Connectivity Augmentation Problem Research Articles

Better-than-$$\frac{4}{3}$$-approximations for leaf-to-leaf tree and connectivity augmentation

AbstractThe Connectivity Augmentation Problem (CAP) together with a well-known special case thereof known as the Tree Augmentation Problem (TAP) are among the most basic Network Design problems. There has been a surge of interest recently to find approximation algorithms with guarantees below 2 for both TAP and CAP, culminating in the currently best approximation factor for both problems of 1.393 through quite sophisticated techniques. We present a new and arguably simple matching-based method for the well-known special case of leaf-to-leaf instances. Combining our work with prior techniques, we readily obtain a $$(4/3+\varepsilon )$$ ( 4 / 3 + ε ) -approximation for Leaf-to-Leaf CAP by returning the better of our solution and one of an existing method. Prior to our work, a $$4/3$$ 4 / 3 -guarantee was only known for Leaf-to-Leaf TAP instances on trees of height 2. Moreover, when combining our technique with a recently introduced stack analysis approach, which is part of the above-mentioned 1.393-approximation, we can further improve the approximation factor to 1.29, obtaining for the first time a factor below $$\frac{4}{3}$$ 4 3 for a nontrivial class of TAP/CAP instances.

Bridging the Gap Between Tree and Connectivity Augmentation: Unified and Stronger Approaches

We consider the Connectivity Augmentation Problem (CAP), a classical problem in the area of Survivable Network Design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a $k$-edge-connected graph $G=(V,E)$ and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to $G$ makes the graph $(k+1)$-edge-connected. If $k$ is odd, the problem is known to reduce to the Tree Augmentation Problem (TAP) -- i.e., $G$ is a spanning tree -- for which significant progress has been achieved recently, leading to approximation factors below $1.5$ (the currently best factor is $1.458$). However, advances on TAP did not carry over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli (STOC 2020) managed to obtain the first approximation factor below $2$ for CAP by presenting a $1.91$-approximation algorithm based on a method that is disjoint from recent advances for TAP. We first bridge the gap between TAP and CAP, by presenting techniques that allow for leveraging insights and methods from TAP to approach CAP. We then introduce a new way to get approximation factors below $1.5$, based on a new analysis technique. Through these ingredients, we obtain a $1.393$-approximation algorithm for CAP, and therefore also TAP. This leads to the currently best approximation result for both problems in a unified way, by significantly improving on the above-mentioned $1.91$-approximation for CAP and also the previously best approximation factor of $1.458$ for TAP by Grandoni, Kalaitzis, and Zenklusen (STOC 2018). Additionally, a feature we inherit from recent TAP advances is that our approach can deal with the weighted setting when the ratio between the largest to smallest cost on extra links is bounded, in which case we obtain approximation factors below $1.5$.

The (2, k)-Connectivity Augmentation Problem: Algorithmic Aspects

Durand de Gevigney and Szigeti (J Gr Theory 91(4):305–325, 2019) have recently given a min–max theorem for the (2, k)-connectivity augmentation problem. This article provides an \(O(n^3(m+ n \text { }\log \text { }n))\) time algorithm to find an optimal solution for this problem.

Sparsest Feedback Selection for Structurally Cyclic Systems With Dedicated Actuators and Sensors in Polynomial Time

This paper deals with optimal feedback selection problem in linear time-invariant (LTI) structured systems for arbitrary pole placement, an important open problem in structured systems. Specifically, we solve the sparsest feedback selection problem for LTI structured systems. In this paper, we consider structurally cyclic systems with dedicated inputs and outputs. For this class of systems, we prove that the sparsest feedback selection problem is equivalent to the strong connectivity augmentation problem in graph theory. We present an $O(n^2)$ algorithm to find a sparsest feedback matrix for structured systems when every state is actuated by a dedicated input and every state is sensed by a dedicated output, where $n$ denotes the number of states in the system. If the inputs and the outputs are such that not every state is actuated by a dedicated input and/or not every state is sensed by a dedicated output, then we provide an $O(n^3)$ algorithm for the sparsest feedback selection problem. These results show that sparsest feedback selection with dedicated i/o is a specific case of the optimal feedback selection problem that is solvable in polynomial time. We later analyze the sparsest feedback selection problem for structurally cyclic systems when both the input and the output are dedicated and the feedback pattern is constrained . When some of the feedback links are forbidden, we prove that the problem is NP-hard. The results in this paper along with the previously known results conclude that the optimal feedback selection problem is polynomial-time solvable only for the dedicated input–output case without forbidden feedback links and also without weights for the feedback links.

