Series-parallel Graphs Research Articles

On mixed linear layouts of series-parallel graphs

A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout. In 2017, Pupyrev disproved this conjecture by demonstrating a planar partial 3-tree that does not admit a mixed 1-stack 1-queue layout.In this work, we strengthen Pupyrev's result by showing that the conjecture does not hold even for 2-trees, also known as series-parallel graphs. We conclude this by means of a stronger result, stating that the conjecture does not hold, even in the more general union and local settings. In the former, crossings (nestings) are allowed in the stack (queue) as long as the involved edges belong to different connected components of the subgraph induced by its edges, while in the latter several stacks and queues may form the layout, but locally every vertex is incident to edges from only one stack and one queue.

Network Design for s - t Effective Resistance

We consider a new problem of designing a network with small s - t effective resistance. In this problem, we are given an undirected graph G = (V,E) , two designated vertices s,t ∈ V , and a budget k . The goal is to choose a subgraph of G with at most k edges to minimize the s - t effective resistance. This problem is an interpolation between the shortest path problem and the minimum cost flow problem and has applications in electrical network design. We present several algorithmic and hardness results for this problem and its variants. On the hardness side, we show that the problem is NP-hard, and the weighted version is hard to approximate within a factor smaller than two assuming the small-set expansion conjecture. On the algorithmic side, we analyze a convex programming relaxation of the problem and design a constant factor approximation algorithm. The key of the rounding algorithm is a randomized path-rounding procedure based on the optimality conditions and a flow decomposition of the fractional solution. We also use dynamic programming to obtain a fully polynomial time approximation scheme when the input graph is a series-parallel graph, with better approximation ratio than the integrality gap of the convex program for these graphs.

Maximum Matching in Almost Linear Time on Graphs of Bounded Clique-Width

Recently, independent groups of researchers have presented algorithms to compute a maximum matching in \(\tilde{\mathcal{O}}(f(k) \cdot (n+m))\) time, for some computable function f, within the graphs where some clique-width upper bound is at most k (e.g., tree-width, modular-width and \(P_4\)-sparseness). However, to the best of our knowledge, the existence of such algorithm within the graphs of bounded clique-width has remained open until this paper. Indeed, we cannot even apply Courcelle’s theorem to this problem directly, because a matching cannot be expressed in \(MSO_1\) logic. Our first contribution is an almost linear-time algorithm to compute a maximum matching in any bounded clique-width graph, being given a corresponding clique-width expression. We also present how to compute the Edmonds-Gallai decomposition in almost linear time by using the same framework. For that, we do apply Courcelle’s theorem but to the classic Tutte-Berge formula, that can easily be expressed as a \(CMSO_1\) optimization problem. Doing so, we can compute the cardinality of a maximum matching, but not the matching itself. To obtain with this approach a maximum matching, we need to combine it with a recursive dissection scheme for bounded clique-width graphs and with a distributed version of Courcelle’s theorem (Courcelle and Vanicat, DAM 2016) – of which we present here a slightly stronger version than the standard one in the literature. Finally, for the bipartite graphs of clique-width at most k, we present an alternative \(\tilde{\mathcal{O}}(k^2\cdot (n+m))\)-time algorithm for the problem. The algorithm is randomized and it is based on a completely different approach than above: combining various reductions to matching and flow problems on bounded tree-width graphs with a very recent result on the parameterized complexity of linear programming (Dong et. al., STOC’21). Our results for bounded clique-width graphs extend many prior works on the complexity of Maximum Matching within cographs, distance-hereditary graphs, series-parallel graphs and other subclasses.

Computing the Clique-Width on Series-Parallel Graphs

The clique-width (cwd) is an invariant of graphs which, similar to other invariants like the tree-width (twd) establishes a parameter for the complexity of a problem. For example, several problems with bounded clique-width can be solved in polynomial time. There is a well known relation between tree-width and clique-width denoted as cwd(G) ≤ 3 · 2 twd(G)−1 . Serial-parallel graphs have tree-width of at most 2, so its clique–width is at most 6 according to the previous relation. In this paper, we improve the bound for this particular case, showing that the clique-width of series-parallel graphs is smaller or equal to 5.

