Parameterized Analysis of the Online Priority and Node-Weighted Steiner Tree Problems

We obtain the first online algorithms for the node-weighted Steiner tree, Steiner forest and group Steiner tree problems that achieve a poly-logarithmic competitive ratio. Our algorithm for the Steiner tree problem runs in polynomial time, while those for the other two problems take quasi-polynomial time. Our algorithms can be viewed as online LP rounding algorithms in the framework of Buchbinder and Naor (Foundations and Trends in Theoretical Computer Science, 2009); however, while the natural LP formulation of these problems do lead to fractional algorithms with a poly-logarithmic competitive ratio, we are unable to round these LPs online without losing a polynomial factor. Therefore, we design new LP formulations for these problems drawing on a combination of paradigms such as spider decompositions, low-depth Steiner trees, generalized group Steiner problems, etc. and use the additional structure provided by these to round the more sophisticated LPs losing only a poly-logarithmic factor in the competitive ratio. As further applications of our techniques, we also design polynomial-time online algorithms with poly-logarithmic competitive ratios for two fundamental network design problems in edge-weighted graphs: the group Steiner forest problem (thereby resolving an open question raised by Chekuri et. al. (SODA 2008)) and the single source ℓ-vertex connectivity problem (which complements similar results for the corresponding edge-connectivity problem due to Gupta et. al. (STOC 2009)).

A bipartite graph G(X, Y) whose vertex set is partitioned into X and Y is a convex bipartite graph, if there is an ordering of $$X=(x_1,\ldots ,x_m)$$ such that for all $$y \in Y$$ , $$N_G(y)$$ is consecutive with respect to the ordering of X, and G is said to have convexity with respect to X. A k-star caterpillar is a tree with a collection of stars with each star having k vertices of degree one whose roots are joined by a path. For a bipartite graph with partitions X and Y, we associate a k-star caterpillar on X such that for each vertex in Y, its neighborhood induces a tree. The minimum Steiner tree problem (STREE) is defined as follows: given a connected graph $$G=(V,E)$$ and a subset of vertices $$R \subseteq V(G)$$ , the objective is to find a minimum cardinality set $$S \subseteq V(G)$$ such that the set $$R \cup S$$ induces a connected subgraph. In this paper, we present the following dichotomy result: we show that STREE is NP-complete for 1-star caterpillar convex bipartite graphs and polynomial-time solvable for 0-star caterpillar convex bipartite graphs (also known as convex bipartite graphs). We also strengthen the well-known result of Müller and Brandstädt (Theoret Comput Sci 53(2-3):257-265, 1987), which says STREE in chordal bipartite graphs is NP-complete (reduction instances are 3-star caterpillar convex bipartite graphs). As an application, we use our STREE results to solve: (i) the classical dominating set problem in convex bipartite graphs, (ii) STREE on interval graphs, linear time.

The following problems are worth studying: Open problems on the Steiner ratio, such as Chung-Gilbert’s conjecture, Graham-Hwang’s conjecture, and Cielick’s conjecture, etc.. Find better approximation for network Steiner trees and establish an explicit lower bound for the approximation performance ratio of network Steiner trees. Close the gap between the lower bound and the upper bound for the approximation performance ratio of Steiner minimum arborescence. Find more efficient approximation algorithms for on-line and dynamic Steiner minimum trees and various Steiner tree packing problems. Find good approximation algorithms for multi-weighted Steiner trees and multiphase Steiner trees and study close relationship between multi-weighted Steiner trees, multiphase Steiner trees, and phylogenetic trees. Make clear whether there exists a polynomial-time approximation scheme for class Steiner tree in the special case with the real world background and highway interconnection problem. Close the gap between the lower bound and the upper bound for the approximation performance ratio of bottleneck Steiner tree in the Euclidean plane and make clear whether there exists a polynomial-time approximation scheme for the Steiner tree with minimum number of Steiner points and bounded edge-length. Implement efficient approximation algorithms to meet the requests from industries.

