Maximum Independent Set Problem Research Articles

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.

Completion Time Minimization in Fog-RANs Using D2D Communications and Rate-Aware Network Coding

The device-to-device communication-aided fog radio access network, referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D2D-aided</i> F-RAN, takes advantage of caching at enhanced remote radio heads (eRRHs) and D2D proximity for improved system performance. For D2D-aided F-RAN, we develop a framework that exploits the cached contents at eRRHs, their transmission rates/powers, and previously received contents by different users to deliver the requesting contents to users with a minimum completion time. Given the intractability of the completion time minimization problem, we formulate it at each transmission by approximating the completion time and decoupling it into two subproblems. In the first subproblem, we minimize the possible completion time in eRRH downlink transmissions, while in the second subproblem, we maximize the number of users to be scheduled on D2D links. We design two theoretical graphs, namely <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">interference-aware</i> instantly decodable network coding (IA-IDNC) and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D2D conflict</i> graphs to reformulate two subproblems as maximum weight clique and maximum independent set problems, respectively. Using these graphs, we heuristically develop joint and coordinated scheduling approaches. Simulation results show that the proposed two approaches achieve a considerable performance gain in terms of the completion time minimization.

Gumbel-softmax-based optimization: a simple general framework for optimization problems on graphs

In computer science, there exist a large number of optimization problems defined on graphs, that is to find a best node state configuration or a network structure, such that the designed objective function is optimized under some constraints. However, these problems are notorious for their hardness to solve, because most of them are NP-hard or NP-complete. Although traditional general methods such as simulated annealing (SA), genetic algorithms (GA), and so forth have been devised to these hard problems, their accuracy and time consumption are not satisfying in practice. In this work, we proposed a simple, fast, and general algorithm framework based on advanced automatic differentiation technique empowered by deep learning frameworks. By introducing Gumbel-softmax technique, we can optimize the objective function directly by gradient descent algorithm regardless of the discrete nature of variables. We also introduce evolution strategy to parallel version of our algorithm. We test our algorithm on four representative optimization problems on graph including modularity optimization from network science, Sherrington–Kirkpatrick (SK) model from statistical physics, maximum independent set (MIS) and minimum vertex cover (MVC) problem from combinatorial optimization on graph, and Influence Maximization problem from computational social science. High-quality solutions can be obtained with much less time-consuming compared to the traditional approaches.

Conflict resolving – A local search algorithm for solving large scale conflict graphs in freight railway timetabling

We consider the problem of planning the annual timetable for all freight trains in Germany simultaneously. That is, for each train, construct a slot through the network such that no two slots of different trains have a conflict. We denote this task by the Train Path Assignment Problem (TPAP) and present a column generation approach where iteratively, the set of possible slots grows. In each iteration, we look for a maximum subset without any conflicts. Modelling the slots as vertices joint by an edge if they have a conflict, this problem is the Maximum Independent Set problem (MIS). Due to the many slots that are constructed, hence variables that are generated, we deal with large scale MIS instances. Therefore, the MIS is solved heuristically with a local search algorithm called Conflict Resolving (CR) that is tailored to the specially structured instances from the application. CR iteratively perturbs the current solution in order to leave local optima and then repeatedly improves the solution by replacing k−1 solution vertices by k non-solution vertices. These steps are embedded in a simulated annealing framework. In this paper, we present the column generation approach and numerically compare three MIS solution methods, CR, a MIP solver and Iterated Local Search (ILS), a state-of-the-art MIS heuristics. It turns out that CR performs best for the instances from real-world timetabling, and is also comparable to the ILS on MIS benchmark instances. With this approach, we can solve the TPAP for more than 5000 freight trains.

From Independent Sets and Vertex Colorings to Isotropic Spaces and Isotropic Decompositions: Another Bridge between Graphs and Alternating Matrix Spaces

In the 1970s, Lovász built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings [Proceedings of FCT, 1979, pp. 565--574]. A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the noncommutative rank problem [A. Garg et al., Proceedings of FOCS, 2016, pp. 109--117; G. Ivanyos, Y. Qiao, and K. V. Subrahmanyam, Comput. Complexity, 26 (2017), pp. 717--763]. In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory. We first show that the maximum independent set problem and the vertex $c$-coloring problem reduce to the maximum isotropic space problem and the isotropic $c$-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration.

