Augmentation Problems Research Articles

Approximation algorithms for node and element connectivity augmentation problems

Approximation algorithms for node and element connectivity augmentation problems

Quantifying the pedestrian access potential of suburban street network retrofits

The United States largely depends on the automobile for personal transportation. This dominance has significant consequences for society over a range of issues, including the environment, public safety, public health, and equity. The issues associated with the dominance of the automobile are most pressing in the suburbs due to their size and curvilinear street network patterns. Thus, any effort to address the negative consequences of automobile dependency in the US needs to consider retrofitting the suburbs and their street networks. We attempt to better understand the potential for street network retrofits to increase suburban pedestrian access. We consider a class of planar graph augmentation problems that attempt to increase pedestrian access to points of interest (POIs) within the study area by adding new pedestrian paths to the street network that follow existing property lines. Our methodology builds on past work on graph dilation and route directness, from the planar graph and street network communities, respectively, to score the pedestrian access disruption of individual blocks. We apply this methodology to a case study of suburban Seattle. We find that, both in the limit of all possible interventions and with a limited number of untargeted interventions, retrofits can meaningfully increase pedestrian access to POIs. Given this promise, the methods we outline present a useful starting point for discussing the potential of street network retrofits to serve non-automobile mobility in suburban communities across the US.

Current Trends in Bone Augmentation Techniques and Dental Implantology: An Editorial Overview

Dental implants and related bone augmentation problems have seen major progress since early protocols were tested in the 1980s [...].

Tight Bounds for Online Weighted Tree Augmentation

The Weighted Tree Augmentation problem (WTAP) is a fundamental problem in network design. In this paper, we consider this problem in the online setting. We are given an n-vertex tree \(T = (V,E)\) and an additional set \(L \subseteq {V \atopwithdelims ()2}\) of edges (called links) with costs. Then, terminal pairs arrive one-by-one and our task is to maintain a low-cost subset of links F such that every terminal pair that has arrived so far is 2-edge-connected in \(T \cup F\). This online problem was first studied by Gupta, Krishnaswamy and Ravi (SICOMP 2012) who used it as a subroutine for the online survivable network design problem. They gave a deterministic \(O(\log ^2 n)\)-competitive algorithm and showed an \(\varOmega (\log n)\) lower bound on the competitive ratio of randomized algorithms. The case when T is a path is also interesting: it is exactly the online interval set cover problem, which also captures as a special case the parking permit problem studied by Meyerson (FOCS 2005). The contribution of this paper is to give tight results for online weighted tree and path augmentation problems. The main result of this work is a deterministic \(O(\log n)\)-competitive algorithm for online WTAP, which is tight up to constant factors.

Complexity and algorithms for constant diameter augmentation problems

We study the following problem: for given integers d,k and graph G, can we obtain a graph with diameter d via at most k edge deletions? We determine the computational complexity of this and related problems for different values of d.

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.

Shortcut sets for the locus of plane Euclidean networks

We study the problem of augmenting the locus Nℓ of a plane Euclidean network N by inserting iteratively a finite set of segments, called shortcut set, while reducing the diameter of the locus of the resulting network. There are two main differences with the classical augmentation problems: the endpoints of the segments are allowed to be points of Nℓ as well as points of the previously inserted segments (instead of only vertices of N), and the notion of diameter is adapted to the fact that we deal with Nℓ instead of N. This increases enormously the hardness of the problem but also its possible practical applications to network design. Among other results, we characterize the existence of shortcut sets, compute them in polynomial time, and analyze the role of the convex hull of Nℓ when inserting a shortcut set. Our main results prove that, while the problem of minimizing the size of a shortcut set is NP-hard, one can always determine in polynomial time whether inserting only one segment suffices to reduce the diameter.

