We initiate a systematic study of utilizing predictions to improve over approximation guarantees of classic algorithms, without increasing the running time. We propose a systematic method for a wide class of optimization problems that ask to select a feasible subset of input items of minimal (or maximal) total weight. This gives simple (near-)linear time algorithms for, e.g., Vertex Cover, Steiner Tree, Min-Weight Perfect Matching, Knapsack, and Clique. Our algorithms produce optimal solutions when provided with perfect predictions and their approximation ratios smoothly degrade with increasing prediction error. With small enough prediction error we achieve approximation guarantees that are beyond reach without predictions in the given time bounds, as exemplified by the NP-hardness and APX-hardness of many of the above problems. Although we show our approach to be optimal for this class of problems as a whole, there is a potential for exploiting specific structural properties of individual problems to obtain improved bounds; we demonstrate this on the Steiner Tree problem. We conclude with an empirical evaluation of our approach.
Considering a graph with unknown weights, can we find the shortest path for a pair of nodes if we know the minimal Steiner trees associated with some subset of nodes? That is, with respect to a fixed latent decision-making system (e.g., a weighted graph), we seek to solve one optimization problem (e.g., the shortest path problem) by leveraging information associated with another optimization problem (e.g., the minimal Steiner tree problem). In this paper, we study such a prototype problem called \textit{query-decision regression with task shifts}, focusing on the shortest path problem and the minimum Steiner tree problem. We provide theoretical insights regarding the design of realizable hypothesis spaces for building scoring models, and present two principled learning frameworks. Our experimental studies show that such problems can be solved to a decent extent with statistical significance.
Spanning tree problems with specialized constraints can be difficult to solve in real-world scenarios, often requiring intricate algorithmic design and exponential time. Recently, there has been growing interest in end-to-end deep neural networks for solving routing problems. However, such methods typically produce sequences of vertices, which makes it difficult to apply them to general combinatorial optimization problems where the solution set consists of edges, as in various spanning tree problems. In this paper, we propose NeuroPrim, a novel framework for solving various spanning tree problems by defining a Markov Decision Process (MDP) for general combinatorial optimization problems on graphs. Our approach reduces the action and state space using Prim's algorithm and trains the resulting model using REINFORCE. We apply our framework to three difficult problems on Euclidean space: the Degree-constrained Minimum Spanning Tree (DCMST) problem, the Minimum Routing Cost Spanning Tree (MRCST) problem, and the Steiner Tree Problem in graphs (STP). Experimental results on literature instances demonstrate that our model outperforms strong heuristics and achieves small optimality gaps of up to 250 vertices. Additionally, we find that our model has strong generalization ability, with no significant degradation observed on problem instances as large as 1000. Our results suggest that our framework can be effective for solving a wide range of combinatorial optimization problems beyond spanning tree problems.
The Euclidean Steiner tree problem seeks the min-cost network to connect a collection of target locations, and it underlies many applications of wireless networks. In this paper, we present a study on solving the Euclidean Steiner tree problem using reinforcement learning enhanced by graph representation learning. Different from the commonly studied connectivity problems like travelling salesman problem or vehicle routing problem where the search space is finite, the Euclidean Steiner tree problem requires to search over the entire Euclidean space, thereby making the existing methods not applicable. In this paper, we design discretization methods by leveraging the unique characteristics of the Steiner tree, and propose new training schemes for handling the dynamic Steiner points emerging during the incremental construction. Our design is examined through a sanity check using experiments on a collection of datasets, with encouraging results demonstrating the utility of our method as an alternative to classic combinatorial methods.
We consider the Steiner tree problem on graphs where we are given a set of nodes and the goal is to find a tree sub-graph of minimum weight that contains all nodes in the given set, potentially including additional nodes. This is a classical NP-hard combinatorial optimisation problem. In recent years, a machine learning framework called learning-to-prune has been successfully used for solving a diverse range of combinatorial optimisation problems. In this paper, we use this learning framework on the Steiner tree problem and show that even on this problem, the learning-to-prune framework results in computing near-optimal solutions at a fraction of the time required by commercial ILP solvers. Our results underscore the potential of the learning-to-prune framework in solving various combinatorial optimisation problems.
