Maximum Weight Independent Set Research Articles

Tree independence number I. (Even hole, diamond, pyramid)‐free graphs

AbstractThe tree‐independence number , first defined and studied by Dallard, Milanič, and Štorgel, is a variant of treewidth tailored to solving the maximum independent set problem. Over a series of papers, Abrishami et al. developed the so‐called central bag method to study induced obstructions to bounded treewidth. Among others, they showed that, in a certain superclass of (even hole, diamond, pyramid)‐free graphs, treewidth is bounded by a function of the clique number. In this paper, we relax the bounded clique number assumption, and show that has bounded . Via existing results, this yields a polynomial‐time algorithm for the Maximum Weight Independent Set problem in this class. Our result also corroborates, for this class of graphs, a conjecture of Dallard, Milanič, and Štorgel that in a hereditary graph class, is bounded if and only if the treewidth is bounded by a function of the clique number.

Submodular Functions and Perfect Graphs

We give a combinatorial polynomial-time algorithm to find a maximum weight independent set in perfect graphs of bounded degree that do not contain a prism or a hole of length four as an induced subgraph. An even pair in a graph is a pair of vertices all induced paths between which are even. An even set is a set of vertices every two of which are an even pair. We show that every perfect graph that does not contain a prism or a hole of length four as an induced subgraph has a balanced separator which is the union of a bounded number of even sets, where the bound depends only on the maximum degree of the graph. This allows us to solve the maximum weight independent set problem using the well-known submodular function minimization algorithm. Funding: This work was supported by the Engineering and Physical Sciences Research Council [Grant EP/V002813/1]; the National Science Foundation [Grants DMS-1763817, DMS-2120644, and DMS-2303251]; and Alexander von Humboldt-Stiftung.

Real-Time Energy Management for Marine Applications Using Markov Approximation

All-electric and hybrid-electric ships have become the centerpiece for reducing greenhouse gas emissions and improving fuel efficiency in the maritime industry. Real-time power management of multiple power sources in both design and operation becomes critical due to the uncertainty and randomness of the vessel's power demand. Furthermore, the vessel operators may have additional requirements for operational flexibility, such as fully recharging the battery before the next trip and limiting engine switch on/off to reduce wear-and-tear, which lead to complex power management systems requiring domain-expert knowledge. In this paper, a hybrid solution for real-time power management is proposed. The model-based approach encapsulates the power system, and the data-driven approach (a machine learning technique called Markov approximation) handles the uncertainty. First, an offline deterministic optimization problem for power management is formulated. Second, the deterministic problem is transformed into a maximum weighted independent set (MWIS) problem. Next, the Markov approximation framework is applied to the transformed problem to utilize the machine learning techniques widely used in wireless communications and computing industries. Finally, a reinforcement learning-based real-time solution is proposed. Extensive simulations are performed on a ferry case study, and the results are within the proven theoretical bounds of machine learning techniques.

Link Scheduling Using Graph Neural Networks

Efficient scheduling of transmissions is a key problem in wireless networks. The main challenge stems from the fact that optimal link scheduling involves solving a maximum weighted independent set (MWIS) problem, which is known to be NP-hard. In practical schedulers, centralized and distributed greedy heuristics are commonly used to approximately solve the MWIS problem. However, most of these greedy heuristics ignore important topological information of the wireless network. To overcome this limitation, we propose fast heuristics based on graph convolutional networks (GCNs) that can be implemented in centralized and distributed manners. Our centralized heuristic is based on tree search guided by a GCN and 1-step rollout. In our distributed MWIS solver, a GCN generates topology-aware node embeddings that are combined with per-link utilities before invoking a distributed greedy solver. Moreover, a novel reinforcement learning scheme is developed to train the GCN in a non-differentiable pipeline. Test results on medium-sized wireless networks show that our centralized heuristic can reach a near-optimal solution quickly, and our distributed heuristic based on a shallow GCN can reduce by nearly half the suboptimality gap of the distributed greedy solver with minimal increase in complexity. The proposed schedulers also exhibit good generalizability across graph and weight distributions.