Solving optimization problems with Rydberg analog quantum computers: Realistic requirements for quantum advantage using noisy simulation and classical benchmarks

Platforms of Rydberg atoms have been proposed as promising candidates to solve some combinatorial optimization problems. Here, we compute quantitative requirements on the system sizes and noise levels that these platforms must fulfill to reach quantum advantage in approximately solving the Unit-Disk Maximum Independent Set problem. Using noisy simulations of Rydberg platforms of up to 26 atoms interacting through realistic van der Waals interactions, we compute the average approximation ratio that can be attained with a simple quantum annealing-based heuristic within a fixed temporal computational budget. Based on estimates of the correlation lengths measured in the engineered quantum state, we extrapolate the results to large atom numbers and compare them to a simple classical approximation heuristic. We find that approximation ratios of at least $\approx 0.84$ are within reach for near-future noise levels. Not taking into account further classical and quantum algorithmic improvements, we estimate that quantum advantage could be reached by attaining a number of controlled atoms of $\sim8,000$ for a time budget of 2 seconds, and $\sim 1,000-1,200$ for a time budget of 0.2 seconds, provided the coherence levels of the system can be improved by a factor 10 while maintaining a constant repetition rate.

A note on the Lasserre hierarchy for different formulations of the maximum independent set problem

In this note, we consider several polynomial optimization formulations of the maximum independent set problem and the use of the Lasserre hierarchy with these different formulations. We demonstrate using computational experiments that the choice of formulation may have a significant impact on the resulting bounds. We also provide theoretical justifications for the observed behavior.

Breaking the linear-memory barrier in [formula omitted]: Fast [formula omitted] on trees with strongly sublinear memory

Recently, studying fundamental graph problems in the Massively Parallel Computation (MPC) framework, inspired by the MapReduce paradigm, has gained a lot of attention. An assumption common to a vast majority of approaches is to allow Ω˜(n) memory per machine, where n is the number of nodes in the graph and Ω˜ hides polylogarithmic factors. However, as pointed out by Karloff et al. [SODA'10] and Czumaj et al. [STOC'18], it might be unrealistic for a single machine to have linear or only slightly sublinear memory.In this paper, we thus study a more practical variant of the MPC model which only requires substantially sublinear or even subpolynomial memory per machine. In contrast to the linear-memory MPC model and also to streaming algorithms, in this low-memory MPC setting, a single machine will only see a small number of nodes in the graph. We introduce a new and strikingly simple technique to cope with this imposed locality.In particular, we show that the Maximal Independent Set (MIS) problem can be solved efficiently, that is, in O(log3⁡log⁡n) rounds, when the input graph is a tree. This constitutes an almost exponential speed-up over the low-memory MPC algorithm in O˜(log⁡n) rounds in a concurrent work by Ghaffari and Uitto [SODA'19] and substantially reduces the local memory from Ω˜(n) required by the recent O(log⁡log⁡n)-round MIS algorithm of Ghaffari et al. [PODC'18] to nε for any ε>0, without incurring a significant loss in the round complexity. Moreover, it demonstrates how to make use of the all-to-all communication in the MPC model to almost exponentially improve on the corresponding bound in the LOCAL and PRAM models by Lenzen and Wattenhofer [PODC'11].

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.

Using synchronized oscillators to compute the maximum independent set

Not all computing problems are created equal. The inherent complexity of processing certain classes of problems using digital computers has inspired the exploration of alternate computing paradigms. Coupled oscillators exhibiting rich spatio-temporal dynamics have been proposed for solving hard optimization problems. However, the physical implementation of such systems has been constrained to small prototypes. Consequently, the computational properties of this paradigm remain inadequately explored. Here, we demonstrate an integrated circuit of thirty oscillators with highly reconfigurable coupling to compute optimal/near-optimal solutions to the archetypally hard Maximum Independent Set problem with over 90% accuracy. This platform uniquely enables us to characterize the dynamical and computational properties of this hardware approach. We show that the Maximum Independent Set is more challenging to compute in sparser graphs than in denser ones. Finally, using simulations we evaluate the scalability of the proposed approach. Our work marks an important step towards enabling application-specific analog computing platforms to solve computationally hard problems.

