Maximum Weight Independent Set Research Articles

Graphs with polynomially many minimal separators

We show that graphs that do not contain a theta, pyramid, prism, or turtle as an induced subgraph have polynomially many minimal separators. This result is the best possible in the sense that there are graphs with exponentially many minimal separators if only three of the four induced subgraphs are excluded. As a consequence, there is a polynomial time algorithm to solve the maximum weight independent set problem for the class of (theta, pyramid, prism, turtle)-free graphs. Since every prism, theta, and turtle contains an even hole, this also implies a polynomial time algorithm to solve the maximum weight independent set problem for the class of (pyramid, even hole)-free graphs.

Dynamic Programming based Low-Latency Schedule (DPLLS) for 6TiSCH networks

Internet of Things (IoT) is a technological concept bringing sustainability and sophistication to our lives and is a significant component of Industry 4.0. The main requirements of Industrial IoT are reliability, stringent latency and energy efficiency. The IEEE 802.15.4e standard has adapted a Medium Access Control (MAC) behavioural mode called Time Slotted Channel Hopping (TSCH) to address the requirements of Industry 4.0. The 6TiSCH (IPv6 over IEEE 802.15.4e TSCH) protocol stack enables us to schedule the transmissions in TSCH network to achieve application-specific guarantees. In this paper, we propose DPLLS, a conflict-free dynamic programming based low-latency scheduling algorithm for TSCH network. In particular, identifying non-interfering transmissions in the network is posed as a maximum weight independent set (MWIS) problem, which is solved using a dynamic programming method. We organise each slotframe into smaller parts called blocks in which the non-interfering transmissions are scheduled simultaneously using either a conservative or a greedy scheme. The blocks are repeated in a slotframe to accommodate the retransmissions of packets to ensure reliability and minimise the latency. The proposed scheme is evaluated using an example to find the suitable block length of the proposed schemes and compared with the existing scheduling algorithms.

Graph-Theoretical Dynamic User Pairing for Downlink NOMA Systems

We propose a novel graph-theoretical dynamic user pairing strategy based on the user rate requirements in cellular networks employing non-orthogonal multiple access (NOMA). The proposed approach relies on first constructing a conflict graph corresponding to all possible user pairings and then reformulating the problem of finding the best user pairs as that of finding the maximum weighted independent set (MWIS) on the conflict graph. This formulation turns the originally NP-hard problem into one that can be solvable in polynomial time thanks to the claw-freeness property of the conflict graph. The proposed user pairing method satisfies the maximum number of user demands with optimal network sum-rate as shown theoretically and as validated by the simulation results.

Combinatorial optimization by weight annealing in memristive hopfield networks

The increasing utility of specialized circuits and growing applications of optimization call for the development of efficient hardware accelerator for solving optimization problems. Hopfield neural network is a promising approach for solving combinatorial optimization problems due to the recent demonstrations of efficient mixed-signal implementation based on emerging non-volatile memory devices. Such mixed-signal accelerators also enable very efficient implementation of various annealing techniques, which are essential for finding optimal solutions. Here we propose a “weight annealing” approach, whose main idea is to ease convergence to the global minima by keeping the network close to its ground state. This is achieved by initially setting all synaptic weights to zero, thus ensuring a quick transition of the Hopfield network to its trivial global minima state and then gradually introducing weights during the annealing process. The extensive numerical simulations show that our approach leads to a better, on average, solutions for several representative combinatorial problems compared to prior Hopfield neural network solvers with chaotic or stochastic annealing. As a proof of concept, a 13-node graph partitioning problem and a 7-node maximum-weight independent set problem are solved experimentally using mixed-signal circuits based on, correspondingly, a 20 × 20 analog-grade TiO2 memristive crossbar and a 12 × 10 eFlash memory array.

Computing inductive vertex orderings

Inductive vertex orderings or edge orientations of graphs, such as inductive k-independence and k-simpliciality orderings, and k-perfect orientation, arise naturally in geometric intersection graphs and have been systematically studied in Ye and Borodin (2012) [4] and Kammer and Tholey (2014) [5]. In general, those orderings and orientations are hard to compute, even approximately. In this note, we show that an inductive k-independence ordering can be computed in k-simplicial graphs, and a k-approximation (that is, a solution with approximation ratio at most k) to the optimal inductive independence ordering can be computed in k-perfectly orientable graphs. As a result, we obtain a k-approximation (2k-approximation) algorithm for maximum weight independent sets in k-simplicial (k-perfectly orientable, respectively) graphs that runs in polynomial time independent of k without prior knowledge of the ordering.

