Maximum Independent Set Problem Research Articles

Distributed (Δ +1)-Coloring in Sublogarithmic Rounds

We give a new randomized distributed algorithm for (Δ +1)-coloring in the LOCAL model, running in O (√ log Δ)+ 2 O (√log log n ) rounds in a graph of maximum degree Δ. This implies that the (Δ +1)-coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds of Ω(min(√/log n log log n , /log Δ log log Δ)) by Kuhn, Moscibroda, and Wattenhofer [PODC’04]. Our algorithm also extends to list-coloring where the palette of each node contains Δ +1 colors. We extend the set of distributed symmetry-breaking techniques by performing a decomposition of graphs into dense and sparse parts.

A linear complementarity based characterization of the weighted independence number and the independent domination number in graphs

The linear complementarity problem is a continuous optimization problem that generalizes convex quadratic programming, characterization of Nash equilibria of bimatrix games and several such problems. This paper presents a continuous optimization formulation for the weighted independence number of a graph by characterizing it as the maximum weighted ℓ1 norm over the solution set of a linear complementarity problem (LCP). The minimum ℓ1 norm of solutions of this LCP forms a lower bound on the independent domination number of the graph. Unlike the case of the maximum ℓ1 norm, this lower bound is in general weak, but we show it to be tight if the graph is a forest. Using methods from the theory of LCPs, we then obtain a stronger variant of the Lovász theta and a new sufficient condition for a graph to be well-covered, i.e., for all maximal independent sets to also be maximum. This latter condition is also shown to be necessary for well-coveredness of forests. Finally, the reduction of the maximum independent set problem to a linear program with (linear) complementarity constraints (LPCC) shows that LPCCs are hard to approximate.

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

As a fundamental subject of theoretical computer science, the maximum independent set (MIS) problem not only is of purely theoretical interest, but also has found wide applications in various fields. However, for a general graph determining the size of a MIS is NP-hard, and exact computation of the number of all MISs is even more difficult. It is thus of significant interest to seek special graphs for which the MIS problem can be exactly solved. In this paper, we address the MIS problem in the pseudofractal scale-free web and the Sierpiński gasket, which have the same number of vertices and edges. For both graphs, we determine exactly the independence number and the number of all possible MISs. The independence number of the pseudofractal scale-free web is as twice as the one of the Sierpiński gasket. Moreover, the pseudofractal scale-free web has a unique MIS, while the number of MISs in the Sierpiński gasket grows exponentially with the number of vertices.

On the analysis of independent sets via multilevel splitting

Counting the number of independent sets is an important problem in graph theory, combinatorics, optimization, and social sciences. However, a polynomial‐time exact calculation, or even a reasonably close approximation, is widely believed to be impossible, since their existence implies an efficient solution to various problems in the non‐deterministic polynomial‐time complexity class. To cope with the approximation challenge, we express the independent set counting problem as a rare‐event estimation problem. We develop a multilevel splitting algorithm which is generally capable of delivering accurate results, while using a manageable computational effort, even when applied to large graphs. We apply the algorithm to both counting and optimization (finding a maximum independent set) problems, and show that it compares favorably with the existing state of the art.

On the Lovász Theta Function for Independent Sets in Sparse Graphs

We consider the maximum independent set problem on sparse graphs with maximum degree d. We show that the Lovasz ϑ-function based semidefinite program (SDP) has an integrality gap of O(d/log3/2 d), improving on the previous best result of O(d/log d). This improvement is based on a new Ramsey-theoretic bound on the independence number of Kr-free graphs for large values of r. We also show that for stronger SDPs, namely, those obtained using polylog(d) levels of the SA+ semidefinite hierarchy, the integrality gap reduces to O(d/log2 d). This matches the best unique-games-based hardness result up to lower-order poly(log log d) factors. Finally, we give an algorithmic version of this SA+-based integrality gap result, albeit using d levels of SA+, via a coloring algorithm of Johansson.

Infinite and Giant Components in the Layers Percolation Model

In this work we continue the investigation launched in [FHR16] of the structural properties of the structural properties of the Layers model, a dependent percolation model. Given an undirected graph $G=(V,E)$ and an integer $k$, let $T_k(G)$ denote the random vertex-induced subgraph of $G$, generated by ordering $V$ according to Uniform$[0,1]$ $\mathrm{i.i.d.}$ clocks and including in $T_k(G)$ those vertices with at most $k-1$ of their neighbors having a faster clock. The distribution of subgraphs sampled in this manner is called the layers model with parameter $k$. The layers model has found applications in the study of $\ell$-degenerate subgraphs, the design of algorithms for the maximum independent set problem and in the study of bootstrap percolation. We prove that every infinite locally finite tree $T$ with no leaves, satisfying that the degree of the vertices grow sub-exponentially in their distance from the root, $T_3(T)$ $\mathrm{a.s.}$ has an infinite connected component. In contrast, we show that for any locally finite graph $G$, $\mathrm{a.s.}$ every connected component of $T_2(G)$ is finite. We also consider random graphs with a given degree sequence and show that if the minimal degree is at least 3 and the maximal degree is bounded, then $\mathrm{w.h.p.}$ $T_3$ has a giant component. Finally, we also consider ${\mathbb{Z}}^{d}$ and show that if $d$ is sufficiently large, then $\mathrm{a.s.}$ $T_4(\mathbb{Z}^d)$ contains an infinite cluster.