Graph data management and querying has many practical applications. When graphs are very heterogeneous and/or users are unfamiliar with their structure, they may need to find how two or more groups of nodes are connected in a graph, even when users are not able to describe the connections. This is only partially supported by existing query languages, which allow searching for paths, but not for trees connecting three or more node groups. The latter is related to the NP-hard Group Steiner Tree problem, and has been previously considered for keyword search in databases. In this work, we formally show how to integrate connecting tree patterns (CTPs, in short) within a graph query language such as SPARQL or Cypher, leading to an Extended Query Language (or EQL, in short). We then study a set of algorithms for evaluating CTPs; we generalize prior keyword search work, most importantly by (i) considering bidirectional edge traversal and (ii) allowing users to select any score function for ranking CTP results. To cope with very large search spaces, we propose an efficient pruning technique and formally establish a large set of cases where our algorithm, MOLESP, is complete even with pruning. Our experiments validate the performance of our CTP and EQL evaluation algorithms on a large set of synthetic and real-world workloads.
This paper optimizes path planning for a trunkand-branch topology network in an irregular 2-dimensional manifold embedded in 3-dimensional Euclidean space with application to submarine cable network planning. We go beyond our earlier focus on the costs of cable construction (including labor, equipment and materials) together with additional cost to enhance cable resilience, to incorporate the overall cost of branching units (again including material, construction and laying) and the choice of submarine cable landing stations, where such a station can be anywhere on the coast in a connected region. These are important issues for the economics of cable laying and significantly change the model and the optimization process. We pose the problem as a variant of the Steiner tree problem, but one in which the Steiner nodes can vary in number, while incurring a penalty. We refer to it as the weighted Steiner node problem. It differs from the Euclidean Steiner tree problem, where Steiner points are forced to have degree three; this is no longer the case, in general, when nodes incur a cost. We are able to prove that our algorithm is applicable to Steiner nodes with degree greater than three, enabling optimization of network costs in this context. The optimal solution is achieved in polynomialtime using dynamic programming.
This paper considers the recently popular beyond-worst-case algorithm analysis model which integrates machine-learned predictions with online algorithm design. We consider the online Steiner tree problem in this model for both directed and undirected graphs. Steiner tree is known to have strong lower bounds in the online setting and any algorithm's worst-case guarantee is far from desirable. This paper considers algorithms that predict which terminal arrives online. The predictions may be incorrect and the algorithms' performance is parameterized by the number of incorrectly predicted terminals. These guarantees ensure that algorithms break through the online lower bounds with good predictions and the competitive ratio gracefully degrades as the prediction error grows. We then observe that the theory is predictive of what will occur empirically. We show on graphs where terminals are drawn from a distribution, the new online algorithms have strong performance even with modestly correct predictions.
Steiner Tree Problem (STP) in graphs aims to find a tree of minimum weight in the graph that connects a given set of vertices. It is a classic NP-hard combinatorial optimization problem and has many real-world applications (e.g., VLSI chip design, transportation network planning and wireless sensor networks). Many exact and approximate algorithms have been developed for STP, but they suffer from high computational complexity and weak worst-case solution guarantees, respectively. Heuristic algorithms are also developed. However, each of them requires application domain knowledge to design and is only suitable for specific scenarios. Motivated by the recently reported observation that instances of the same NP-hard combinatorial problem may maintain the same or similar combinatorial structure but mainly differ in their data, we investigate the feasibility and benefits of applying machine learning techniques to solving STP. To this end, we design a novel model Vulcan based on novel graph neural networks and deep reinforcement learning. The core of Vulcan is a novel, compact graph embedding that transforms highdimensional graph structure data (i.e., path-changed information) into a low-dimensional vector representation. Given an STP instance, Vulcan uses this embedding to encode its pathrelated information and sends the encoded graph to a deep reinforcement learning component based on a double deep Q network (DDQN) to find solutions. In addition to STP, Vulcan can also find solutions to a wide range of NP-hard problems (e.g., SAT, MVC and X3C) by reducing them to STP. We implement a prototype of Vulcan and demonstrate its efficacy and efficiency with extensive experiments using real-world and synthetic datasets.