Structured Connectivity Augmentation

We initiate the algorithmic study of the following “structured augmentation” question: is it possible to increase the connectivity of a given graph $G$ by superposing it with another given graph $H$? More precisely, graph $F$ is the superposition of $G$ and $H$ with respect to injective mapping $\varphi\colon V(H)\rightarrow V(G)$ if every edge $uv$ of $F$ is either an edge of $G$ or $\varphi^{-1}(u)\varphi^{-1}(v)$ is an edge of $H.$ Thus $F$ contains both $G$ and $H$ as subgraphs, and the edge set of $F$ is the union of the edge sets of $G$ and $\varphi(H).$ We consider the following optimization problem. Given graphs $G,$ $H,$ and a weight function $\omega$ assigning nonnegative weights to pairs of vertices of $V(G),$ the task is to find $\varphi$ of minimum weight $\omega(\varphi)= \sum_{xy\in E(H)}\omega(\varphi(x)\varphi(y))$ such that the edge connectivity of the superposition $F$ of $G$ and $H$ with respect to $\varphi$ is higher than the edge connectivity of $G$. Our main result is the following “dichotomy” complexity classification. We say that a class of graphs $\mathcal{C}$ has bounded vertex-cover number if there is a constant $t$ depending on $\mathcal{C}$ only such that the vertex-cover number of every graph from $\mathcal{C}$ does not exceed $t.$ We show that for every class of graphs $\mathcal{C}$ with bounded vertex-cover number, the problems of superposing into a connected graph $F$ and to 2-edge connected graph $F$ are solvable in polynomial time when $H\in\mathcal{C}.$ On the other hand, for any hereditary class $\mathcal{C}$ with unbounded vertex-cover number, both problems are NP-hard when $H\in\mathcal{C}.$ For the unweighted variants of structured augmentation problems, i.e., the problems where the task is to identify whether there is a superposition of graphs of required connectivity, we provide necessary and sufficient combinatorial conditions on the existence of such superpositions. These conditions imply polynomial time algorithms solving the unweighted variants of the problems.

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.

Complexity of Network Design for Private Communication and the P-vs-NP Question

We investigate infeasibility issues arising along net- work design for information-theoretically secure cryptography. In particular, we consider the problem of communication in perfect privacy and formally relate it to graph augmentation problems and the P-vs-NP-question. Based on a game-theoretic privacy measure, we consider two optimization problems related to secure infrastructure design with constraints on computational efforts and limited budget to build a transmission network. It turns out that information-theoretic security, although not drawing its strength from computational infeasibility, still can run into complexity-theoretic difficulties at the stage of physical network design. Even worse, if we measure (quantify) secrecy by the probability of information-leakage, we can prove that approximations of a network design towards maximal security are computationally equivalent to the exact solutions to the same problem, both of which are again equivalent to asserting that P = NP. In other words, the death of public-key cryptosystems upon P = NP may become the birth of feasible network design algorithms towards information-theoretically confidential communication. Index Terms—Complexity; NP; Privacy; Security; Game The- ory; Graph Augmentation

Augmenting Edge-Connectivity between Vertex Subsets

Given a graph G = (V, E) and a requirement function r: W1 x W2 → R+ for two families W1, W2 ⊆ 2V - {θ}, we consider the problem (called area-to-area edge-connectivity augmentation problem) of augmenting G by a smallest number of new edges so that the resulting graph G satisfies δG (X) ≥ r(W1, W2) for all X ⊆ V, W1 ∈ W1, and W2 ∈ W2 with W1 ⊆ X ⊆ V - W2, where δG(X) denotes the degree of a vertex set X in G. This problem can be regarded as a natural generalization of the global, local, and node-to-area edge-connectivity augmentation problems. In this paper, we show that there exists a constant c such that the problem is inapproximable within a ratio of clog α(W1, W2), unless P=NP, even restricted to the directed global node-to-area edge-connectivity augmentation or undirected local node-to-area edge-connectivity augmentation, where α(W1, W2) denotes the number of pairs W1 ∈ W1 and W2 ∈ W2 with r(W1, W2) > 0. We also provide an O(log α (W1, W2))-approximation algorithm for the area-to-area edge-connectivity augmentation problem. This together with the negative result implies that the problem is Θ(log α (W1, W2))-approximable, unless P=NP, which solves open problems for node-to-area edge-connectivity augmentation (Ishii et al. 2008, Ishii and Hagiwara 2006, Miwa and Ito 2004). Furthermore, we characterize the node-to-area and area-to-area edge-connectivity augmentation problems as the augmentation problems with modulotone and extended modulotone functions.