On the Possibility of Network Scheduling With Polynomial Complexity and Delay

Considering the collection of all networks with independent set interference model, Shah, Tse, and Tsitsiklis showed that there exist scheduling algorithms with polynomial complexity and delay, only if the maximum independent set problem can be solved in polynomial time (equivalently, P=NP). In this paper, we extend this result to arbitrary collections of networks and present a clear-cut criterion for the existence of polynomial complexity and delay scheduling algorithms relative to a given collection of networks with arbitrary interference models, not confined to independent set interference or SINR models, and not necessarily encompassing all network topologies. This amounts to the equivalence of polynomial scheduling and effective approximation of maximum weighted actions .

The complexity of the Clar number problem and an exact algorithm

The Clar number of a (hydro)carbon molecule, introduced by Clar (The aromatic sextet, 1972), is the maximum number of mutually disjoint resonant hexagons in the molecule. Calculating the Clar number can be formulated as an optimization problem on 2-connected plane graphs. Namely, it is the maximum number of mutually disjoint even faces a perfect matching can simultaneously alternate on. It was proved by Abeledo and Atkinson (Linear Algebra Appl 420(2):441–448, 2007) that the Clar number can be computed in polynomial time if the plane graph has even faces only. We prove that calculating the Clar number in general 2-connected plane graphs is \(\mathsf {NP}\)-hard. We also prove \(\mathsf {NP}\)-hardness of the maximum independent set problem for 2-connected plane graphs with odd faces only, which may be of independent interest. Finally, we give an exact algorithm that determines the Clar number of a given 2-connected plane graph. The algorithm has a polynomial running time if the length of the shortest odd join in the planar dual graph is fixed, which gives an efficient algorithm for some fullerene classes, such as carbon nanotubes.

Faster approximation for maximum independent set on unit disk graph

Maximum independent set from a given set D of unit disks intersecting a horizontal line can be solved in O(n2) time and O(n2) space. As a corollary, we design a factor 2 approximation algorithm for the maximum independent set problem on unit disk graph which takes both time and space of O(n2). The best known factor 2 approximation algorithm for this problem runs in O(n2log⁡n) time and takes O(n2) space [1,2].

Exact algorithms for maximum independent set

We show that the maximum independent set problem on an n-vertex graph can be solved in 1.1996nnO(1) time and polynomial space, which even is faster than Robson's 1.2109nnO(1)-time exponential-space algorithm published in 1986. We also obtain improved algorithms for MIS in graphs with maximum degree 6 and 7, which run in time of 1.1893nnO(1) and 1.1970nnO(1), respectively. Our algorithms are obtained by using fast algorithms for MIS in low-degree graphs in a hierarchical way and making a careful analysis on the structure of bounded-degree graphs.

Finding near-optimal independent sets at scale

The maximum independent set problem is NP-hard and particularly difficult to solve in sparse graphs, which typically take exponential time to solve exactly using the best-known exact algorithms. In this paper, we present two new novel heuristic algorithms for computing large independent sets on huge sparse graphs, which are intractable in practice. First, we develop an advanced evolutionary algorithm that uses fast graph partitioning with local search algorithms to implement efficient combine operations that exchange whole blocks of given independent sets. Though the evolutionary algorithm itself is highly competitive with existing heuristic algorithms on large social networks, we further show that it can be effectively used as an oracle to guess vertices that are likely to be in large independent sets. We then show how to combine these guesses with kernelization techniques in a branch-and-reduce-like algorithm to compute high-quality independent sets quickly in huge complex networks. Our experiments against a recent (and fast) exact algorithm for large sparse graphs show that our technique always computes an optimal solution when the exact solution is known, and it further computes consistent results on even larger instances where the solution is unknown. Ultimately, we show that identifying and removing vertices likely to be in large independent sets opens up the reduction space--which not only speeds up the computation of large independent sets drastically, but also enables us to compute high-quality independent sets on much larger instances than previously reported in the literature.

An Exact Algorithm Exhibiting RS–RSB/Easy–Hard Correspondence for the Maximum Independent Set Problem

A recently proposed exact algorithm for the maximum independent set problem is analyzed. The typical running time is improved exponentially in some parameter regions compared to simple binary search. The algorithm also overcomes the core transition point, where the conventional leaf removal algorithm fails, and works up to the replica symmetry breaking (RSB) transition point. This suggests that a leaf removal core itself is not enough for typical hardness in the random maximum independent set problem, providing further evidence for RSB being the obstacle for algorithms in general.

