A distributed proof (also known as local certification, or proof-labeling scheme) is a mechanism to certify that the solution to a graph problem is correct. It takes the form of an assignment of labels to the nodes, that can be checked locally. There exists such a proof for the minimum spanning tree problem, using $O(\log n \log W)$ bit labels (where $n$ is the number of nodes in the graph, and $W$ is the largest weight of an edge). This is due to Korman and Kutten who describe it in concise and formal manner in [Korman and Kutten 07]. In this note, we propose a more intuitive description of t…
We present a new algorithm, which solves the problem of distributively finding a minimum diameter spanning tree of any (non-negatively) real-weighted graph $G = (V,E,ω)$. As an intermediate step, we use a new, fast, linear-time all-pairs shortest paths distributed algorithm to find an absolute center of $G$. The resulting distributed algorithm is asynchronous, it works for named asynchronous arbitrary networks and achieves $\mathcal{O}(|V|)$ time complexity and $\mathcal{O}\left(|V|\,|E|\right)$
We consider the minimum spanning tree problem with predictions, using the weight-arrival model, i.e., the graph is given, together with predictions for the weights of all edges. Then the actual weights arrive one at a time and an irrevocable decision must be made regarding whether or not the edge should be included into the spanning tree. In order to assess the quality of our algorithms, we define an appropriate error measure and analyze the performance of the algorithms as a function of the error. We prove that, according to competitive analysis, the simplest algorithm, Follow-the-Predictions…
The Stackelberg Minimum Spanning Tree Game is a two-level combinatorial pricing problem played on a graph representing a network. Its edges are colored either red or blue, and the red edges have a given fixed cost, representing the competitor's prices. The first player chooses an assignment of prices to the blue edges, and the second player then buys the cheapest spanning tree, using any combination of red and blue edges. The goal of the first player is to maximize the total price of purchased blue edges. We study this problem in the cases of planar and bounded-treewidth graphs. We show that t…
We consider the minimum spanning tree problem in a setting where information about the edge weights of the given graph is uncertain. Initially, for each edge $e$ of the graph only a set $A_e$, called an uncertainty area, that contains the actual edge weight $w_e$ is known. The algorithm can `update' $e$ to obtain the edge weight $w_e \in A_e$. The task is to output the edge set of a minimum spanning tree after a minimum number of updates. An algorithm is $k$-update competitive if it makes at most $k$ times as many updates as the optimum. We present a 2-update competitive algorithm if all areas…
We present an implementation and an experimental evaluation of an algorithm that, given a connected graph G (represented by adjacency lists), estimates in sublinear time, with a relative error, the Minimum Spanning Tree Weight of G; the original algorithm has been presented in "Approximating the minimum spanning tree weight in sublinear time", by Bernard Chazelle, Ronitt Rubinfeld, and Luca Trevisan (published with SIAM, DOI 10.1137/S0097539702403244). Since the theoretical performances have already been shown and demonstrated in the above-mentioned paper, our goal is, exclusively, to experime…
Given a directed graph $G$ on $n$ vertices with a special vertex $s$, the directed minimum degree spanning tree problem requires computing a incoming spanning tree rooted at $s$ whose maximum tree in-degree is the smallest among all such trees. The problem is known to be NP-hard, since it generalizes the Hamiltonian path problem. The best LP-based polynomial time algorithm can achieve an approximation of $Δ^*+2$ [Bansal et al, 2009], where $Δ^*$ denotes the optimal maximum tree in-degree. As for purely combinatorial algorithms (algorithms that do not use LP), the best approximation is $O(Δ^*+\…
We consider problems related to the combinatorial game (Free-)Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of flooding operations. We show that the minimum number of moves required to flood any given graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T. This result is then applied to give two polynomial-time algorithms for flood-filling problems. Firstly, we can compute in polynomial time the minimum number of moves required to flood a graph with only a polynomial number of co…
We give a Las Vegas data structure which maintains a minimum spanning forest in an n-vertex edge-weighted dynamic graph undergoing updates consisting of any mixture of edge insertions and deletions. Each update is supported in O(n^{1/2 - c}) expected worst-case time for some constant c > 0 and this worst-case bound holds with probability at least 1 - n^{-d} where d is a constant that can be made arbitrarily large. This is the first data structure achieving an improvement over the O(n^{1/2}) deterministic worst-case update time of Eppstein et al., a bound that has been standing for nearly 25 ye…
The minimum-cost $k$-edge-connected spanning subgraph ($k$-ECSS) problem is a generalization and strengthening of the well-studied minimum-cost spanning tree (MST) problem. While the round complexity of distributedly computing the latter has been well-understood, the former remains mostly open, especially as soon as $k\geq 3$. In this paper, we present the first distributed algorithm that computes an approximation of $k$-ECSS in sublinear time for general $k$. Concretely, we describe a randomized distributed algorithm that, in $\tilde{O}(k(D+k\sqrt{n}))$ rounds, computes a $k$-edge-connected s…
The minimum spanning tree (MST) construction is a classical problem in Distributed Computing for creating a globally minimized structure distributedly. Self-stabilization is versatile technique for forward recovery that permits to handle any kind of transient faults in a unified manner. The loop-free property provides interesting safety assurance in dynamic networks where edge-cost changes during operation of the protocol. We present a new self-stabilizing MST protocol that improves on previous known ap- proaches in several ways. First, it makes fewer system hypotheses as the size of the netwo…