Independent Sets in (P_4+P_4,Triangle)-Free Graphs

The Maximum Weight Independent Set Problem (WIS) is a well-known NP-hard problem. A popular way to study WIS is to detect graph classes for which WIS can be solved in polynomial time, with particular reference to hereditary graph classes, i.e., defined by a hereditary graph property or equivalently by forbidding one or more induced subgraphs. Given two graphs G and H, G+H denotes the disjoint union of G and H. This manuscript shows that (i) WIS can be solved for (P_4+P_4, Triangle)-free graphs in polynomial time, where a P_4 is an induced path of four vertices and a Triangle is a cycle of three vertices, and that in particular it turns out that (ii) for every (P_4+P_4, Triangle)-free graph G there is a family {{mathcal {S}}} of subsets of V(G) inducing (complete) bipartite subgraphs of G, which contains polynomially many members and can be computed in polynomial time, such that every maximal independent set of G is contained in some member of {mathcal {S}}. These results seem to be harmonic with respect to other polynomial results for WIS on [subclasses of] certain S_{i,j,k}-free graphs and to other structure results on [subclasses of] Triangle-free graphs.

Maximum weight independent sets for (S1,2,4,triangle)-free graphs in polynomial time

The Maximum Weight Independent Set (MWIS) problem on finite undirected graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum weight sum. MWIS is one of the most investigated and most important algorithmic graph problems; it is well known to be NP-complete, and it remains NP-complete even under various strong restrictions such as for triangle-free graphs. Its complexity for Pk-free graphs, k≥7, is an open problem. In [7], it is shown that MWIS can be solved in polynomial time for (P7,triangle)-free graphs. This result is extended by Maffray and Pastor [22] showing that MWIS can be solved in polynomial time for (P7,bull)-free graphs. In the same paper, they also showed that MWIS can be solved in polynomial time for (S1,2,3,bull)-free graphs.In this paper, using a similar approach as in [7], we show that MWIS can be solved in polynomial time for (S1,2,4,triangle)-free graphs which generalizes the result for (P7,triangle)-free graphs.

Approximation algorithms for maximum weight k-coverings of graphs by packings

Let [Formula: see text] be a family of graphs and let [Formula: see text] be a set of connected graphs, each with at most [Formula: see text] vertices, [Formula: see text] fixed. A [Formula: see text]-packing of a graph GA is a vertex induced subgraph of GA with every connected component isomorphic to a member of [Formula: see text]. A maximum weight [Formula: see text]-covering of a graph GA by [Formula: see text]-packings, is a maximum weight subgraph of GA exactly covered by [Formula: see text] vertex disjoint [Formula: see text]-packings. For a graph [Formula: see text] let [Formula: see text](GA) be a graph, every vertex [Formula: see text] of which corresponds to a vertex subgraph [Formula: see text] of GA isomorphic to a member of [Formula: see text], two vertices [Formula: see text] of [Formula: see text](GA) being adjacent if and only if [Formula: see text] and [Formula: see text] have common vertices or interconnecting edges. The closed neighborhoods containment graph [Formula: see text] of a graph [Formula: see text], is the graph with vertex set [Formula: see text] and edges directed from vertices [Formula: see text] to [Formula: see text] if and only if they are adjacent in GA and the closed neighborhood of [Formula: see text] is contained in the closed neighborhood of [Formula: see text]. A graph [Formula: see text] is a [Formula: see text] reduced graph if it can be obtained from a graph [Formula: see text] by deleting the edges of a transitive subgraph [Formula: see text] of CNCG(GA). We describe 1.582-approximation algorithms for maximum weight [Formula: see text]-coverings by [Formula: see text]-packings of [Formula: see text] and [Formula: see text] reduced graphs when [Formula: see text] is vertex hereditary, has an algorithm for maximum weight independent set and [Formula: see text]. These algorithms can be applied to families of interval filament, subtree filament, weakly chordal, AT-free and circle graphs, to find 1.582 approximate maximum weight [Formula: see text]-coverings by vertex disjoint induced matchings, dissociation sets, forests whose subtrees have at most [Formula: see text] vertices, etc.