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.

Restless legs syndrome (RLS) augmentation associated with dopamine agonist and levodopa usage in a community sample

Objective Assess the rate of augmentation as it occurs during standard long-term dopaminergic treatment of RLS, potential risk factors or predictors of augmentation, the relationship between treatment duration and augmentation, and the clinical impact of augmentation on subjects’ health outcomes. Methods Two hundred sixty-six patients with dopamine-treated RLS completed a one-time online survey. All subjects were recruited by their PCP/neurologist and were 18 or older. Augmentation was assessed using NIH guidelines and an augmentation classification system was developed through this research. Results Overall, 20% of the patients were classified as having definitive or highly suggestive clinical indications of augmentation. Five factors were considered likely to reflect increased risk of developing augmentation, including more frequent RLS symptoms pre-treatment, greater discomfort with RLS symptoms before treatment, and longer treatment duration. RLS augmentation occurred at a rate of about 8% each year for at least the first 8 years of dopamine treatment. Subjects reporting definite or highly suggestive clinical indicators of augmentation had an average IRLS score of 23.6, indicating generally inadequate treatment with generally poor clinical outcomes. Only 25% of the patients reported no indications of augmentation and they were the only group to show on average a low (<15) IRLS score and good clinical outcomes. Conclusions As currently used, long term dopaminergic treatment for an average ± SD of 2.7 ± 2.4 years produced significant augmentation problems in at least 20% of the patients and only 25% of the patients were totally free of this problem. It is important for physicians to carefully screen patients for changes in RLS symptoms for as long as they are on dopamine agents, with particular attention paid to those patients who present with the most severe RLS symptoms prior to treatment initiation. Given the marked increase in suffering with augmentation, a method for early detection and intervention would be an important contribution to the effective management and treatment of RLS.

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 &gt; 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.

Smallest Bipartite Bridge-Connectivity Augmentation

This paper addresses two augmentation problems related to bipartite graphs. The first, a fundamental graph-theoretical problem, is how to add a set of edges with the smallest possible cardinality so that the resulting graph is 2-edge-connected, i.e., bridge-connected, and still bipartite. The second problem, which arises naturally from research on the security of statistical data, is how to add edges so that the resulting graph is simple and does not contain any bridges. In both cases, after adding edges, the graph can be either a simple graph or, if necessary, a multi-graph. Our approach then determines whether or not such an augmentation is possible. We propose a number of simple linear-time algorithms to solve both problems. Given the well-known bridge-block data structure for an input graph, the algorithms run in O(log n) parallel time on an EREW PRAM using a linear number of processors, where n is the number of vertices in the input graph. We note that there is already a polynomial time algorithm that solves the first augmentation problem related to graphs with a given general partition constraint in O(n(m+nlog n)log n) time, where m is the number of distinct edges in the input graph. We are unaware of any results for the second problem.

Edge-splittings preserving local edge-connectivity of graphs

Let G = ( V + s , E ) be a 2-edge-connected graph with a designated vertex s. A pair of edges rs , st is called admissible if splitting off these edges (replacing rs and st by rt) preserves the local edge-connectivity (the maximum number of pairwise edge disjoint paths) between each pair of vertices in V . The operation splitting off is very useful in graph theory, it is especially powerful in the solution of edge-connectivity augmentation problems as it was shown by Frank [Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Math. 5(1) (1992) 22–53]. Mader [A reduction method for edge-connectivity in graphs, Ann. Discrete Math. 3 (1978) 145–164] proved that if d ( s ) ≠ 3 then there exists an admissible pair incident to s. We generalize this result by showing that if d ( s ) ⩾ 4 then there exists an edge incident to s that belongs to at least ⌊ d ( s ) / 3 ⌋ admissible pairs. An infinite family of graphs shows that this bound is best possible. We also refine a result of Frank [On a theorem of Mader, Discrete Math. 101 (1992) 49–57] by describing the structure of the graph if an edge incident to s belongs to no admissible pairs. This provides a new proof for Mader's theorem.

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 .