A graph G is said to be a bisplit graph if its vertex set can be partitioned into a stable set and a complete bipartite graph. The minimum Steiner tree problem (STREE) is defined as follows: given a connected graph G and a subset of vertices \(R\subseteq V(G)\), the objective is to find a minimum cardinality set \(S\subset V (G)\) such that the set \(R \cup S\) induces a connected subgraph. In this paper, we present an interesting dichotomy result for STREE on bisplit graphs, we show that STREE is polynomial-time solvable for chordal bipartite bisplit graphs, and NP-complete otherwise. Further, we show that for chordal bisplit graphs, the problem is polynomial-time solvable. A revisit of our NP-complete reduction instances reveals that the instances are diameter at most 5 bipartite graphs. We also obtain one more dichotomy result for STREE on bisplit graphs which says that for diameter 5 the problem is NP-complete and polynomial-time solvable for diameter at most 4. On the parameterized complexity front, we show that the parameterized version of Steiner tree problem on bisplit graphs is fixed-parameter tractable when the parameter is the biclique size and is W[2]-hard on bisplit graphs if the parameter is the solution size. We conclude this paper by presenting structural results of bisplit graphs which will be of use to solve other combinatorial problems.KeywordsBisplit graphsThe Steiner tree problemW-hardnessDichotomy results

As the supporting architecture of cryptocurrencies, blockchain is also showing its promising potential to be applied to mobile devices. Being WiFi supported, mobile devices nowadays are able to perform ad-hoc local communications in multi-hop manners, thus can potentially extend cryptocurrency service into areas without stable Internet connection. However, traditional blockchain architecture suffers from the intolerably high block propagation latency and communication cost which will be even trickier if adopted in wireless multi-hop networks (WMNs). This paper aims to reduce this latency and cost by leveraging node weights to distinguish the feedback speed of different nodes and constructing a low-length multicast tree to select a subset of nodes with higher feedback speed to participate in the block verification. Thus, block propagation is depicted by the minimum length multicast tree, intrinsically the Steiner Tree Problem. The primary challenge lies in that the WMNs only allow local communication and the distributed consensus protocol of the blockchain makes a predetermination of receiver nodes impossible. We design our algorithm via a ``toward source'' Steiner tree approach in favor of the distributed environment. Tree construction proceeds by progressively enlarging the searching areas until the accumulated weight of receiver nodes reaches a threshold. Our algorithm provably returns a tree length fairly close to that of the optimal Steiner tree with latency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(\sqrt{n\log n})$</tex-math></inline-formula> (where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> represents the number of nodes in the network), the best so far. Furthermore, our algorithm is empirically validated to provide instructive insights for efficient block propagation with applications in Bitcoin network.

Online algorithms are a hallmark of worst case optimization under uncertainty. On the other hand, in practice, the input is often far from worst case, and has some predictable characteristics. A recent line of work has shown how to use machine learned predictions to circumvent strong lower bounds on competitive ratios in classic online problems such as ski rental and caching. We study how predictive techniques can be used to break through worst case barriers in online scheduling. The makespan minimization problem with restricted assignments is a classic problem in online scheduling theory. Worst case analysis of this problem gives Ω(log m) lower bounds on the competitive ratio in the online setting. We identify a robust quantity that can be predicted and then used to guide online algorithms to achieve better performance. Our predictions are compact in size, having dimension linear in the number of machines, and can be learned using standard off the shelf methods. The performance guarantees of our algorithms depend on the accuracy of the predictions, given predictions with error η, we show how to construct O(log η) competitive fractional assignments. We then give an online algorithm that rounds any fractional assignment into an integral schedule. Our algorithm is O((log log m)3)-competitive and we give a nearly matching Ω(log log m) lower bound for online rounding algorithms.1 Altogether, we give algorithms that, equipped with predictions with error η, achieve O(log η (log log m)3) competitive ratios, breaking the Ω(log m) lower bound even for moderately accurate predictions.