On Edge Pruning of Communication Networks under an Age-of-Information Framework

Effective non-repetitive routing among nodes in a network is an essential function in communication networks. To achieve that, pruning the links of the network is helpful with the trade-off of making the network less robust in transmitting messages while reducing redundancy to increase flow with limited network bandwidth, so we enhance the quality of service (QoS). In our paper, we study the case that if a link removal has either no effect or an insignificant effect on the Age of Information (AoI) of the messages delivered in the communication network. The pruning of such links can be applied to the k least significant links in terms of their impact on the AoI of the messages transmitted in the system. The objective of our work is to study the effect of pruning a number of links in a network on the AoI, in order to reduce the redundancy of the messages that may be received at the destination many times while transmitted only once. In our context, the objective of the communication system would be to maintain the information from the source as fresh as possible when it arrives at the destination while reducing the redundancy of messages. In this work, we introduce an efficient reduction method designed for series-parallel networks with links of exponentially distributed wait times. In addition, we consider the deterministic case and present the pruning technique when link removal would not affect the AoI. Lastly, we present a comprehensive simulation to study the effect of pruning the links on the AoI of the network and the redundancy of messages received by the destination.

Search Strategies for Topological Network Optimization

We consider an application of combinatorial search to the optimization of topologies in series-parallel networks. We propose a recursive search over the space of decomposition trees, in which partial solutions are obtained by exploring k-way partitionings of expandable nodes. We present two complementary pruning techniques that bound the value of intermediate solutions from above and below, applying monotonic operations to the contents of unresolved leaves. We also develop a means to exploit the convexity of our objective function, so as to prevent the redundant recomputation of subcircuit configurations. Finally, we evaluate our approach on a parameterized benchmark suite of electrical circuits, demonstrating over an order of magnitude improvement in performance as compared to a baseline implementation.

The Connected Critical Node Problem

The Critical Node Problem is a well-known optimisation problem that aims to find the subset of nodes in a graph whose removal impacts the graph connectivity as much as possible according to a specific connectivity measure. In this work, we study a new version of the Critical Node Problem, which we call the Connected Critical Node Problem, where the set of the removed nodes has to form a connected subgraph. We consider three connectivity measures and provide complexity results and solution approaches for general graphs and specific classes of graphs such as graphs with bounded treewidth, trees and series-parallel graphs. We consider the Connected Critical Node Problem where the pairwise connectivity (related to the number of pairs of vertices still connected in the graph) is minimised. We prove that this problem is strongly NP-hard and inapproximable on general graphs and is polynomial-time solvable on graphs with bounded treewidth and with unit connection costs. Further, we prove the NP-hardness of the problem with arbitrary connection costs over trees and series-parallel graphs and derive dynamic programming algorithms. We extend our results to the problem variants that consider the minimisation of the largest connected component and the maximisation of the number of connected components (also called K-way Vertex Cut Problem). As side results, we provide new complexity results for the classic Critical Node Problem on series-parallel graphs.

A tight relation between series-parallel graphs and bipartite distance hereditary graphs

Bandelt and Mulder’s structural characterization of bipartite distance hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by repeatedly adding either pendant vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffin’s structural characterization of 2–connected series–parallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we give an elementary proof that the two constructions are the same construction when bipartite graphs are viewed as the fundamental graphs of a graphic matroid. We then apply the result to re-prove known results concerning bipartite distance hereditary graphs and series–parallel graphs and to provide a new class of polynomially-solvable instances for the integer multi-commodity flow of maximum value.

Structural identifiability of series-parallel LCR systems

We consider the identifiability problem for the parameters of series-parallel LCR circuit networks. We prove that for networks with only two classes of components (inductor-capacitor (LC), inductor-resistor (LR), and capacitor-resistor (RC)), the parameters are identifiable if and only if the number of non-monic coefficients of the constitutive equations equals the number of parameters. The notion of the “type” of the constitutive equations plays a key role in the identifiability of LC, LR, and RC networks. We also investigate the general series-parallel LCR circuits (with all three classes of components), and classify the types of constitutive equations that can arise, showing that there are 22 different types. However, we produce an example that shows that the basic notion of type that works to classify identifiability of two class networks is not sufficient to classify the identifiability of general series-parallel LCR circuits.

On Mixed Linear Layouts of Series-Parallel Graphs

A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and of a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout. Recently, Pupyrev disproved this conjectured by demonstrating a planar partial 3-tree that does not admit a 1-stack 1-queue layout. In this note, we strengthen Pupyrev's result by showing that the conjecture does not hold even for 2-trees, also known as series-parallel graphs.