Fast Link Scheduling in Wireless Networks Using Regularized Off-Policy Reinforcement Learning

The centralized-link-scheduling problem in a wireless network graph involves solving the maximum-weighted-independent-set (MWIS) problem on the conflict graph. In this paper, we propose a novel regularized off-policy reinforcement learning-based MWIS solver and use for the scheduling problem. The proposed MWIS algorithm achieves 17% improvement over state-of-the-art heuristic solver KaMIS, 60% over greedy solver, 16% and 17% over RL-based solvers LwD and S2V-DQN, respectively. We show that our scheduler achieves stable throughput values 14% and 22% higher than LwD and a distributed greedy scheduler, respectively. We demonstrate the flexibility of our RL algorithm by modifying it to create a time-since-last-service-aware scheduler.

Rydberg-Blockade-Based Parity Quantum Optimization.

We present a scalable architecture for solving higher-order constrained binary optimization (HCBO) problems on current neutral-atom hardware operating in the Rydberg blockade regime. In particular, we formulate the recently developed parity encoding of arbitrary connected HCBO problems as a maximum-weight independent set (MWIS) problem on disk graphs, that are directly encodable on such devices. Our architecture builds from small MWIS modules in a problem-independent way, crucial for practical scalability.

Eigenvalue-Invariant Transformation of Ising Problem for Anti-Crossing Mitigation in Quantum Annealing

We have proposed the energy landscape transformation of Ising problems (ELTIP), which changes the combination of the state and eigenvalue without changing all the original eigenvalues [arXiv:2202.05927]. We study how the ELTIP affects the anti-crossing between two levels of the ground and first excited states during quantum annealing. We use a 5-spin maximum-weighted independent set for the problem to numerically investigate the anti-crossing. For comparison, we introduce a non-stoquastic Hamiltonian that adds antiferromagnetic interaction to the normal transverse magnetic field. Annealing with the non-stoquastic Hamiltonian is effective for difficult problems. The non-stoquastic Hamiltonian mitigates the anti-crossing when only the energy gap between the ground state and the first excited state of the final state is small. When the ELTIP is used, the anti-crossing disappears. For the problems investigated in this paper, the ELTIP shortens the annealing time to guarantee adiabatic change more than the non-stoquastic Hamiltonian.

Adaptive Selection and Clustering of Partial Reconfiguration Modules for Modern FPGA Design Flow

Dynamic Partially Reconfiguration (DPR) on FPGA has attracted significant research interest in recent years since it provides benefits such as reduced area and flexible functionality. However, due to the lack of supporting synthesis tools in the current DPR design flow, leveraging benefits from DPR requires specific design expertise with laborious manual design effort. Considering the complicated concurrency relations among various functions, it is challenging to select appropriate Partial Reconfiguration Modules (PR Modules) and cluster them into proper groups with a proper reconfiguration schedule so that the hardware modules can be swapped in and out correctly during the run time. Furthermore, the design of PR Modules also impacts reconfiguration latency and resource utilization greatly. In this paper, we propose a Maximum-Weight Independent Set model to formulate the PR Module selection and clustering problem so that the original manual exploration can be solved efficiently and automatically. We also propose a step-wise adjustment configuration prefetching strategy incorporated in our model to generate optimized reconfiguration schedules. Our proposed approach not only supports various design constraints but also can consider multiple objectives such as area and reconfiguration delay. Experimental results show that our approach can optimize resource utilization and reduce reconfiguration delay with good scalability. Especially, the implementation of the real design case shows that our approach can be embedded in Xilinx's DPR design flow successfully.

Coordinated Scheduling and Decentralized Federated Learning Using Conflict Clustering Graphs in Fog-Assisted IoD Networks

Despite the advantages of fog-assisted internet of drones (IoD) networks for federated learning (FL) model aggregations, it is restricted by the limited battery capacity of drones and wireless channel conditions between a fog and drones. In this paper, we address the deployment problem of decentralized FL and the secrecy rate maximization problem in fog-assisted IoD networks. In particular, we leverage the overlapped zones of drones to spread their local aggregated models to the network. As such, the global model aggregation is achieved without any participation from the fog. Then, aiming to secure the proposed decentralized FL-based framework, we consider the secrecy rate maximization problem, which can be addressed by scheduling users to drones and their available radio resource blocks (RRBs). This coordinated scheduling problem of users-drones/RRBs is NP-hard. To efficiently solve it using graph theory techniques, we construct a centralized conflict clustering graph and reformulate the considered scheduling problem as a maximum-weight independent set (MWIS) problem in the constructed conflict graph. To alleviate the need for the coordinated scheduling policy at the fog, we propose to solve the coordinated scheduling problem in a distributed manner. Our proposed distributed schemes rely on the configuration of the constructed conflict clustering graph used to solve the MWIS problem. We conduct extensive simulations to verify the effectiveness of the proposed schemes over existing schemes. Selected numerical results show that the proposed distributed scheduling schemes provide similar performances in terms of secrecy rate and energy consumption as compared with the centralized scheme, with reasonable amount of information exchanges among the drones.