Covering Minimal Separators and Potential Maximal Cliques in $P_t$-Free Graphs

A graph is called $P_t$-free if it does not contain a $t$-vertex path as an induced subgraph. While $P_4$-free graphs are exactly cographs, the structure of $P_t$-free graphs for $t \geqslant 5$ remains little understood. On one hand, classic computational problems such as Maximum Weight Independent Set (MWIS) and $3$-Coloring are not known to be NP-hard on $P_t$-free graphs for any fixed $t$. On the other hand, despite significant effort, polynomial-time algorithms for MWIS in $P_6$-free graphs~[SODA 2019] and $3$-Coloring in $P_7$-free graphs~[Combinatorica 2018] have been found only recently. In both cases, the algorithms rely on deep structural insights into the considered graph classes.
 One of the main tools in the algorithms for MWIS in $P_5$-free graphs~[SODA 2014] and in $P_6$-free graphs~[SODA 2019] is the so-called Separator Covering Lemma that asserts that every minimal separator in the graph can be covered by the union of neighborhoods of a constant number of vertices. In this note we show that such a statement generalizes to $P_7$-free graphs and is false in $P_8$-free graphs. We also discuss analogues of such a statement for covering potential maximal cliques with unions of neighborhoods.

A General Framework for Intersection Traffic Control With Backpressure Routing

Traffic congestion has long been a worldwide difficult problem in metropolitan transportation networks, incurring tremendous time waste and exhaust pollution. Extensive efforts have been made to address this problem, among which traffic control at intersections is known to be especially crucial while challenging. Also, it is helpful to the recent advent of autopilot technologies by enhancing the schedule accuracy and reducing the infrastructure cost. In practice, however, existing researches can hardly work well in a pervasive manner since they are essentially limited to two ideal assumptions: 1) each intersection comprises four ways; 2) each way is homogeneously composed of two lanes. Through an in-depth examination of their basic models, we find that the two-fold ideal assumptions are largely compelled by the surprisingly high complexity of converting a practical intersection topology into a theoretical conflict graph. Moreover, existing works seldom consider the modeling of complex intersections in the scene of cooperative control of multiply intersections. Driven by the above understandings, our first effort towards a general framework (for handling multiple-way heterogeneous intersections) is to carefully transform a practical intersection topology into a homomorphic, regular conflict graph which is suited to theoretical modeling and further processing with an affordable complexity. Besides, a maximum weight independent set (MWIS) based approach is proposed to minimize the average waiting time of vehicles at an isolated intersection. In addition, we apply a backpressure-based algorithm to our framework to further optimize the global average waiting time of vehicles in a whole road network. Simulation results have demonstrated that the average waiting time achieved by our approach is merely 80 seconds while traditional traffic light control reaches 370 seconds.

MetaVaR: Introducing metavariant species models for reference-free metagenomic-based population genomics

The availability of large metagenomic data offers great opportunities for the population genomic analysis of uncultured organisms, which represent a large part of the unexplored biosphere and play a key ecological role. However, the majority of these organisms lack a reference genome or transcriptome, which constitutes a technical obstacle for classical population genomic analyses. We introduce the metavariant species (MVS) model, in which a species is represented only by intra-species nucleotide polymorphism. We designed a method combining reference-free variant calling, multiple density-based clustering and maximum-weighted independent set algorithms to cluster intra-species variants into MVSs directly from multisample metagenomic raw reads without a reference genome or read assembly. The frequencies of the MVS variants are then used to compute population genomic statistics such as FST, in order to estimate genomic differentiation between populations and to identify loci under natural selection. The MVS construction was tested on simulated and real metagenomic data. MVSs showed the required quality for robust population genomics and allowed an accurate estimation of genomic differentiation (ΔFST < 0.0001 and <0.03 on simulated and real data respectively). Loci predicted under natural selection on real data were all detected by MVSs. MVSs represent a new paradigm that may simplify and enhance holistic approaches for population genomics and the evolution of microorganisms.