Carousel greedy: A generalized greedy algorithm with applications in optimization

In this paper, we introduce carousel greedy, an enhanced greedy algorithm which seeks to overcome the traditional weaknesses of greedy approaches. We have applied carousel greedy to a variety of well-known problems in combinatorial optimization such as the minimum label spanning tree problem, the minimum vertex cover problem, the maximum independent set problem, and the minimum weight vertex cover problem. In all cases, the results are very promising. Since carousel greedy is very fast, it can be used to solve very large problems. In addition, it can be combined with other approaches to create a powerful, new metaheuristic. Our goal in this paper is to motivate and explain the new approach and present extensive computational results.

A hybrid iterated local search heuristic for the maximum weight independent set problem

This paper presents a hybrid iterated local search (ILS) algorithm for the maximum weight independent set (MWIS) problem, a generalization of the classical maximum independent set problem. Two efficient neighborhood structures are proposed and they are explored using the variable neighborhood descent procedure. Moreover, we devise a perturbation mechanism that dynamically adjusts the balance between intensification and diversification during the search. The proposed algorithm was tested on two well-known benchmarks (DIMACS-W and BHOSLIB-W) and the results obtained were compared with those found by state-of-the-art heuristics and exact methods. Our heuristic outperforms the best-known heuristic for the MWIS as well as the best heuristics for the maximum weight clique problem. The results also show that the hybrid ILS was capable of finding all known optimal solutions in milliseconds.

A refined algorithm for maximum independent set in degree-4 graphs

The maximum independent set problem is one of the most important problems in theoretical analysis on time and space complexities of exact algorithms. Theoretical improvement on upper bounds on time...

On a family of 0/1-polytopes with an NP-complete criterion for vertex nonadjacency relation

В 1995 году Т. Мацуи рассмотрел специальное семейство 0/1-многогранников с NP-полным критерием несмежности вершин. В 2012 году автор настоящей работы установил, что все многогранники этого семейства присутствуют в качестве граней в многогранниках, ассоциированных со следующими NP-полными задачами: задачей коммивояжера, задачей 3-выполнимость, задачей о рюкзаке, задачей о покрытии множества, задачей о частичном упорядочивании, задачей о кубическом подграфе и некоторыми другими. В работе показано, что ни один из многогранников упомянутого выше специального семейства, за исключением одномерного отрезка, не может быть гранью многогранников, ассоциированных с задачами о независимом множестве в графе, об упаковке и разбиении множества и с трехиндексной задачей о назначениях.

On sequential heurestic methods for the maximum independent set problem

On sequential heurestic methods for the maximum independent set problem

Polynomial and Extensible Solutions in Lock-Chart Solving

ABSTRACTLock-chart (also known as master-key system) solving is a process for designing mechanical keys and locks so that every key can open and be blocked in a user-defined set of locks. We present several algorithms as a result of our 3 years industrial collaboration project, chosen by their asymptotic time complexity and extensibility, which is the ability to add new keys and locks and satisfy the previously unforeseen blocking requirements. Our strongest result is a linear-time algorithm to find the largest diagonal lock-chart with a condition on mechanical properties of keys. Without these restrictions, but given a solution for master keys, the problem is translated into a maximum independent set problem and a greedy approximation is proposed. We evaluate the algorithms on a generated dataset using a non-backtracking procedure to simulate minimal extensions. Both approaches are suggested as heuristics or subroutines for a backtracking constraint satisfaction problem-based (CSP) solver.

Independent sets in some classes of $$S_{i, j, k}$$ S i , j , k -free graphs

The maximum weight independent set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. In 1982, Alekseev (Comb Algebraic Methods Appl Math 132:3---13, 1982) showed that the M(W)IS problem remains NP-complete on H-free graphs, whenever H is connected, but neither a path nor a subdivision of the claw. We will focus on graphs without a subdivision of a claw. For integers $$i, j, k \ge 1$$i,j,kź1, let $$S_{i, j, k}$$Si,j,k denote a tree with exactly three vertices of degree one, being at distance i, j and k from the unique vertex of degree three. Note that $$S_{i,j, k}$$Si,j,k is a subdivision of a claw. The computational complexity of the MWIS problem for the class of $$S_{1, 2, 2}$$S1,2,2-free graphs, and for the class of $$S_{1, 1, 3}$$S1,1,3-free graphs are open. In this paper, we show that the MWIS problem can be solved in polynomial time for ($$S_{1, 2, 2}, S_{1, 1, 3}$$S1,2,2,S1,1,3, co-chair)-free graphs, by analyzing the structure of the subclasses of this class of graphs. This also extends some known results in the literature.

A subexponential-time algorithm for the Maximum Independent Set Problem in [formula omitted]-free graphs

The Maximum Independent Set Problem is known to be NP-hard in general. In the last decades lots of effort were spent to find polynomial-time algorithms for Pt-free graphs. A recent result presents such an algorithm for P5-free graphs. On the other hand, the complexity status for larger t is still unknown. In this paper we present a subexponential algorithm for the Maximum (Weighted) Independent Set Problem in Pt-free graphs.