Maximum Weight Independent Set Problem 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.

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.

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.

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.

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.

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.

Solving Robust Weighted Independent Set Problems on Trees and under Interval Uncertainty

The maximum weighted independent set (MWIS) problem is important since it occurs in various applications, such as facility location, selection of non-overlapping time slots, labeling of digital maps, etc. However, in real-life situations, input parameters within those models are often loosely defined or subject to change. For such reasons, this paper studies robust variants of the MWIS problem. The study is restricted to cases where the involved graph is a tree. Uncertainty of vertex weights is represented by intervals. First, it is observed that the max–min variant of the problem can be solved in linear time. Next, as the most important original contribution, it is proved that the min–max regret variant is NP-hard. Finally, two mutually related approximation algorithms for the min–max regret variant are proposed. The first of them is already known, but adjusted to the considered situation, while the second one is completely new. Both algorithms are analyzed and evaluated experimentally.

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.

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.

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.

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.

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.

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.

Grundy dominating sequences on [formula omitted]-join product

In this paper we study the Grundy domination number on the X-join product G↩R of a graph G and a family of graphs R={Gv:v∈V(G)}. The results led us to extend the few known families of graphs where this parameter can be efficiently computed. We prove that if, for all v∈V(G), the Grundy domination number of Gv is given, and G is a power of a cycle, a power of a path, or a split graph, computing the Grundy domination number of G↩R can be done in polynomial time. In particular, our results for powers of cycles and paths are derived from a polynomial reduction to the Maximum Weight Independent Set problem on these graphs.As a consequence, we derive closed formulas to compute the Grundy domination number of the lexicographic product G∘H when G is a power of a cycle, a power of a path or a split graph, generalizing the results on cycles and paths given by Brešar et al. in 2016. Moreover, our results on the X-join product when G is a split graph also provide polynomial-time algorithms to compute the Grundy domination number for (q,q−4) graphs, partner limited graphs and extended P4-laden graphs, graph classes that are high in the hierarchy of few P4’s graphs.

Solving Robust Variants of the Maximum Weighted Independent Set Problem on Trees

This paper deals with the maximum weighted independent set (MWIS) problem. We consider several robust variants of the MWIS problem on trees and prove that most of them are NP-hard. We propose a heuristic for solving the considered robust MWIS variants, which is customized for trees. We demonstrate by experiments that our algorithm produces high-quality solutions and runs much faster than a general-purpose optimization software.

The effects of the problem Hamiltonian parameters on the minimum spectral gap in adiabatic quantum optimization

We study the relation between the Ising problem Hamiltonian parameters and the minimum spectral gap (min-gap) of the system Hamiltonian in the Ising-based quantum annealer. The main argument we use in this paper to assess the performance of a QA algorithm is the presence or absence of an anti-crossing during quantum evolution. For this purpose, we introduce a new parametrization definition of the anti-crossing. Using the Maximum-weighted Independent Set (MIS) problem in which there are flexible parameters (energy penalties J between pairs of edges) in an Ising formulation as the model problem, we construct examples to show that by changing the value of J, we can change the quantum evolution from one that has an anti-crossing (that results in an exponential small min-gap) to one that does not have, or the other way around, and thus drastically change (increase or decrease) the min-gap. However, we also show that by changing the value of $J$ alone, one can not avoid the anti-crossing. We recall a polynomial reduction from an Ising problem to an MIS problem to show that the flexibility of changing parameters without changing the problem to be solved can be applied to any Ising problem. As an example, we show that by such a reduction alone, it is possible to remove the anti-crossing and thus increase the min-gap. Our anti-crossing definition is necessarily scaling invariant as scaling the problem Hamiltonian does not change the nature (i.e. presence or absence) of an anti-crossing. As a side note, we show exactly how the min-gap is scaled if we scale the problem Hamiltonian by a constant factor.