Network design problems often involve scenarios where there exist many possible routes between fixed vertex locations (e.g. building new roads between cities, deploying communications and power lines). Many possible routes in a network result in graph representations with many edges, which can lead to difficulty running computationally demanding optimization algorithms on. While producing a subgraph with a reduced number of edges would in itself be useful, it is also important to preserve the ability to interconnect vertices in a cost effective manner. Suppose there is a set of target vertices in a graph that needs to be part of any size-reduced subgraph. Given an edge-weighted, undirected graph with n vertices, set of target vertices T, and parameter \(k \ge 1\), we introduce an algorithm that produces a subgraph with the number of edges bounded by \(\min \{O(n\frac{n^{1/k}}{\left| {T}\right| }), O(n\left| {T}\right| )\}\) times optimal, while guaranteeing that, for any subset of the target vertices, their minimum Steiner tree in the subgraph costs at most 2k times the cost of their minimum Steiner tree in the original graph. We evaluate our approach against existing algorithms using data from Carbon Capture and Storage studies and find that in addition to its theoretical guarantees, our approach also performs well in practice.

The node-weighted Steiner tree problem is a variation of classical Steiner minimum tree problem. Given a graph G = (V ,E ) with node weight function C :V ***R + and a subset X of V , the node-weighted Steiner tree problem is to find a Steiner tree for the set X such that its total weight is minimum. In this paper, we study this problem in unit disk graphs and present a (1+*** )-approximation algorithm for any *** > 0, when the given set of vertices is c -local. As an application, we use node-weighted Steiner tree to solve the node-weighted connected dominating set problem in unit disk graphs and obtain a (5 + *** )-approximation algorithm.

We consider a model for online computation in which the online algorithm receives, together with each request, some information regarding the future, referred to as advice . The advice provided to the online algorithm may allow an improvement in its performance, compared to the classical model of complete lack of information regarding the future. We are interested in the impact of such advice on the competitive ratio, and in particular, in the relation between the size b of the advice, measured in terms of bits of information per request, and the (improved) competitive ratio. Since b = 0 corresponds to the classical online model, and $ b = \lceil \log |\mathcal{A}| \rceil $, where $\mathcal{A}$ is the algorithm's action space, corresponds to the optimal (offline) one, our model spans a spectrum of settings ranging from classical online algorithms to offline ones. In this paper we propose the above model and illustrate its applicability by considering two of the most extensively studied online problems, namely, metrical task systems (MTS) and the k-server problem. For MTS we establish tight (up to constant factors) upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms with advice for any choice of 1 ≤ b ≤ *** (logn ) , where n is the number of states in the system: we prove that any randomized online algorithm for MTS has competitive ratio *** ( log(n ) / b ) and we present a deterministic online algorithm for MTS with competitive ratio O (log(n ) / b ) . For the k -server problem we construct a deterministic online algorithm for general metric spaces with competitive ratio k O (1 / b ) for any choice of *** (1) ≤ b ≤ logk .

The Node Weighted Steiner Tree Problem (NW-STP) is a generalization of the Steiner Tree Problem. A lagrangean heuristic presented in EngevallS: StrLBN: 98, and based on the work in Lucena: 92, solves the problem by relaxing an exponential family of generalized subtour elimination constraints and taking into account only the violated ones as the computation proceeds. In EngevallS: StrLBN: 98 the computational results refer to complete graphs up to one hundred vertices. In this paper, we present a branch-and-bound algorithm based on this formulation. Its performance on the instances from the literature confirms the effectiveness of the approach. The experimentation on a newly generated set of benchmark problems, more similar to the real-world applications, shows that the approach is still valid, provided that suitable refinements on the bounding procedures and a preprocessing phase are introduced. The algorithm solves to optimality all of the considered instances up to one thousand vertices, with the exception of 11 hard instances, derived from the literature of a similar problem, the Prize Collecting Steiner Tree Problem.

