Exact algorithm for solving the problem of maximum independent set

An 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 an independent set with the largest possible number of nodes. Since the MIS problem is NP-hard, finding optimal solutions for large graphs is quite difficult and approximation or heuristic algorithms are used to find independent sets as large as possible. The main contribution of this paper is to present an annealing-based parallel heuristic algorithm for finding approximate solutions for the MIS problem. We have implemented it to run on a computing server with multiple GPUs. The performance of our GPU implementation is compared with the other approaches including a GPU QUBO solver, Gurobi optimizer running on an Intel multicore server, and D-wave 2000Q quantum annealer. The experimental results for random graphs show that our GPU implementation for the MIS problem can find optimal or better solutions than the other approaches. In particular, for small random graphs with 128 nodes or less, our GPU implementation can find optimal solutions of the MIS problem in less than 20µs, while the other approaches take more than 10ms. Further, for large regular random graphs with 32M nodes, our GPU implementation can obtain 22.6%-38.5% larger independent sets. Thus, our parallel GPU implementation for solving the MIS problem is a potent approach to solve the MIS problem.

The Eigenvalue Method for Extremal Problems on Infinite Vertex-Transitive Graphs

Line of Sight (LoS) networks were designed to model wireless networks in settings which may contain obstacles restricting visibility of sensors. A graph \(G=(V,E)\) is a 2-dimensional LoS network if it can be embedded in an \(n \times k\) rectangular point set such that a pair of vertices in V are adjacent if and only if the embedded vertices are placed on the same row or column and are at a distance less than \(\omega \). We study the Maximum Independent Set (MIS) problem in restricted LoS networks where k is a constant. It has been shown in the unrestricted case when \(n=k\) and \(n\rightarrow \infty \) that the MIS problem is NP-hard when \( \omega > 2\) is fixed or when \(\omega =O(n^{1-\epsilon })\) grows as a function of n for fixed \(0<\epsilon <1\). In this paper we develop a dynamic programming (DP) algorithm which shows that in the restricted case the MIS problem is solvable in polynomial time for all \(\omega \). We then generalise the DP algorithm to solve three additional problems which involve two versions of the Maximum Weighted Independent Set (MWIS) problem and a scheduling problem which exhibits LoS properties in one dimension. We use the initial DP algorithm to develop an efficient polynomial time approximation scheme (EPTAS) for the MIS problem in restricted LoS networks. This has important applications, as it provides a semi-online solution to a particular instance of the scheduling problem. Finally we extend the EPTAS result to the MWIS problem.

The Maximum Independent Set (MIS) problem is a well-known problem where the aim is to find the maximum cardinality independent set in an associated graph. Map labeling problems can often be modeled as a MIS problem in a conflict graph, where labels are selected to be placed near graphical features not allowing overlaps (conflicts) between labels or between labels and features. However, the MIS problem is NP-hard and exact techniques present difficulties for solving some instances. Thus, this paper presents a Lagrangean decomposition to solve a map labeling problem, the Point-Feature Label Placement problem. We treated the problem by a conflict graph that is partitioned into small sub-problems and copies of some decision variables are done. These copies are used in the sub-problems constraints and in some constraints to ensure the equality between the original variables and their copies. After these steps, we relax these copy constraints in a Lagrangean way. Using instances proposed in the literature, our approach was able to prove the optimality for all of them, except one, and the results were better than the ones provided by a commercial solver.

Finding maximum independent set based on multi-stage simulated quantum adiabatic evolution

The learning with errors (LWE) problem is a problem derived from machine learning that is believed to be intractable for quantum computers. This paper proposes a method that can reduce an LWE problem to a set of maximum independent set (MIS) problems, which are graph problems that are suitable for a quantum annealing machine to solve. The reduction algorithm can reduce an n-dimensional LWE problem to several small MIS problems with at most 2^{O(sqrt{n})} nodes when the lattice-reduction algorithm used in the LWE-reduction method successfully finds short vectors. The algorithm is useful for solving LWE problems in a quantum-classical hybrid manner by using an existing quantum algorithm to solve the MIS problems. For example, the smallest LWE challenge problem is reduced to MIS problems with about 40,000 vertices. This result means that the smallest LWE challenge problem will be within the scope of a real quantum computer in the near future.

The notion of augmenting graphs generalizes Berge’s idea of augmenting chains, which was used by Edmonds in his celebrated so-called Blossom Algorithm for the maximum matching problem (Edmonds, Can J Math 17:449–467, 1965). This method then was developed for more general maximum independent set (MIS) problem, first for claw-free graphs (Minty, J Comb Theory Ser B 28(3):284–304, 1980; Sbihi, Discret Math 29(1):53–76, 1980). Then the method was used extensively for various special cases, for example, S1,1,2 (or fork)-free graphs (Alekseev, Discret Anal Oper Res Ser 1:3–19, 1999), subclasses of P5 free graphs (Boliac and Lozin, Discret Appl Math 131(3):567–575, 2003; Gerber et al., Discret Appl Math 132(1–3):109–119, 2004; Mosca, Discret Appl Math 132(1–3):175–183, 2004; Lozin and Mosca, Inf Process Lett 109(6):319–324, 2009), P6-free graphs (Mosca, Discuss Math Graph Theory 32(3):387–401, 2012), S1,2,l-free graphs (Hertz et al., Inf Process Lett 86(3):311–316, 2003; Hertz and Lozin, The maximum independent set problem and augmenting graphs. In: Graph theory and combinatorial optimization. Springer Science and Business Media, New York, pp. 69–99, 2005), S1,2,5-free graphs (Lozin and Milanic, Discret Appl Math 156(13):2517–2529, 2008), and for S1,1,3-free graphs (Dabrowski et al., Graphs Comb 32(4):1339–1352, 2016). In this paper, we will extend the method for some more general graph classes. Our objective is combining these approaches to apply this technique to develop polynomial time algorithms for the MIS problem in some special subclasses of S2,2,5-free graphs, extending in this way different known results. Moreover, we also consider the augmenting technique for some other combinatorial graph-theoretical problems, for example maximum induced matching, maximum multi-partite induced subgraphs, maximum dissociative set, etc.

