Independent Set Problem Research Articles

Twin-width and Polynomial Kernels

We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for k -Dominating Set on graphs of twin-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for Connected k -Dominating Set and Total k -Dominating Set (albeit with a worse upper bound on the twin-width). The k -Independent Set problem admits the same lower bound by a much simpler argument, previously observed [ICALP ’21], which extends to k -Independent Dominating Set, k -Path, k -Induced Path, k -Induced Matching, etc. On the positive side, we obtain a simple quadratic vertex kernel for Connected k -Vertex Cover and Capacitated k -Vertex Cover on graphs of bounded twin-width. Interestingly the kernel applies to graphs of Vapnik–Chervonenkis density 1, and does not require a witness sequence. We also present a more intricate \(O(k^{1.5})\) vertex kernel for Connected k -Vertex Cover. Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most optimization/decision graph problems can be solved in polynomial time on graphs of twin-width at most 1.

Combinatorial optimization with physics-inspired graph neural networks

Combinatorial optimization problems are pervasive across science and industry. Modern deep learning tools are poised to solve these problems at unprecedented scales, but a unifying framework that incorporates insights from statistical physics is still outstanding. Here we demonstrate how graph neural networks can be used to solve combinatorial optimization problems. Our approach is broadly applicable to canonical NP-hard problems in the form of quadratic unconstrained binary optimization problems, such as maximum cut, minimum vertex cover, maximum independent set, as well as Ising spin glasses and higher-order generalizations thereof in the form of polynomial unconstrained binary optimization problems. We apply a relaxation strategy to the problem Hamiltonian to generate a differentiable loss function with which we train the graph neural network and apply a simple projection to integer variables once the unsupervised training process has completed. We showcase our approach with numerical results for the canonical maximum cut and maximum independent set problems. We find that the graph neural network optimizer performs on par or outperforms existing solvers, with the ability to scale beyond the state of the art to problems with millions of variables.

Polynomial-time Algorithm for Maximum Weight Independent Set on P 6 -free Graphs

In the classic Maximum Weight Independent Set problem, we are given a graph G with a nonnegative weight function on its vertices, and the goal is to find an independent set in G of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any P 6 -free graph, that is, a graph that has no path on 6 vertices as an induced subgraph. This improves the polynomial-time algorithm on P 5 -free graphs of Lokshtanov et al. [ 15 ] and the quasipolynomial-time algorithm on P 6 -free graphs of Lokshtanov et al. [ 14 ]. The main technical contribution leading to our main result is enumeration of a polynomial-size family ℱ of vertex subsets with the following property: For every maximal independent set I in the graph, ℱ contains all maximal cliques of some minimal chordal completion of G that does not add any edge incident to a vertex of I .

High dimensional Hoffman bound and applications in extremal combinatorics

The n-th tensor power of a graph with vertex set V is the graph on the vertex set V n , where two vertices are connected by an edge if they are connected in each coordinate. One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. In this paper we introduce the problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many prominent open problems in extremal combinatorics, such as the Turán problem for graphs and hypergraphs, can be encoded as special cases of this problem. We generalize the Hoffman bound to hypergraphs, and give several applications.

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.

Time-free Solution to Independent Set Problem using P Systems with Active Membranes

Membrane computing is a branch of natural computing aiming to abstract computing models from the structure and functioning of living cells. The computation models obtained in the field of membrane computing are usually called P systems. P systems have been used to solve computationally hard problems efficiently on the assumption that the execution of each rule is completed in exactly one time-unit (a global clock is assumed for timing and synchronizing the execution of rules). However, in biological reality, different biological processes take different times to be completed, which can also be influenced by many environmental factors. In this work, with this biological reality, we give a time-free solution to independent set problem using P systems with active membranes, which solve the problem independent of the execution time of the involved rules.

On approximating MIS over B1-VPG graphs*

In this paper, we present an approximation algorithm for the maximum independent set (MIS) problem over the class of [Formula: see text]-VPG graphs when the input is specified by a [Formula: see text]-VPG representation. We obtain a [Formula: see text]-approximation algorithm running in [Formula: see text] time. This is an improvement over the previously best [Formula: see text]-approximation algorithm [J. Fox and J. Pach, Computing the independence number of intersection graphs, in Proc. Twenty-Second Annual ACM-SIAM Symp. Discrete Algorithms (SODA 2011), 2011, pp. 1161–1165, doi:10.1137/1.9781611973082.87] (for some fixed [Formula: see text]) designed for some subclasses of string graphs, on [Formula: see text]-VPG graphs.

Simple iterative trial search for the maximum independent set problem optimized for the GPUs

AbstractAn independent set of a graph is a subset of the nodes such that no two nodes in it are adjacent. The maximum independent set (MIS) problem is an optimization problem to find a largest independent set. The main contribution of this article is to introduce a generic iterative trial search algorithm that we call iMIS for finding approximate solutions for the MIS problem. The generic algorithm iMIS is designed so that it can be implemented to run on GPUs very efficiently. Since the performance of the algorithm varies depending on its search strategy, we present a hybrid algorithm that combines three strategies of the iMIS such that best one of them for an input graph is automatically selected. We have implemented our hybrid algorithm to run on a multi‐GPU server with eight NVIDIA A100 GPUs. And evaluated the performance for 66 DIMACS benchmark graphs and 78 random graphs with up to 256M nodes. We have also evaluated the performance of three previously published algorithms for the MIS problem and an approach using Gurobi linear programming solver. The experimental results show that our hybrid algorithm can find larger independent sets for all 144 graphs compared to other methods.

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.