On some combinatorial problems in cographs

The family of graphs that can be constructed from isolated vertices by disjoint union and graph join operations are called cographs. These graphs can be represented in a tree-like representation termed parse tree or cotree. In this paper, we study some popular combinatorial problems restricted to cographs. We first present a structural characterization of minimal vertex separators in cographs. Further, we show that listing all minimal vertex separators and the complexity of some constrained vertex separators are polynomial-time solvable in cographs. We propose polynomial-time algorithms for connectivity augmentation problems and its variants in cographs, preserving the cograph property. Finally, using the dynamic programming paradigm, we present a generic framework to solve classical optimization problems such as the longest path, the Steiner path and the minimum leaf spanning tree problems restricted to cographs, our framework yields polynomial-time algorithms for all three problems.

On ‐connected graphs

AbstractA graph is called ‐connected if is ‐edge‐connected and is ‐edge‐connected for every vertex . The study of ‐connected graphs is motivated by a theorem of Thomassen [J. Combin. Theory Ser. A 110 (2015), pp. 67–78] (that was a conjecture of Frank [SIAM J. Discrete Math. 5 (1992), no. 1, pp. 25–53]), which states that a graph has a ‐vertex‐connected orientation if and only if it is (2,2)‐connected. In this paper, we provide a construction of the family of ‐connected graphs for even, which generalizes the construction given by Jordán [J. Graph Theory 52 (2006), pp. 217–229] for (2,2)‐connected graphs. We also solve the corresponding connectivity augmentation problem: given a graph and an integer , what is the minimum number of edges to be added to make ‐connected. Both these results are based on a new splitting‐off theorem for ‐connected graphs.

Minimum Cardinality Point-to-point Connectivity Augmentation Problem

Minimum Cardinality Point-to-point Connectivity Augmentation Problem

Improved Approximation Algorithms for Minimum Cost Node-Connectivity Augmentation Problems

Let κ G (s, t) denote the maximum number of pairwise internally disjoint st-paths in a graph G = (V, E). For a set $T \subseteq V$ of terminals, G is k-T-connected if κ G (s, t) ≥ k for all s, t ∈ T; if T = V then G is k-connected. Given a root node s, G is k- (T, s)-connected if κ G (t, s) ≥ k for all t ∈ T. We consider the corresponding min-cost connectivity augmentation problems, where we are given a graph G = (V, E) of connectivity k, and an additional edge set $\hat E$ on V with costs. The goal is to compute a minimum cost edge set $J \subseteq \hat {E}$ such that $G \cup J$ has connectivity k + 1. For the k-T-Connectivity Augmentation problem when $\hat {E}$ is an edge set on T we obtain ratio $O\left (\ln \frac {|T|}{|T|-k}\right )$ , improving the ratio $O\left (\frac {|T|}{|T|-k} \cdot \ln \frac {|T|}{|T|-k}\right )$ of Nutov (Combinatorica, 34(1), 95–114, 2014). For the k -Connectivity Augmentation problem we obtain the following approximation ratios. For n ≥ 3k − 5, we obtain ratio 3 for directed graphs and 4 for undirected graphs, improving the previous ratio 5 of Nutov (Combinatorica, 34(1), 95–114, 2014). For directed graphs and k = 1, or k = 2 and n odd, we further improve to 2.5 the previous ratios 3 and 4, respectively. For the undirected 2-(T, s)-Connectivity Augmentation problem we achieve ratio $4\frac {2}{3}$ , improving the previous best ratio 12 of Nutov (ACM Trans. Algorithms, 9(1), 1, 2014). For the special case when all the edges in $\hat E$ are incident to s, we give a polynomial time algorithm, improving the ratio $4\frac {17}{30}$ of Kortsarz and Nutov, (2015) and Nutov (Algorithmica, 63(1-2), 398–410, 2012) for this variant.