Upward planar drawings with two slopes

In an upward planar 2-slope drawing of a digraph, edges are drawn as straight-line segments in the upward direction without crossings using only two different slopes. We investigate whether a given upward planar digraph admits such a drawing and, if so, how to construct it. For the fixed embedding scenario, we give a simple characterisation and a linear-time construction by adopting algorithms from orthogonal drawings. For the variable embedding scenario, we describe a linear-time algorithm for single-source digraphs, a quartic-time algorithm for series-parallel digraphs, and a fixed-parameter tractable algorithm for general digraphs. For the latter two classes, we make use of SPQR-trees and the notion of upward spirality. As an application of this drawing style, we show how to draw an upward planar phylogenetic network with two slopes such that all leaves lie on a horizontal line.

Robust minimum cost flow problem under consistent flow constraints

The robust minimum cost flow problem under consistent flow constraints (RobMCFequiv ) is a new extension of the minimum cost flow (MCF) problem. In the RobMCFequiv problem, we consider demand and supply that are subject to uncertainty. For all demand realizations, however, we require that the flow value on an arc needs to be equal if it is included in the predetermined arc set given. The objective is to find feasible flows that satisfy the equal flow requirements while minimizing the maximum occurring cost among all demand realizations. In the case of a finite discrete set of scenarios, we derive structural results which point out the differences with the polynomial time solvable MCF problem in networks with integral demands, supplies, and capacities. In particular, the Integral Flow Theorem of Dantzig and Fulkerson does not hold. For this reason, we require integral flows in the entire paper. We show that the RobMCFequiv problem is strongly mathcal {NP}-hard on acyclic digraphs by a reduction from the (3, B2)-Sat problem. Further, we demonstrate that the RobMCFequiv problem is weakly mathcal {NP}-hard on series-parallel digraphs by providing a reduction from Partition. If in addition the number of scenarios is constant, we propose a pseudo-polynomial algorithm based on dynamic programming. Finally, we present a special case on series-parallel digraphs for which we can solve the RobMCFequiv problem in polynomial time.

The knapsack problem with special neighbor constraints

The knapsack problem is one of the simplest and most fundamental NP-hard problems in combinatorial optimization. We consider two knapsack problems which contain additional constraints in the form of directed graphs whose vertex set corresponds to the item set. In the one-neighbor knapsack problem, an item can be chosen only if at least one of its neighbors is chosen. In the all-neighbors knapsack problem, an item can be chosen only if all its neighbors are chosen. For both problems, we consider uniform and general profits and weights. We prove upper bounds for the time complexity of these problems when restricting the graph constraints to special sets of digraphs. We discuss directed co-graphs, minimal series-parallel digraphs, and directed trees.

The circular chromatic numbers of signed series-parallel graphs

It is known that any signed series-parallel graph (G,σ) has circular chromatic number at most 10/3. This paper proves that for each rational r∈[2,10/3], there is a signed series-parallel graph (G,σ) whose circular chromatic number equals r.

Orthogonal planarity testing of bounded treewidth graphs

Given a graph G and an integer b, OrthogonalPlanarity is the problem of testing whether G admits a planar orthogonal drawing with at most b bends. OrthogonalPlanarity is known to be NP-complete. We show that this problem belongs to the XP class when parameterized by treewidth. The proof exploits a fixed-parameter tractable approach that uses two more parameters besides treewidth, namely the natural parameter b and the number of vertices with degree at most two of G. Such approach is based on the new concept of sketched orthogonal representation, which synthetically describes a family of equivalent orthogonal drawings. The approach is general enough to be applicable to other related problems, namely HV-Planarity and FlexDraw. Also, in the special case of series-parallel graphs we obtain that both OrthogonalPlanarity and HV-Planarity can be solved in O(n3log⁡n) time, which improves on the previous O(n4) bounds for these two.