We study network design with a cost structure motivated by redundancy in data traffic. We are given a graph, g groups of terminals, and a universe of data packets. Each group of terminals desires a subset of the packets from its respective source. The cost of routing traffic on any edge in the network is proportional to the total size of the distinct packets that the edge carries. Our goal is to find a minimum cost routing. We focus on two settings. In the first, the collection of packet sets desired by source-sink pairs is laminar. For this setting, we present a primal-dual based 2-approximation, improving upon a logarithmic approximation due to Barman and Chawla (2012){BC12}. In the second setting, packet sets can have non-trivial intersection. We focus on the case where each packet is desired by either a single terminal group or by all of the groups. This setting does not admit an O(log^{{1}/{4} - gamma} g)-approximation for any constant gamma under a standard assumption; we present an O(log g)-approximation when the graph is unweighted. Our approximation for the second setting is based on a novel spanner-type construction in unweighted graphs that, given a collection of g vertex subsets, finds a subgraph of cost only a constant factor more than the minimum spanning tree of the graph, such that every subset in the collection has a Steiner tree in the subgraph of cost at most O(log g) that of its minimum Steiner tree in the original graph. We call such a subgraph a group spanner.

In this article, we study the online Euclidean spanners problem for points in \(\mathbb{R}^{d}\) . Given a set \(S\) of \(n\) points in \(\mathbb{R}^{d}\) , a \(t\) -spanner on \(S\) is a subgraph of the underlying complete graph \(G=(S,\binom{S}{2})\) , that preserves the pairwise Euclidean distances between points in \(S\) to within a factor of \(t\) , that is the stretch factor . Suppose we are given a sequence of \(n\) points \((s_{1},s_{2},\ldots,s_{n})\) in \(\mathbb{R}^{d}\) , where point \(s_{i}\) is presented in step \(i\) for \(i=1,\ldots,n\) . The objective of an online algorithm is to maintain a geometric \(t\) -spanner on \(S_{i}=\{s_{1},\ldots,s_{i}\}\) for each step \(i\) . The algorithm is allowed to add new edges to the spanner when a new point is presented but cannot remove any edge from the spanner. The performance of an online algorithm is measured by its competitive ratio, which is the supremum, over all sequences of points, of the ratio between the weight of the spanner constructed by the algorithm and the weight of an optimum spanner. Here, the weight of a spanner is the sum of all edge weights. First, we establish a lower bound of \(\Omega(\varepsilon^{-1}\log n/\log\varepsilon^{-1})\) for the competitive ratio of any online \((1+\varepsilon)\) -spanner algorithm, for a sequence of \(n\) points in 1-dimension. We show that this bound is tight, and there is an online algorithm that can maintain a \((1+\varepsilon)\) -spanner with competitive ratio \(O(\varepsilon^{-1}\log n/\log\varepsilon^{-1})\) . Next, we design online algorithms for sequences of points in \(\mathbb{R}^{d}\) , for any constant \(d\geq 2\) , under the \(L_{2}\) norm. We show that previously known incremental algorithms achieve a competitive ratio \(O(\varepsilon^{-(d+1)}\log n)\) . However, if the algorithm is allowed to use additional points (Steiner points), then it is possible to substantially improve the competitive ratio in terms of \(\varepsilon\) . We describe an online Steiner \((1+\varepsilon)\) -spanner algorithm with competitive ratio \(O(\varepsilon^{(1-d)/2}\log n)\) . As a counterpart, we show that the dependence on \(n\) cannot be eliminated in dimensions \(d\geq 2\) . In particular, we prove that any online spanner algorithm for a sequence of \(n\) points in \(\mathbb{R}^{d}\) under the \(L_{2}\) norm has competitive ratio \(\Omega(f(n))\) , where \(\lim_{n\rightarrow\infty}f(n)=\infty\) . Finally, we provide improved lower bounds under the \(L_{1}\) norm: \(\Omega(\varepsilon^{-2}/\log\varepsilon^{-1})\) in the plane and \(\Omega(\varepsilon^{-d})\) in \(\mathbb{R}^{d}\) for \(d\geq 3\) .