Maximum independent sets in subcubic graphs: New results

The maximum independent set problem is known to be NP-hard in the class of subcubic graphs, i.e. graphs of vertex degree at most 3. We study complexity of the problem on hereditary subclasses of subcubic graphs. Each such subclass can be described by means of forbidden induced subgraphs. In case of finitely many forbidden induced subgraphs a necessary condition for polynomial-time solvability of the problem in subcubic graphs (unless P=NP) is the exclusion of the graph Si,j,k, which is a tree with three leaves of distance i,j,k from the only vertex of degree 3. Whether this condition is also sufficient is an open question, which was previously answered only for S1,k,k-free subcubic graphs and S2,2,2-free subcubic graphs. Combining various algorithmic techniques, in the present paper we generalize both results and show that the problem can be solved in polynomial time for S2,k,k-free subcubic graphs, for any fixed value of k.

Computing graph gonality is hard

There are several notions of gonality for graphs. The divisorial gonality dgon(G) of a graph G is the smallest degree of a divisor of positive rank in the sense of Baker–Norine. The stable gonality sgon(G) of a graph G is the minimum degree of a finite harmonic morphism from a refinement of G to a tree, as defined by Cornelissen, Kato and Kool. We show that computing dgon(G) and sgon(G) are NP-hard by a reduction from the maximum independent set problem and the vertex cover problem, respectively. Both constructions show that computing gonality is moreover APX-hard.

A $$(1.4 + \epsilon )$$-approximation algorithm for the 2-Max-Duo problem

The maximum duo-preservation string mapping (Max-Duo) problem is the complement of the well studied minimum common string partition problem, both of which have applications in many fields including text compression and bioinformatics. k-Max-Duo is the restricted version of Max-Duo, where every letter of the alphabet occurs at most k times in each of the strings, which is readily reduced into the well known maximum independent set (MIS) problem on a graph of maximum degree $$\Delta \le 6(k-1)$$ . In particular, 2-Max-Duo can then be approximated arbitrarily close to 1.8 using the state-of-the-art approximation algorithm for the MIS problem on bounded-degree graphs. 2-Max-Duo was proved APX-hard and very recently a $$(1.6 + \epsilon )$$ -approximation algorithm was claimed, for any $$\epsilon > 0$$ . In this paper, we present a vertex-degree reduction technique, based on which, we show that 2-Max-Duo can be approximated arbitrarily close to 1.4.

A Graph-Based Approach for the DNA Word Design Problem.

The aim of this paper is to improve the best known solution of an important problem, the DNA Word Design problem, which has its roots in Bioinformatics and Coding Theory. The problem is to design DNA codes that satisfy some combinatorial constraints. The constraints considered are: minimum Hamming distance, fixed GC content and the reverse complement Hamming distance. The problem is modeled as a maximum independent set problem. Existing complex approaches for the maximum independent set problem, suitable for large graphs, were tested. In order to tackle large instances, libraries for external memory computations and sampling techniques were investigated. Eventually, we succeed in finding good lower bounds for the instances that were analyzed.

Distributed Self-Stabilizing Capacitated Maximal Independent Set Construction in Wireless Sensor Networks