Solving the Set Packing Problem via a Maximum Weighted Independent Set Heuristic

The set packing problem (SPP) is a significant NP-hard combinatorial optimization problem with extensive applications. In this paper, we encode the set packing problem as the maximum weighted independent set (MWIS) problem and solve the encoded problem with an efficient algorithm designed to the MWIS problem. We compare the independent set-based method with the state-of-the-art algorithms for the set packing problem on the 64 standard benchmark instances. The experimental results show that the independent set-based method is superior to the existing algorithms in terms of the quality of the solutions and running time obtained the solutions.

On Lagrangian relaxation for constrained maximization and reoptimization problems

We prove a general result demonstrating the power of Lagrangian relaxation in solving constrained maximization problems with arbitrary objective functions. This yields a unified approach for solving a wide class of subset selection problems with linear constraints. Given a problem in this class and some small ε∈(0,1), we show that if there exists an r-approximation algorithm for the Lagrangian relaxation of the problem, for some r∈(0,1), then our technique achieves a ratio of rr+1−ε to the optimal, and this ratio is tight.Using the technique we obtain (re)approximation algorithms for natural (reoptimization) variants of classic subset selection problems, including real-time scheduling, the maximum generalized assignment problem (GAP) and maximum weight independent set. For all of the problems studied in this paper, the number of calls to the r-approximation algorithm, used by our algorithms, is linear in the input size and in log(1∕ε) for inputs with a cardinality constraint, and polynomial in the input size and in log(1∕ε) for inputs with an arbitrary linear constraint.

Energy-Efficient Cluster Head Selection via Quantum Approximate Optimization

This paper proposes an energy-efficient cluster head selection method in the wireless ad hoc network by using a hybrid quantum-classical approach. The wireless ad hoc network is divided into several clusters via cluster head selection, and the performance of the network topology depends on the distribution of these clusters. For an energy-efficient network topology, none of the selected cluster heads should be neighbors. In addition, all the selected cluster heads should have high energy-consumption efficiency. Accordingly, an energy-efficient cluster head selection policy can be defined as a maximum weight independent set (MWIS) formulation. The cluster head selection policy formulated with MWIS is solved by using the quantum approximate optimization algorithm (QAOA), which is a hybrid quantum-classical algorithm. The accuracy of the proposed energy-efficient cluster head selection via QAOA is verified via simulations.

Quantum Approximation for Wireless Scheduling

This paper proposes an application algorithm based on a quantum approximate optimization algorithm (QAOA) for wireless scheduling problems. QAOA is one of the promising hybrid quantum-classical algorithms to solve combinatorial optimization problems and it provides great approximate solutions to non-deterministic polynomial-time (NP) hard problems. QAOA maps the given problem into Hilbert space, and then it generates the Hamiltonian for the given objective and constraint. Then, QAOA finds proper parameters from the classical optimization loop in order to optimize the expectation value of the generated Hamiltonian. Based on the parameters, the optimal solution to the given problem can be obtained from the optimum of the expectation value of the Hamiltonian. Inspired by QAOA, a quantum approximate optimization for scheduling (QAOS) algorithm is proposed. The proposed QAOS designs the Hamiltonian of the wireless scheduling problem which is formulated by the maximum weight independent set (MWIS). The designed Hamiltonian is converted into a unitary operator and implemented as a quantum gate operation. After that, the iterative QAOS sequence solves the wireless scheduling problem. The novelty of QAOS is verified with simulation results implemented via Cirq and TensorFlow-Quantum.

Iterated local search for the generalized independent set problem

The generalized independent set problem (GISP) can be conceived as a relaxation of the maximum weight independent set problem. GISP has a number of practical applications, such as forest harvesting and handling geographic uncertainty in spatial information. This work presents an iterated local search (ILS) heuristic for solving GISP. The proposed heuristic relies on two new neighborhood structures, which are explored using a variable neighborhood descent procedure. Experimental results on a well-known GISP benchmark indicate our proposal outperforms the best existing heuristic for the problem. In particular, our ILS approach was able to find all known optimal solutions and to present new improved best solutions.