The problem of sharing the cost of a common infrastructure among a set of strategic and cooperating players has been the subject of intensive research in recent years. However, most of these studies consider cooperative cost sharing games in an offline setting, i.e., the mechanism knows all players and their respective input data in advance. In this paper, we consider cooperative cost sharing games in an online setting: Upon the arrival of a new player, the mechanism has to take instantaneous and irreversible decisions without any knowledge about players that arrive in the future. We propose an online model for general demand cost sharing games and give a complete characterization of both weakly group-strategyproof and group-strategyproof online cost sharing mechanisms for this model. Moreover, we present a simple method to derive incremental online cost sharing mechanisms from online algorithms such that the competitive ratio is preserved. Based on our general results, we develop online cost sharing mechanisms for several binary demand and general demand cost sharing games.

The Group Steiner Tree (GST) problem is a classical problem in combinatorial optimization and theoretical computer science. In the Edge-Weighted Group Steiner Tree (EW-GST) problem, we are given an undirected graph G = (V, E) on n vertices with edge costs c : E → ℝ≥ 0, a source vertex s and a collection of subsets of vertices, called groups, S1, . . . , Sk ⊆ V. The goal is to find a minimum-cost tree H ⊆ G that connects s to some vertex from each group Si, for all i = 1, 2, . . . , k. The Node-Weighted Group Steiner Tree (NW-GST) problem has the same setting, but the costs are associated with nodes. The goal is to find a minimum-cost node set X ⊆ V such that G[X] connects every group to the source.When G is a tree, both EW-GST and NW-GST admit a polynomial-time O(log n log k) approximation algorithm due to the seminal result of [Garg et al., SODA'98 and J. Algorithm]. The matching hardness of log2−ϵn is known even for tree instances of EW-GST and NW-GST [Halperin and Krauthgamer STOC'03]. In general graphs, most of polynomial-time approximation algorithms for EW-GST reduce the problem to a tree instance using the metric-tree embedding, incurring a loss of O(log n) on the approximation factor [Bartal, FOCS'96; Fakcharoenphol et al., FOCS'03 and JCSS]. This yields an approximation ratio of O(log2n log k) for EW-GST. Using metric-tree embedding, this factor cannot be improved: The loss of Ω(log n) is necessary on some input graphs (e.g., grids and expanders). There are alternative approaches that avoid metric-tree embedding, e.g., the algorithm of [Chekuri and Pal, FOCS'05], which gives a tight approximation ratio, but none of which achieves polylogarithmic approximation in polynomial-time. This state of the art shows a clear lack of understanding of GST in general graphs beyond the metric-tree embedding technique. For NW-GST (for which the metric-tree embedding does not apply), not even a polynomial-time polylogarithmic approximation algorithm is known.In this paper, we present O(log n log k) approximation algorithms that run in time nO(tw(G)2) for both NW-GST and EW-GST1, where tw(G) denotes the treewidth of graph G. The key to both results is a different type of "tree-embedding" that produces a tree of much bigger size, but does not cause any loss on the approximation factor. Our embedding is inspired by dynamic programming, a technique which is typically not applicable to Group Steiner problems.

The peak-demand charge motivates large-load customers to flatten their demand curves, while their self-owned renewable generations aggravate demand fluctuations. Thus, it is attractive to utilize energy storage for shaping real-time loads and reducing electricity bills. In this paper, we propose the first peak-aware competitive online algorithm for leveraging stored energy (e.g., in fuel cells) to minimize peak-demand charges. Our algorithm decides the discharging quantity slot by slot to maintain the optimal worst-case performance guarantee (namely, competitive ratio) among all deterministic online algorithms. Interestingly, we show that the best competitive ratio can be computed by solving a linear number of linear-fractional problems. We can also extend our competitive algorithm and analysis to improve the average-case performance and consider short-term prediction.