Quantum Optimization with Arbitrary Connectivity Using Rydberg Atom Arrays

Programmable quantum systems based on Rydberg atom arrays have recently been used for hardware-efficient tests of quantum optimization algorithms [Ebadi et al., Science, 376, 1209 (2022)] with hundreds of qubits. In particular, the maximum independent set problem on so-called unit-disk graphs, was shown to be efficiently encodable in such a quantum system. Here, we extend the classes of problems that can be efficiently encoded in Rydberg arrays by constructing explicit mappings from a wide class of problems to maximum-weighted independent set problems on unit-disk graphs, with at most a quadratic overhead in the number of qubits. We analyze several examples, including maximum-weighted independent set on graphs with arbitrary connectivity, quadratic unconstrained binary optimization problems with arbitrary or restricted connectivity, and integer factorization. Numerical simulations on small system sizes indicate that the adiabatic time scale for solving the mapped problems is strongly correlated with that of the original problems. Our work provides a blueprint for using Rydberg atom arrays to solve a wide range of combinatorial optimization problems with arbitrary connectivity, beyond the restrictions imposed by the hardware geometry.5 MoreReceived 14 September 2022Revised 14 December 2022Accepted 4 January 2023DOI:https://doi.org/10.1103/PRXQuantum.4.010316Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBoolean satisfiability problemNP-hard problemsQuantum information architectures & platformsQuantum softwarePhysical SystemsAtomsRydberg atoms & moleculesQuantum InformationAtomic, Molecular & OpticalStatistical Physics

Grid induced minor theorem for graphs of small degree

A graph H is an induced minor of a graph G if H can be obtained from G by vertex deletions and edge contractions. We show that there is a function f(k,d)=O(k10+2d5) so that if a graph has treewidth at least f(k,d) and maximum degree at most d, then it contains a k×k-grid as an induced minor. This proves the conjecture of Aboulker, Adler, Kim, Sintiari, and Trotignon (2021) [1] that any graph with large treewidth and bounded maximum degree contains a large wall or the line graph of a large wall as an induced subgraph. It also implies that for any fixed planar graph H, there is a subexponential time algorithm for maximum weight independent set on H-induced-minor-free graphs.

Network-Level Traffic Signal Cooperation: A Higher-Order Conflict Graph Approach

Traffic signal control and cooperation are extremely important to alleviate traffic congestion in a large traffic network. This study develops a higher-order conflict graph approach for network-wide traffic signal control and cooperation. A conflict graph is applied to model the traffic signal configurations, which identifies the conflict and unconflicted movements for each intersection. In conflict graph, the node represents each movement. The weight of each node can be defined as traffic volume, queue length, fuel consumption, or any weighted combinations of these measurements. The calculation of the optimal green light duration and green light sequence (for different movements) is equivalent to sequentially finding the maximum weight independent set (MWIS) in the conflict graph. The conflict graph also provides a uniform and efficient way to connect traffic signal operations among nearby intersections spatially. Then, we introduced the concept of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -th order neighborhood to model the degree of connectivity between each movement to the movements at upstream or downstream intersections. The weight of each node in the higher-order conflict graph not only represents its own congestion level, but also relates to the traffic conditions of nearby intersections. Through this approach, the cooperation of multiple intersections can be realized by incorporating their spatial connectivity into conflict graph and solving the MWIS problem. A simulation network is built in SUMO to test the effectiveness of the proposed method. Results suggested that the proposed model outperformed other state-of-the-art signal control methods. Also, the scheme maintains good performance under varying traffic demands.

The submodularity of two-stage stochastic maximum-weight independent set problems

In this paper, we extend the maximal independent set problem to two-stage stochastic case: given an independence system associated with one deterministic weight function and a random weight function, the goal is to find two nonoverlapping independent subsets from these two stages with the maximum total weight. In this paper, we study the submodularity of three kinds of two-stage independent set problems with max-weight. When the independent set problem is a matroid constraint, we can show its submodularity. However, neither submodular nor supermodular maximization problem can be obtained for the knapsack independent set problem by designing a counterexample. At last, we show that the robust two-stage stochastic maximum-weight uniform matroid problem can be formulated as a γ-submodular problem with cardinality constraint and also give a lower bound for γ.