A Novel Method for Scheduling of Wireless Ad Hoc Networks in Polynomial Time

In this article, we address the scheduling problem in wireless ad hoc networks by exploiting the computational advantage that comes when scheduling problems can be represented by claw-free conflict graphs where we consider a wireless broadcast medium. It is possible to formulate a scheduling problem of broadcast transmissions as finding the maximum weighted independent set (MWIS) in the conflict graph of the network. Finding the MWIS of a general graph is NP-hard leading to an NP-hard complexity of scheduling. In a claw-free conflict graph, MWIS may be found in polynomial time leading to a throughput-optimal scheduling. We show that the conflict graphs of certain wireless ad hoc networks are claw-free. In order to obtain claw-free conflict graphs in general networks, we suggest introducing additional conflicts (edges) with the aim of keeping the decrease in MWIS size minimal. To this end, we introduce an iterative optimization problem to decide where to introduce edges and investigate its efficient implementation. We conclude that the claw breaking method by adding extra edges can perform very close to optimal scenario and better than the polynomial time maximal independent set scheduling benchmark under the necessary assumptions.

You Calculate and I Provision: A DRL-Assisted Service Framework to Realize Distributed and Tenant-Driven Virtual Network Slicing

This article studies the problem of virtual network (VNT) slicing in datacenter interconnections (DCIs), and proposes a novel service framework to better balance the tradeoff between cost-effectiveness, and time-efficiency. Our idea is to partition a DCI into non-overlapped subgraphs, divide the VNT slicing in each subgraph into four collaborative steps, and get tenants involved in the calculation of virtual network embedding (VNE) schemes. With our proposal, an agent of infrastructure provider (InP) leverages deep reinforcement learning (DRL) to price, and advertise the substrate resources in one subgraph, motivates tenants to request resources in a load-balanced manner, and accepts VNE schemes from the tenants to avoid resource conflicts. Meanwhile, the tenants' task is to compute their own VNE schemes independently, and distributedly according to the resource information (i.e., the available resources, and their prices) advertised by the agent. We first design the DRL model based on the deep deterministic policy gradient (DDPG), and develop a VNT compression method based on auto-encoder (AE) to generalize the DRL's operation. Then, we study how to resolve resource conflicts among the distributedly-calculated VNE schemes, build a conflict graph (CG) to transform the VNE selection into finding the maximum weighted independent set (MWIS) in the CG, and design a polynomial-time approximation algorithm to solve the problem. Extensive simulations confirm that compared with the centralized service framework relying solely on the InP for VNE calculation, our proposed DRL-assisted distributed framework provisions VNT requests with significantly shorter computation time, and comparable blocking performance.

Two-Stage Scalable Air Traffic Flow Management Model Under Uncertainty

In order to efficiently balance the current and future air traffic demands with the system capacity, a proper Air Traffic Flow Management (ATFM) approach is required. The current focus of ATFM is generally on optimally utilizing the available airspace and airport capacities, while maintaining the required safety separation between aircraft. Yet, only a minor focus is given to the inherent uncertainty in the Air Transportation System (ATS), especially to its adverse effect on safety and day-to-day operations. To this end, we propose an ATFM framework scrutinizing the stochastic nature of ATS through a chance-constraint-based probabilistic approach. Moreover, anticipating the high volumes in air traffic in the future, we propose to split the model into two stages, in which the first stage scrutinizes the behavior of a set of flights as a flow, while the second stage transforms them into individual flight plans, enhancing scalability. The two models are formulated as an Integer Linear Programming (ILP) problem, and a Mixed Integer Linear Programming (MILP) problem at stages I and II, respectively. The NP-hard nature of the overall problem is minimized by transforming the problem into a Maximum Weighted Independent Set (MWIS) finding problem.

Clearing Kidney Exchanges via Graph Neural Network Guided Tree Search (Student Abstract)

Kidney exchange is an organized barter market that allows patients with end-stage renal disease to trade willing donors—and thus kidneys—with other patient-donor pairs. The central clearing problem is to find an arrangement of swaps that maximizes the number of transplants. It is known to be NP-hard in almost all cases. Most existing approaches have modeled this problem as a mixed integer program (MIP), using classical branch-and-price-based tree search techniques to optimize. In this paper, we frame the clearing problem as a Maximum Weighted Independent Set (MWIS) problem, and use a Graph Neural Network guided Monte Carlo Tree Search to find a solution. Our initial results show that this approach outperforms baseline (non-optimal but scalable) algorithms. We believe that a learning-based optimization algorithm can improve upon existing approaches to the kidney exchange clearing problem.