Recently, Balliu, Brandt, and Olivetti [FOCS '20] showed the first $ω(\log^* n)$ lower bound for the maximal independent set (MIS) problem in trees. In this work we prove lower bounds for a much more relaxed family of distributed symmetry breaking problems. As a by-product, we obtain improved lower bounds for the distributed MIS problem in trees. For a parameter $k$ and an orientation of the edges of a graph $G$, we say that a subset $S$ of the nodes of $G$ is a $k$-outdegree dominating set if $S$ is a dominating set of $G$ and if in the induced subgraph $G[S]$, every node in $S$ has outdegree at most $k$. Note that for $k=0$, this definition coincides with the definition of an MIS. For a given $k$, we consider the problem of computing a $k$-outdegree dominating set. We show that, even in regular trees of degree at most $Δ$, in the standard \LOCAL model, there exists a constant $ε&gt;0$ such that for $k\leq Δ^ε$, for the problem of computing a $k$-outdegree dominating set, any randomized algorithm requires at least $Ω(\min\{\logΔ,\sqrt{\log\log n}\})$ rounds and any deterministic algorithm requires at least $Ω(\min\{\logΔ,\sqrt{\log n}\})$ rounds. The proof of our lower bounds is based on the recently highly successful round elimination technique. We provide a novel way to do simplifications for round elimination, which we expect to be of independent interest. Our new proof is considerably simpler than the lower bound proof in [FOCS '20]. In particular, our round elimination proof uses a family of problems that can be described by only a constant number of labels. The existence of such a proof for the MIS problem was believed impossible by the authors of [FOCS '20].

Independence number and the number of maximum independent sets in pseudofractal scale-free web and Sierpiński gasket

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.

Efficient decomposition algorithms for the weighted maximum independent set, minimum coloring, and minimum clique cover problems on planar perfect graphs are presented. These planar graphs can also be characterized by the absence of induced odd cycles of length greater than 3 (odd holes). The algorithm in this paper is based on decomposing these graphs into essentially two special classes of inseparable component graphs whose optimization problems are easy to solve, finding the solutions for these components and combining them to form a solution for the original graph. These two classes are (i) planar comparability graphs and (ii) planar line graphs of those planar bipartite graphs whose maximum degrees are no greater than three. The same techniques can be applied to other classes of perfect graphs, provided that efficient algorithms are available for their inseparable component graphs.

The maximum duo-preservation string mapping (Max-Duo) problem is the complement of the well studied minimum common string partition (MCSP) 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. 2-Max-Duo was proved APX-hard and very recently a (1.6 + \epsilon)-approximation 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.

We introduce a structural parameter of a graph, the longest directed path length (LDPL), and express the boundary of the difficulty of the problems along this parameter. The parallel complexity of the lexicographically first maximal independent set (LFMIS) problem gradually increases as the value of LDPL grows; the LFMIS problem on a graph of LDPL O( log kn) can be solved in O( log kn) time in parallel; and the problem is P-complete on a graph of LDPL Θ(n∊). Computing the LDPL itself is in NC2. This is important in the sense that a "measure" is valid only if measuring the complexity of a problem is easier than solving it. On the other hand, we also show the limits of the measure. We reduce the LFMIS problem to a kind of the lexicographically first maximal subgraph (LFMS) problems on a graph of LDPL 1. This implies that, even on a graph of LDPL 1, the LFMS problem is P-complete. Finally we discuss the probability that a random graph has LDPL l and show that a random graph of which each edge exists with probability p has LDPL Θ(np) with high probability. This implies the limit of the LDPL on the average case, because the LFMIS problem on a random graph can be efficiently solved in parallel.

Parallelism in the theoretical computation mainly depends on the particular paradigm or computational environment considered, and its importance has been confirmed with the emergence of each novel computing technique. Programmable tile assembly is a novel computing technique to tackle computationally difficult problems, in which computing time grows exponentially corresponding to problematic size. The Maximum independent set (MIS) problem is a typical nondeterministic polynomial problem, which is often associated with strategy applications. In this paper, a novel approach dealing with the MIS problem is proposed based on the abstract tile assembly model, which is believed to be better than the conventional silicon-based computing in solving the same problem. The method can get the solutions of the MIS problem in θ( m + n) time complexity based on θ( mn) distinct tile types.

In this paper, ant colony optimization heuristic is extended for solving maximum independent set (MIS) problems. MIS problems are quite different from the travelling salesman problems (TSP) etc., in which no concept of "path or order" exists in its solutions. Based on such characteristics, the ant colony optimization heuristic is modified in this paper in the following ways: (i) a new computation method for heuristic information is adapted; (ii) the pheromone update rule is augmented; (iii) a complement solution construction process is designed. The simulation shows that the proposed ant colony optimization heuristic is effective and efficient for MIS problems.