We create a novel Physarum Steiner algorithm designed to solve the Euclidean Steiner tree problem. Physarum is a unicellular slime mold with the ability to form networks and fuse with other Physarum organisms. We use the simplicity and fusion of Physarum to create large swarms which independently operate to solve the Steiner problem. The Physarum Steiner tree algorithm then utilizes a swarm of Physarum organisms which gradually find terminals and fuse with each other, sharing intelligence. The algorithm is also highly capable of solving the obstacle avoidance Steiner tree problem and is a strong alternative to the current leading algorithm. The algorithm is of particular interest due to its novel approach, rectilinear properties, and ability to run on varying shapes and topological surfaces.
The expansion of Fiber-To-The-Home (FTTH) networks creates high costs due to expensive excavation procedures. Optimizing the planning process and minimizing the cost of the earth excavation work therefore lead to large savings. Mathematically, the FTTH network problem can be described as a minimum Steiner Tree problem. Even though the Steiner Tree problem has already been investigated intensively in the last decades, it might be further optimized with the help of new computing paradigms and emerging approaches. This work studies upcoming technologies, such as Quantum Annealing, Simulated Annealing and nature-inspired methods like Evolutionary Algorithms or slime-mold-based optimization. Additionally, we investigate partitioning and simplifying methods. Evaluated on several real-life problem instances, we could outperform a traditional, widely-used baseline (NetworkX Approximate Solver) on most of the domains. Prior partitioning of the initial graph and the presented slime-mold-based approach were especially valuable for a cost-efficient approximation. Quantum Annealing seems promising, but was limited by the number of available qubits.
Graph neural networks have been successful in many learning problems and real-world applications. A recent line of research explores the power of graph neural networks to solve combinatorial and graph algorithmic problems such as subgraph isomorphism, detecting cliques, and the traveling salesman problem. However, many NP-complete problems are as of yet unexplored using this method. In this paper, we tackle the Steiner Tree Problem. We employ four learning frameworks to compute low cost Steiner trees: feed-forward neural networks, graph neural networks, graph convolutional networks, and a graph attention model. We use these frameworks in two fundamentally different ways: 1) to train the models to learn the actual Steiner tree nodes, 2) to train the model to learn good Steiner point candidates to be connected to the constructed tree using a shortest path in a greedy fashion. We illustrate the robustness of our heuristics on several random graph generation models as well as the SteinLib data library. Our finding suggests that the out-of-the-box application of GNN methods does worse than the classic 2-approximation method. However, when combined with a greedy shortest path construction, it even does slightly better than the 2-approximation algorithm. This result sheds light on the fundamental capabilities and limitations of graph learning techniques on classical NP-complete problems.
The Steiner tree problem is a well-known problem in network design, routing, and VLSI design. Given a graph, edge costs, and a set of dedicated vertices (terminals), the Steiner tree problem asks to output a sub-graph that connects all terminals at minimum cost. A state-of-the-art algorithm to solve the Steiner tree problem by means of dynamic programming is the Dijkstra-Steiner algorithm. The algorithm builds a Steiner tree of the entire instance by systematically searching for smaller instances, based on subsets of the terminals, and combining Steiner trees for these smaller instances. The search heavily relies on a guiding heuristic function in order to prune the search space. However, to ensure correctness, this algorithm allows only for limited heuristic functions, namely, those that satisfy a so-called consistency condition. In this paper, we enhance the Dijkstra-Steiner algorithm and establish a revisited algorithm, called DS*. The DS* algorithm allows for arbitrary lower bounds as heuristics relaxing the previous condition on the heuristic function. Notably, we can now use linear programming based lower bounds. Further, we capture new requirements for a heuristic function in a condition, which we call admissibility. We show that admissibility is indeed weaker than consistency and establish correctness of the DS* algorithm when using an admissible heuristic function. We implement DS* and combine it with modern preprocessing, resulting in an open-source solver (DS* Solve). Finally, we compare its performance on standard benchmarks and observe a competitive behavior.