Wireless sensor networks (WSNs) are composed of a large number of wireless self-organized sensor nodes connected through a wireless decentralized distributed network without the aid of a predefined infrastructure. Fault-tolerance and power management are fundamental challenges in WSNs. A WSN is self-stabilizing if it can initially start at any state and obtain a legitimate state in a finite time without any external intervention. Self-stabilization is an important method for providing fault-tolerance in WSNs. Maximal independent set (MIS) is an extensively used structure for many important applications such as clustering (Randhawa and Jain in Wirel Personal Commun 97(3):3355, 2017. https://doi.org/10.1007/s11277-017-4674-5 ) and routing (Attea et al. in Wirel Personal Commun 81(2):819, 2015. https://doi.org/10.1007/s11277-014-2159-3 ; Lipinski in Wirel Personal Commun 101(1):251, 2018. https://doi.org/10.1007/s11277-018-5686-5 ) in WSNs. The capacitated MIS (CapMIS) problem is an extension of MIS in that each node has a capacity that determines the number of nodes it may dominate. In this paper, we propose a distributed self-stabilizing capacitated maximal independent set algorithm (CapMIS) in order to reduce energy consumption and support load balancing in WSNs. To the best of our knowledge, this is the first algorithm in this manner. The algorithm is validated through theoretical analysis as well as testbed implementations and simulations.

On the Maximum Independent Set Problem in Graphs of Bounded Maximum Degree

We consider the maximum independent set (MIS) problem, i.e., the problem asking for a vertex subset of maximum cardinality of a graph such that no two vertices in this set are adjacent. The problem is known to be NP-hard in general, even if restricted on graphs of maximum degree at most Δ for a given integer Δ ≥ 3, i.e., every vertex is of degree at most Δ. We try to figure out some bounded maximum degree graph classes, under which the problem can be solved in polynomial time.

Max-independent set and the quantum alternating operator ansatz

The maximum-independent set (MIS) problem of graph theory using the quantum alternating operator ansatz is studied. We perform simulations on the Rigetti Forest simulator for the square ring, [Formula: see text], and [Formula: see text] graphs and analyze the dependence of the algorithm on the depth of the circuit and initial states. The probability distribution of observation of the feasible states representing maximum-independent sets is observed to be asymmetric for the MIS problem, which is unlike the Max-Cut problem where the probability distribution of feasible states is symmetric. For asymmetric graphs, it is shown that the algorithm clearly favors the independent set with the larger number of elements even for finite circuit depth. We also compare the approximation ratios for the algorithm when we choose different initial states for the square ring graph and show that it is dependent on the choice of the initial state.

A multiagent reinforcement learning algorithm to solve the maximum independent set problem

A multiagent reinforcement learning algorithm to solve the maximum independent set problem

Integrated integer programming and decision diagram search tree with an application to the maximum independent set problem

We propose an optimization framework which integrates decision diagrams (DDs) and integer linear programming (ILP) to solve combinatorial optimization problems. The hybrid DD-ILP approach explores the solution space based on a recursive compilation of relaxed DDs and incorporates ILP calls to solve subproblems associated with DD nodes. The selection of DD nodes to be explored by ILP technology is a significant component of the approach. We show how supervised machine learning can be useful to detect, on-the-fly, a subproblem structure for ILP technology. We use the maximum independent set problem as a case study. Computational experiments show that, in presence of suitable problem structure, the integrated DD-ILP approach can exploit complementary strengths and improve upon the performance of both a stand-alone DD solver and an ILP solver in terms of solution time and number of solved instances.

Framework and algorithms for identifying honest blocks in blockchain.

Blockchain technology gains more and more attention in the past decades and has been applied in many areas. The main bottleneck for the development and application of blockchain is its limited scalability. Blockchain with directed acyclic graph structure (BlockDAG) is proposed in order to alleviate the scalability problem. One of the key technical problems in BlockDAG is the identification of honest blocks which are very important for establishing a stable and invulnerable total order of all the blocks. The stability and security of BlockDAG largely depends on the precision of honest block identification. This paper presents a novel universal framework based on graph theory, called MaxCord, for identifying the honest blocks in BlockDAG. By introducing the concept of discord, the honest block identification is modelled as a generalized maximum independent set problem. Several algorithms are developed, including exact, greedy and iterative filtering algorithms. The extensive comparisons between proposed algorithms and the existing method were conducted on the simulated BlockDAG data to show that the proposed iterative filtering algorithm identifies the honest blocks both efficiently and effectively. The proposed MaxCord framework and algorithms can set the solid foundation for the BlockDAG technology.