Experiments with two heuristic algorithms for the Maximum Algebraic Connectivity Augmentation Problem

In this work we present a heuristic algorithm to solve the Maximum Algebraic Connectivity Augmentation Problem (MACAP). This is an NP-complete problem (proved by Mosk-Aoyama in 2008) and consists in, given a graph, determining the smallest set of edges not belonging to it in such a way that the value of the algebraic connectivity of the augmented graph is maximum. In 2006, Ghosh and Boyd presented a heuristic procedure to solve this problem. This heuristic is an iterative method that selects one edge at a time based on the values of the components of a Fiedler vector of the graph. Our goal is to increase the value of the algebraic connectivity of a given graph by inserting edges based on the eccentricity of vertices. In order to evaluate our algorithm, computational tests comparing it with the Ghosh and Boyd procedure are presented.

Fixed-Parameter Algorithms for Minimum-Cost Edge-Connectivity Augmentation

We consider connectivity-augmentation problems in a setting where each potential new edge has a non-negative cost associated with it, and the task is to achieve a certain connectivity target with at most p new edges of minimum total cost. The main result is that the minimum cost augmentation of edge-connectivity from k − 1 to k with at most p new edges is fixed-parameter tractable parameterized by p and admits a polynomial kernel. We also prove the fixed-parameter tractability of increasing edge connectivity from 0 to 2 and increasing node connectivity from 1 to 2.

A unifying approach to splitting-off

We study the behaviour of splitting-off algorithms when applied to the problem of covering a symmetric skew-supermodular set function by a graph. This hard problem is a natural generalization of many solved connectivity augmentation problems, such as local edge-connectivity augmentation of graphs, global arc-connectivity augmentation of mixed graphs with undirected edges, or the node-to-area connectivity augmentation problem in graphs. Using a simple lemma we characterize the situation when a splitting-off algorithm can get stuck. This characterization enables us to give very simple proofs for the results mentioned above. Finally we apply our observations on generalizations of the above problems: we consider three connectivity augmentation problems in hypergraphs where the objective is to use hyperedges of minimum total size without increasing the rank. The first is local edge-connectivity augmentation of undirected hypergraphs. The second is global arc-connectivity augmentation of mixed hypergraphs with undirected hyperedges. The third is a hypergraphic generalization of the node-to-area connectivity augmentation problem. We show that a greedy approach (almost) works in these cases.

Augmenting the Connectivity of Planar and Geometric Graphs

In this paper we study connectivity augmentation problems. Given a connected graph G with some desirable property, we want to make G 2-vertex connected (or 2-edge connected) by adding edges such that the resulting graph keeps the property. The aim is to add as few edges as possible. The property that we consider is planarity, both in an abstract graph-theoretic and in a geometric setting, where vertices correspond to points in the plane and edges to straight-line segments. We show that it is NP-hard to nd a minimum-cardinality augmentation that makes a planar graph 2-edge connected. For making a planar graph 2-vertex connected this was known. We further show that both problems are hard in the geometric setting, even when restricted to trees. The problems remain hard for higher degrees of connectivity. On the other hand we give polynomial-time algorithms for the special case of convex geometric graphs. We also study the following related problem. Given a planar (plane geometric) graph G, two vertices s and t of G, and an integer c, how many edges have to be added to G such that G is still planar (plane geometric) and contains c edge- (or vertex-) disjoint s{t paths? For the planar case we give a linear-time algorithm for c = 2. For the plane geometric case we give optimal worst-case bounds for c = 2; for c = 3 we characterize the cases that have a solution.

An algorithm for [formula omitted]-connectivity augmentation problem: Jump system approach

We consider the problem of making a given (k−1)-connected graph k-connected by adding a minimum number of new edges, which we call the k-connectivity augmentation problem. In this paper, we deal with the problem when k=n−3 where n is the number of vertices of the input graph. By considering the complement graph, the (n−3)-connectivity augmentation problem can be reduced to the problem of finding a maximum square-free 2-matching in a simple graph with maximum degree at most three.We give a polynomial-time algorithm to find a maximum square-free 2-matching in a simple graph with maximum degree at most three, which yields a polynomial-time algorithm for the (n−3)-connectivity augmentation problem. Our algorithm is based on the fact that the square-free 2-matchings are endowed with a matroid structure called a jump system. We also show that the weighted (n−3)-connectivity augmentation problem can be solved in polynomial time if the weights are induced by a function on the vertex set, whereas the problem is NP-hard for general weights.