This paper characterizes and discusses devolutionary genetic algorithms and evaluates their performances in solving the minimum labeling Steiner tree (MLST) problem. We define devolutionary algorithms as the process of reaching a feasible solution by devolving a population of super-optimal unfeasible solutions over time. We claim that distinguishing them from the widely used evolutionary algorithms is relevant. The most important distinction lies in the fact that in the former type of processes, the value function decreases over successive generation of solutions, thus providing a natural stopping condition for the computation process. We show how classical evolutionary concepts, such as crossing, mutation and fitness can be adapted to aim at reaching an optimal or close-to-optimal solution among the first generations of feasible solutions. We additionally introduce a novel integer linear programming formulation of the MLST problem and a valid constraint used for speeding up the devolutionary process. Finally, we conduct an experiment comparing the performances of devolutionary algorithms to those of state of the art approaches used for solving randomly generated instances of the MLST problem. Results of this experiment support the use of devolutionary algorithms for the MLST problem and their development for other NP-hard combinatorial optimization problems.
Resource allocation problems are a family of problems in which resources must be selected to satisfy given demands. This paper focuses on the two-stage stochastic generalization of resource allocation problems where future demands are expressed in a finite number of possible scenarios. The goal is to select cost effective resources to be acquired in the present time (first stage), and to implement a complete solution for each scenario (second stage), while minimizing the total expected cost of the choices in both stages. We propose an evolutionary framework for solving general two-stage stochastic resource allocation problems. In each iteration of our framework, a local search algorithm selects resources to be acquired in the first stage. A genetic metaheuristic then completes the solutions for each scenario and relevant information is passed onto the next iteration, thereby supporting the acquisition of promising resources in the following first stage. Experimentation on numerous instances of the two-stage stochastic Steiner tree problem suggests that our evolutionary framework is powerful enough to address large instances of a wide variety of two-stage stochastic resource allocation problems.
A vibrant theoretical research area are efficient exact parameterized algorithms. Very recent solving competitions such as the PACE challenge show that there is also increasing practical interest in the parameterized algorithms community. An important research question is whether dedicated parameterized exact algorithms exhibit certain practical relevance and one can even beat well-established problem solvers. We consider the logic-based declarative modeling language and problem solving framework Answer Set Programming (ASP). State-of-the-art ASP solvers rely considerably on Sat-based algorithms. An ASP solver (DynASP2), which is based on a classical dynamic programming on tree decompositions, has been published very recently. Unfortunately, DynASP2 can outperform modern ASP solvers on programs of small treewidth only if the question of interest is to count the number of solutions. In this paper, we describe underlying concepts of our new implementation (DynASP2.5) that shows competitive behavior to state-of-the-art ASP solvers even for finding just one solution when solving problems as the Steiner tree problem that have been modeled in ASP on graphs with low treewidth. Our implementation is based on a novel approach that we call multi-pass dynamic programming (M-DPSINC).
The Steiner Tree Problem (STP) in graphs is an important problem with various applications in many areas such as design of integrated circuits, evolution theory, networking, etc. In this paper, we propose an algorithm to solve the STP. The algorithm includes a reducer and a solver using Variable Neighborhood Descent (VND), interacting with each other during the search. New constructive heuristics and a vertex score system for intensification purpose are proposed. The algorithm is tested on a set of benchmarks which shows encouraging results.