New Results on Independent Sets in Extensions of 2K_2-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 is solvable 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. For any two graphs G and H, G+H denotes the disjoint union of G and H. A lK_2 is the disjoint union of l edges. A Y_{m,m} is the disjoint union of two stars of m+1 vertices plus one vertex that is adjacent only to the centers of such stars. For any graph family {{{mathcal {Y}}}}, the class of {mathcal Y}-free graphs is formed by graphs which are Y-free for every Y in {{{mathcal {Y}}}}, and the class of lK_2+{{{mathcal {Y}}}}-free graphs is formed by graph which are lK_2+Y-free for every Y in {mathcal Y}. The main result of this manuscript is the following: For any constant m and for any graph family {{{mathcal {Y}}}} which contains an induced subgraph of Y_{m,m}, if WIS is solvable in polynomial time for {{{mathcal {Y}}}}-free graphs, then WIS is solvable in polynomial time for lK_2+{{{mathcal {Y}}}}-free graphs for any constant l. That extends some known polynomial results, namely, when {{{mathcal {Y}}}} = {Y} and Y is a fork or is a P_5. The proof of the main result is based on Farber’s approach to prove that every 2K_2-free graph has O(n^2) maximal independent sets (Farber in Discrete Math 73:249–260, 1989), which directly leads to a polynomial time algorithm to solve WIS for 2K_2-free graphs through a dynamic programming approach, and on some extensions of Farber’s approach.

Drug repurposing based on a quantum-inspired method versus classical fingerprinting uncovers potential antivirals against SARS-CoV-2.

The COVID-19 pandemic has accelerated the need to identify new antiviral therapeutics at pace, including through drug repurposing. We employed a Quadratic Unbounded Binary Optimization (QUBO) model, to search for compounds similar to Remdesivir, the first antiviral against SARS-CoV-2 approved for human use, using a quantum-inspired device. We modelled Remdesivir and compounds present in the DrugBank database as graphs, established the optimal parameters in our algorithm and resolved the Maximum Weighted Independent Set problem within the conflict graph generated. We also employed a traditional Tanimoto fingerprint model. The two methods yielded different lists of lead compounds, with some overlap. While GS-6620 was the top compound predicted by both models, the QUBO model predicted BMS-986094 as second best. The Tanimoto model predicted different forms of cobalamin, also known as vitamin B12. We then determined the half maximal inhibitory concentration (IC50) values in cell culture models of SARS-CoV-2 infection and assessed cytotoxicity. We also demonstrated efficacy against several variants including SARS-CoV-2 Strain England 2 (England 02/2020/407073), B.1.1.7 (Alpha), B.1.351 (Beta) and B.1.617.2 (Delta). Lastly, we employed an in vitro polymerization assay to demonstrate that these compounds directly inhibit the RNA-dependent RNA polymerase (RdRP) of SARS-CoV-2. Together, our data reveal that our QUBO model performs accurate comparisons (BMS-986094) that differed from those predicted by Tanimoto (different forms of vitamin B12); all compounds inhibited replication of SARS-CoV-2 via direct action on RdRP, with both models being useful. While Tanimoto may be employed when performing relatively small comparisons, QUBO is also accurate and may be well suited for very complex problems where computational resources may limit the number and/or complexity of possible combinations to evaluate. Our quantum-inspired screening method can therefore be employed in future searches for novel pharmacologic inhibitors, thus providing an approach for accelerating drug deployment.

A faster algorithm for maximum independent set on interval filament graphs

We provide an algorithm requiring only $O(N^2)$ time to compute the maximum weight independent set of interval filament graphs. This also implies an $O(N^4)$ algorithm to compute the maximum weight induced matching of interval filament graphs. Both algorithms significantly improve upon the previous best complexities for these problems. Previously, the maximum weight independent set and maximum weight induced matching problems required $O(N^3)$ and $O(N^6)$ time respectively.

Scanning the Issue

<bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Two-Stage Scalable Air Traffic Flow Management Model Under Uncertainty</b> <p xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>G. G. N. Sandamali, R. Su, K. L. K. Sudheera, Y. Zhang, and Y. Zhang</i> <p xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">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, the authors propose an ATFM framework for scrutinizing the stochastic nature of ATS through a chance-constraint-based probabilistic approach. Moreover, anticipating the high volumes in air traffic in the future, the authors 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.

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.