Approximating Node-Connectivity Augmentation Problems

We consider the (undirected) Node Connectivity Augmentation (NCA) problem: given a graph J=(V,E J ) and connectivity requirements $\{r(u,v): u,v \in V\}$, find a minimum size set I of new edges (any edge is allowed) such that the graph J∪I contains r(u,v) internally-disjoint uv-paths, for all u,v∈V. In Rooted NCA there is s∈V such that r(u,v)>0 implies u=s or v=s. For large values of k=max u,v∈V r(u,v), NCA is at least as hard to approximate as Label-Cover and thus it is unlikely to admit an approximation ratio polylogarithmic in k. Rooted NCA is at least as hard to approximate as Hitting-Set. The previously best approximation ratios for the problem were O(kln n) for NCA and O(ln n) for Rooted NCA. In this paper we give an approximation algorithm with ratios O(kln 2 k) for NCA and O(ln 2 k) for Rooted NCA. This is the first approximation algorithm with ratio independent of n, and thus is a constant for any fixed k. Our algorithm is based on the following new structural result which is of independent interest. If $\mathcal{D}$ is a set of node pairs in a graph J, then the maximum degree in the hypergraph formed by the inclusion minimal tight sets separating at least one pair in $\mathcal{D}$ is O(l 2), where l is the maximum connectivity in J of a pair in $\mathcal{D}$.

Approximating connectivity augmentation problems

Let G = ( V , E ) be an undirected graph and let S ⊆ V . The S-connectivity λ S G ( u , v ) of a node pair ( u , v ) in G is the maximum number of uv -paths that no two of them have an edge or a node in S - { u , v } in common. The corresponding Connectivity Augmentation (CA) problem is: given a graph G = ( V , E ), a node subset S ⊆ V , and a nonnegative integer requirement function r ( u , v ) on V × V , add a minimum size set F of new edges to G so that λ S G + F ( u , v ) ≥ r ( u , v ) for all ( u , v ) ∈ V × V . Three extensively studied particular cases are: the Edge-CA ( S = ∅), the Node-CA ( S = V ), and the Element-CA ( r ( u , v )= 0 whenever u ∈ S or v ∈ S ). A polynomial-time algorithm for Edge-CA was developed by Frank. In this article we consider the Element-CA and the Node-CA, that are NP-hard even for r ( u , v ) ∈ {0,2}. The best known ratios for these problems were: 2 for Element-CA and O ( r max ṡ ln n ) for Node-CA, where r max = max u , v ∈ V r ( u , v ) and n = | V |. Our main result is a 7/4-approximation algorithm for the Element-CA, improving the previously best known 2-approximation. For Element-CA with r ( u , v ) ∈ {0,1,2} we give a 3/2-approximation algorithm. These approximation ratios are based on a new splitting-off theorem, which implies an improved lower bound on the number of edges needed to cover a skew-supermodular set function. For Node-CA we establish the following approximation threshold: Node-CA with r ( u , v ) ∈ {0, k } cannot be approximated within O (2 log 1-ϵ n ) for any fixed ϵ > 0, unless NP ⊆ DTIME( n polylog( n ) ).

Kernelization and complexity results for connectivity augmentation problems

Connectivity augmentation problems ask for adding a set of at most k edges (called links) whose insertion makes a given graph satisfy a specified connectivity property, such as bridge-connectivity or biconnectivity. A bridge-connected (biconnected) graph is a connected graph that does not possess an edge (a vertex) whose removal results in a disconnected graph. We show that, for bridge-connectivity and biconnectivity, the respective connectivity augmentation problems admit problem kernels with O(k2) vertices and links. Moreover, we study partial connectivity augmentation problems, naturally generalizing connectivity augmentation problems. Here, we do not require that, after adding the edges, the entire graph should satisfy the connectivity property, but a large subgraph. In this setting, three polynomial-time solvable connectivity augmentation problems behave differently, namely, the partial bridge-connectivity augmentation problem and the partial biconnectivity augmentation problem remain polynomial-time solvable, whereas the partial strong connectivity augmentation problem becomes W[2]-hard with respect to k. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