Dispersing and grouping points on planar segments

Motivated by (continuous) facility location, we study the problem of dispersing and grouping points on a set of segments (of streets) in the plane. In the former problem, given a set of n disjoint line segments in the plane, we investigate the problem of computing a point on each of the n segments such that the minimum Euclidean distance between any two of these points is maximized. We prove that this 2D dispersion problem is NP-hard, in fact, it is NP-hard even if all the segments are parallel and are of unit length. This is in contrast to the polynomial solvability of the corresponding 1D problem by Li and Wang (2016), where the intervals are in 1D and are all disjoint. With this result, we also show that the Independent Set problem on Colored Linear Unit Disk Graph (meaning the convex hulls of points with the same color form disjoint line segments) remains NP-hard, and the parameterized version of it is in W[2]. In the latter problem, given a set of n disjoint line segments in the plane we study the problem of computing a point on each of the n segments such that the maximum Euclidean distance between any two of these points is minimized. We present a factor-1.1547 approximation algorithm which runs in O(nlog⁡n) time. Our results can be generalized to the Manhattan distance.

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.

Learning Pseudo-Backdoors for Mixed Integer Programs

We propose a machine learning approach for quickly solving Mixed Integer Programs (MIP) by learning to prioritize a set of decision variables, which we call pseudo-backdoors, for branching that results in faster solution times. Learning-based approaches have seen success in the area of solving combinatorial optimization problems by being able to flexibly leverage common structures in a given distribution of problems. Our approach takes inspiration from the concept of strong backdoors, which corresponds to a small set of variables such that only branching on these variables yields an optimal integral solution and a proof of optimality. Our notion of pseudo-backdoors corresponds to a small set of variables such that only branching on them leads to faster solve time (which can be solver dependent). A key advantage of pseudo-backdoors over strong backdoors is that they are much amenable to data-driven identification or prediction. Our proposed method learns to estimate the solver performance of a proposed pseudo-backdoor, using a labeled dataset collected on a set of training MIP instances. This model can then be used to identify high-quality pseudo-backdoors on new MIP instances from the same distribution. We evaluate our method on the generalized independent set problems and find that our approach can efficiently identify high-quality pseudo-backdoors. In addition, we compare our learned approach against Gurobi, a state-of-the-art MIP solver, demonstrating that our method can be used to improve solver performance.

Quantum Speedup and Mathematical Solutions of Implementing Bio-Molecular Solutions for the Independent Set Problem on IBM Quantum Computers.

In this paper, we propose a bio-molecular algorithm with O( n2 + m ) biological operations, O( 2n ) DNA strands, O( n ) tubes and the longest DNA strand, O( n ), for solving the independent-set problem for any graph G with m edges and n vertices. Next, we show that a new kind of the straightforward Boolean circuit yielded from the bio-molecular solutions with m NAND gates, ( m +n × ( n + 1 )) AND gates and (( n × ( n + 1 ))/2) NOT gates can find the maximal independent-set(s) to the independent-set problem for any graph G with m edges and n vertices. We show that a new kind of the proposed quantum-molecular algorithm can find the maximal independent set(s) with the lower bound Ω ( 2n/2 ) queries and the upper bound O( 2n/2 ) queries. This work offers an obvious evidence for that to solve the independent-set problem in any graph G with m edges and n vertices, bio-molecular computers are able to generate a new kind of the straightforward Boolean circuit such that by means of implementing it quantum computers can give a quadratic speed-up. This work also offers one obvious evidence that quantum computers can significantly accelerate the speed and enhance the scalability of bio-molecular computers. Next, the element distinctness problem with input of n bits is to determine whether the given 2n real numbers are distinct or not. The quantum lower bound of solving the element distinctness problem is Ω ( 2n×(2/3) ) queries in the case of a quantum walk algorithm. We further show that the proposed quantum-molecular algorithm reduces the quantum lower bound to Ω (( 2n/2 )/( [Formula: see text]) queries. Furthermore, to justify the feasibility of the proposed quantum-molecular algorithm, we successfully solve a typical independent set problem for a graph G with two vertices and one edge by carrying out experiments on the backend ibmqx4 with five quantum bits and the backend simulator with 32 quantum bits on IBM's quantum computer.

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.

On the analysis of ant colony optimization for the maximum independent set problem

On the analysis of ant colony optimization for the maximum independent set problem

Collision-free routing problem with restricted L-path

Given a set of vehicles that are allowed to move in a plane along a predefined directed rectilinear path, the collision-free routing problem seeks a maximum number of vehicles that can move without collision. This problem is known to be NP-hard (Ajaykumar et al., 2016). Here we study a variant of this problem called the constrained collision-free routing problem (in short, CCRP ). In this problem, each vehicle is allowed only to move in a directed L-shaped path. First, we show that CCRP is NP-hard. Further, we prove that any β -approximation algorithm for the maximum independent set problem in B 1 -VPG graphs would produce a β -approximation for CCRP . Also, we propose a 2-approximation algorithm for the maximum independent set problem on intersection graph of n unit-L frames (the union of a unit-length vertical and a unit-length horizontal line segment that shares an endpoint) with O ( n ) space and O ( n 2 ) time.

A note on the fine-grained complexity of MIS on regular graphs

We show that there is no subexponential time algorithm for computing the exact solution of the maximum independent set problem in d-regular graphs, unless ETH fails. We expand our method to show that it helps to provide lowerbounds for other covering problems such as vertex cover and clique. We utilize the construction to show the NP-hardness of MIS on 5-regular planar graphs, closing the exact complexity status of the problem on regular planar graphs.

Combinatorial Optimization and Nonlinear Optimization

The objective of this paper is to locate a superior method that merges quicker of maximal independent set problem (MIS). One methodology is to the correlation between The Penalty and Augmented Lagrangian Methods. We have likewise built up the hypothetical combination properties of these methods.