Computation of Dynamic Equilibria in Series-Parallel Networks

We consider dynamic equilibria for flows over time under the fluid queuing model. In this model, queues on the links of a network take care of flow propagation. Flow enters the network at a single source and leaves at a single sink. In a dynamic equilibrium, every infinitesimally small flow particle reaches the sink as early as possible given the pattern of the rest of the flow. Although this model has been examined for many decades, progress has been relatively recent. In particular, the derivatives of dynamic equilibria have been characterized as thin flows with resetting, which allows for more structural results. Our two main results are based on the formulation of thin flows with resetting as a linear complementarity problem and its analysis. We present a constructive proof of existence for dynamic equilibria if the inflow rate is right-monotone. The complexity of computing thin flows with resetting, which occurs as a subproblem in this method, is still open. We settle it for the class of two-terminal, series-parallel networks by giving a recursive algorithm that solves the problem for all flow values simultaneously in polynomial time.

Market equilibrium in multi‐tier supply chain networks

AbstractWe model multistage competition in supply chains by a sequential game theoretical model in a network. We show the uniqueness of equilibrium for series parallel networks and provide a linear time algorithm to compute the equilibrium. We study the impact of the network structure on the firms' payoffs, the total trade flow, and the overall social welfare.

Efficient computation of the oriented chromatic number of recursively defined digraphs

In this paper we consider colorings of oriented graphs, i.e. digraphs without cycles of length 2. Given some oriented graph G=(V,E), an oriented r-coloring for G is a partition of the vertex set V into r independent sets, such that all the arcs between two of these sets have the same direction. The oriented chromatic number of G is the smallest integer r such that G permits an oriented r-coloring. Deciding whether an acyclic digraph has an oriented 4-coloring is NP-hard, which motivates to consider the problem on special graph classes.In this paper we consider the Oriented Chromatic Number problem on classes of recursively defined oriented graphs. Oriented co-graphs (short for oriented complement reducible graphs) can be recursively defined from the single vertex graph by applying the disjoint union and order composition. This recursive structure allows to compute an optimal oriented coloring and the oriented chromatic number in linear time. We generalize this result using the concept of perfect orderable graphs. Therefore, we show that for acyclic transitive digraphs every greedy coloring along a topological ordering leads to an optimal oriented coloring. Msp-digraphs (short for minimal series-parallel digraphs) can be defined from the single vertex graph by applying the parallel composition and series composition. We prove an upper bound of 7 for the oriented chromatic number for msp-digraphs and we give an example to show that this is bound best possible. We apply this bound and the recursive structure of msp-digraphs to obtain a linear time solution for computing the oriented chromatic number of msp-digraphs.In order to generalize the results on computing the oriented chromatic number of special graph classes, we consider the parameterized complexity of the Oriented Chromatic Number problem by so-called structural parameters, which are measuring the difficulty of decomposing a graph into a special tree-structure.

Nonempty intersection of longest paths in graphs without forbidden pairs

In 1966, Gallai asked whether all longest paths in a connected graph have a nonempty intersection. The answer to this question is not true in general and various counterexamples have been found. However, there is a positive solution to Gallai’s question for many well-known classes of graphs such as split graphs, series–parallel graphs, and 2K2-free graphs. Split graphs, series–parallel graphs and 2K2-free graphs were all proven to be Hamiltonian under given toughness conditions. This observation motivates us to investigate Gallai’s question in graphs that are close to have certain Hamiltonicity properties. Let {R,S} be a pair of connected graphs. In particular, in this paper, we show that Gallai’s question has an affirmative answer for all connected {R,S}-free graphs such that every 2-connected {R,S}-free graph is Hamiltonian. These pairs {R,S} were completely characterized in the 1990s.

A mathematical programming model to select maintenance strategies in railway networks

This paper presents a nonlinear integer programming model to support the selection of maintenance strategies to implement on different segments of a railway network. Strategies are selected which collectively minimise the impact of sections’ conditions on service, given network availability and budget constraints. Different metrics related to the network topology, sections’ availability, service frequency, performance requirements and maintenance costs, are combined into a quantitative approach with a holistic view. The main contribution is to provide a simple yet effective modelling approach and solution method which are suitable for large networks and make use of standard solvers. Both an ad hoc heuristic solution and relaxation methods are developed, the latter enabling the quality of the heuristic solution to be estimated. The availability of railway lines is computed by exploiting the analogy with series–parallel networks. By varying the model parameters, a scenario analysis is performed to give insight into the influence of the system parameters on the selection of strategies, thus enabling more informed decisions. For its simple structure, the model is versatile to address similar problems arising in the maintenance of other types of networks, such as road and bridges networks, when deciding on the strategic allocation of maintenance efforts.