Augmenting the Connectivity of Planar and Geometric Graphs

In this paper we study connectivity augmentation problems. Given a connected graph G with some desirable property, we want to make G 2-vertex connected (or 2-edge connected) by adding edges such that the resulting graph keeps the property. The aim is to add as few edges as possible. The property that we consider is planarity, both in an abstract graph-theoretic and in a geometric setting, where vertices correspond to points in the plane and edges to straight-line segments. We show that it is NP-hard to nd a minimum-cardinality augmentation that makes a planar graph 2-edge connected. For making a planar graph 2-vertex connected this was known. We further show that both problems are hard in the geometric setting, even when restricted to trees. The problems remain hard for higher degrees of connectivity. On the other hand we give polynomial-time algorithms for the special case of convex geometric graphs. We also study the following related problem. Given a planar (plane geometric) graph G, two vertices s and t of G, and an integer c, how many edges have to be added to G such that G is still planar (plane geometric) and contains c edge- (or vertex-) disjoint s{t paths? For the planar case we give a linear-time algorithm for c = 2. For the plane geometric case we give optimal worst-case bounds for c = 2; for c = 3 we characterize the cases that have a solution.

Tight approximation algorithm for connectivity augmentation problems

The S-connectivity λ G S ( u , v ) of ( u , v ) in a graph G is the maximum number of uv-paths that no two of them have an edge or a node in S − { u , v } in common. The corresponding Connectivity Augmentation ( CA ) problem is: given a graph G 0 = ( V , E 0 ) , S ⊆ V , and requirements r ( u , v ) on V × V , find a minimum size set F of new edges (any edge is allowed) so that λ G 0 + F S ( u , v ) ⩾ r ( u , v ) for all u , v ∈ V . Extensively studied particular choices of S are the edge- CA (when S = ∅ ) and the node- CA (when S = V ). A. Frank gave a polynomial algorithm for undirected edge- CA and observed that the directed case even with rooted { 0 , 1 } -requirements is at least as hard as the Set-Cover problem (in rooted requirements there is s ∈ V − S so that if r ( u , v ) > 0 then: u = s for directed graphs, and u = s or v = s for undirected graphs). Both directed and undirected node- CA have approximation threshold Ω ( 2 log 1 − ε n ) . The only polylogarithmic approximation ratio known for CA was for rooted requirements— O ( log n ⋅ log r max ) = O ( log 2 n ) , where r max = max u , v ∈ V r ( u , v ) . No nontrivial approximation algorithms were known for directed CA even for r ( u , v ) ∈ { 0 , 1 } , nor for undirected CA with S arbitrary. We give an approximation algorithm for the general case that matches the known approximation thresholds. For both directed and undirected CA with arbitrary requirements our approximation ratio is: O ( log n ) for S ≠ V arbitrary, and O ( r max ⋅ log n ) for S = V .

Approximating Rooted Connectivity Augmentation Problems

Approximating Rooted Connectivity Augmentation Problems

Splitting off edges between two subsets preserving the edge-connectivity of the graph

Splitting off a pair of edges su, sv in a graph G means replacing these two edges by a new edge uv. This operation is well known in graph theory. Let G=( V+ s, E+ F) be a graph which is k-edge-connected in V and suppose that | F| is even. Here F denotes the set of edges incident with s. Lovász (Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979) proved that if k⩾2 then the edges in F can be split off in pairs preserving the k-edge-connectivity in V. This result was recently extended to the case where a bipartition R∪ Q= V is given and every split edge must connect R and Q (SIAM J. Discrete Math. 12 (2) (1999) 160). In this paper, we investigate an even more general problem, where two disjoint subsets R, Q⊂ V are given and the goal is to split off (the largest possible subset of) the edges of F preserving k-edge-connectivity in V in such a way that every split edge incident with a vertex from R has the other end-vertex in Q. Motivated by connectivity augmentation problems, we introduce another extension, the so-called split completion version of our problem. Here, the smallest set F ∗ of edges incident to s has to be found for which all the edges of F+F ∗ can be split off in the augmented graph G=(V+s, E+F+F ∗) preserving k-edge-connectivity and in such a way that every split edge incident with a vertex from R has the other end-vertex in Q. We solve each of the above extensions when k is even: we give min–max formulae and polynomial algorithms to find the optima. For the case when k is odd we show how to find a solution to the split completion problem using at most two edges more than the optimum.