Set Problem In Graphs Research Articles

Max Weight Independent Set in Graphs with No Long Claws: An Analog of the Gyárfás’ Path Argument

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw S t,t,t as an induced subgraph and provide a subexponential-time algorithm with improved running time \(2^{\mathcal {O}(\sqrt {nt}\log n)}\) and a quasipolynomial-time approximation scheme with improved running time \(2^{\mathcal {O}(\varepsilon ^{-1}t \log ^{5} n)}\) . The Gyárfás’ path argument, a powerful tool that is the main building block for many algorithms in P t -free graphs, ensures that given an n -vertex P t -free graph, in polynomial time we can find a set P of at most t -1 vertices such that every connected component of G-N[P] has at most n /2 vertices. Our main technical contribution is an analog of this result for S t,t,t -free graphs: given an n -vertex S t,t,t -free graph, in polynomial time we can find a set P of \(\mathcal {O}(t \log n)\) vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of G-N[P] such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most n /2 vertices.

Covering Problems and Core Percolations on Hypergraphs.

We introduce two generalizations of core percolation in graphs to hypergraphs, related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs, respectively. We offer analytical solutions of these two core percolations for uncorrelated random hypergraphs whose vertex degree and hyperedge cardinality distributions are arbitrary but have nondiverging moments. We find that for several real-world hypergraphs their two cores tend to be much smaller than those of their null models, suggesting that covering problems in those real-world hypergraphs can actually be solved in polynomial time.

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.

On the Maximum Weight Independent Set Problem in Graphs without Induced Cycles of Length at Least Five

A hole in a graph is an induced cycle of length at least 4, and an antihole is the complement of an induced cycle of length at least 4. A hole or antihole is long if its length is at least 5. For a...

New Cases of the Polynomial Solvability of the Independent Set Problem for Graphs with Forbidden Paths

The independent set problem is solvable in polynomial time for the graphs not containing the path P k for any fixed k. If the induced path P k is forbidden then the complexity of this problem is unknown for k > 6. We consider the intermediate cases that the induced path P k and some of its spanning supergraphs are forbidden. We prove the solvability of the independent set problem in polynomial time for the following cases: (1) the supergraphs whose minimal degree is less than k/2 are forbidden; (2) the supergraphs whose complementary graph has more than k/2 edges are forbidden; (3) the supergraphs from which we can obtain P k by means of graph intersection are forbidden.

Polyhedral results on the stable set problem in graphs containing even or odd pairs

Even and odd pairs are important tools in the study of perfect graphs and were instrumental in the proof of the Strong Perfect Graph Theorem. We suggest that such pairs impose a lot of structure also in arbitrary, not just perfect graphs. To this end, we show that the presence of even or odd pairs in graphs imply a special structure of the stable set polytope. In fact, we give a polyhedral characterization of even and odd pairs.

Streaming Algorithms for Independent Sets in Sparse Hypergraphs

We give the first treatment of the classic independent set problem in graphs and hypergraphs in the streaming setting. The objective is to find space-efficient algorithms that output independent sets that are "combinatorially optimal", that is, with size guarantee in terms of the degree sequence alone. Our main result is a randomized algorithm that achieves this using space in bits that is linear in the number of vertices. We use this to examine assumptions about the streaming model, and advocate the study of output-efficient algorithms that measure space usage relative to the size of the output solution. In that sense, our main algorithm uses space linear in the output size. We also examine algorithms that use little or no space in addition to the bits storing the output. Our algorithms fall also into an online streaming model, where output-changes can go only in one direction. In particular a feasible solution must be maintained at all times, and items that are removed from the solution can never reenter. We obtain tight bounds on deterministic algorithms for independent sets in graphs in that model.

On the complexity of the minimum outer-connected dominating set problem in graphs

For a graph $$G=(V,E)$$G=(V,E), a dominating set is a set $$D\subseteq V$$D⊆V such that every vertex $$v\in V\setminus D$$v?V\D has a neighbor in $$D$$D. The minimum outer-connected dominating set (Min-Outer-Connected-Dom-Set) problem for a graph $$G$$G is to find a dominating set $$D$$D of $$G$$G such that $$G[V\setminus D]$$G[V\D], the induced subgraph by $$G$$G on $$V\setminus D$$V\D, is connected and the cardinality of $$D$$D is minimized. In this paper, we consider the complexity of the Min-Outer-Connected-Dom-Set problem. In particular, we show that the decision version of the Min-Outer-Connected-Dom-Set problem is NP-complete for split graphs, a well known subclass of chordal graphs. We also consider the approximability of the Min-Outer-Connected-Dom-Set problem. We show that the Min-Outer-Connected-Dom-Set problem cannot be approximated within a factor of $$(1-\varepsilon ) \ln |V|$$(1-?)ln|V| for any $$\varepsilon >0$$?>0, unless NP $$\subseteq $$⊆ DTIME($$|V|^{\log \log |V|}$$|V|loglog|V|). For sufficiently large values of $$\varDelta $$Δ, we show that for graphs with maximum degree $$\varDelta $$Δ, the Min-Outer-Connected-Dom-Set problem cannot be approximated within a factor of $$\ln \varDelta -C \ln \ln \varDelta $$lnΔ-ClnlnΔ for some constant $$C$$C, unless P $$=$$= NP. On the positive side, we present a $$\ln (\varDelta +1)+1$$ln(Δ+1)+1-factor approximation algorithm for the Min-Outer-Connected-Dom-Set problem for general graphs. We show that the Min-Outer-Connected-Dom-Set problem is APX-complete for graphs of maximum degree 4.

Solving the Minimum Independent Domination Set Problem in Graphs by Exact Algorithm and Greedy Heuristic

In this paper we present a new approach to solve the Minimum Independent Dominating Set problem in general graphs which is one of the hardest optimization problem. We propose a method using a clique partition of the graph, partition that can be obtained greedily. We provide conditions under which our method has a better complexity than the complexity of the previously known algorithms. Based on our theoretical method, we design in the second part of this paper an efficient algorithm by including cuts in the search process. We then experiment it and show that it is able to solve almost all instances up to 50 vertices in reasonable time and some instances up to several hundreds of vertices. To go further and to treat larger graphs, we analyze a greedy heuristic. We show that it often gives good (sometimes optimal) results in large instances up to 60 000 vertices in less than 20 s. That sort of heuristic is a good approach to get an initial solution for our exact method. We also describe and analyze some of its worst cases.

Exact Algorithms for Annotated Edge Dominating Set in Graphs with Degree Bounded by 3

Given a graph G = (V, E) together with a nonnegative integer requirement on vertices r : V → Z+, the annotated edge dominating set problem is to find a minimum set M ⊆ E such that, each edge in E − M is adjacent to some edge in M, and M contains at least r(v) edges incident on each vertex v ∈ V . The annotated edge dominating set problem is a natural extension of the classical edge dominating set problem, in which the requirement on vertices is zero. The edge dominating set problem is an important graph problem and has been extensively studied. It is well known that the problem is NP-hard, even when the graph is restricted to a planar or bipartite graph with maximum degree 3. In this paper, we show that the annotated edge dominating set problem in graphs with maximum degree 3 can be solved in O∗(1.2721n) time and polynomial space, where n is the number of vertices in the graph. We also show that there is an O∗(2.2306k)-time polynomial-space algorithm to decide whether a graph with maximum degree 3 has an annotated edge dominating set of size k or not. key words: edge dominating sets, exact algorithms, cubic graphs

Graph Neural Network for Minimum Dominating Set

The dominating set concept in graphs has been used in many applications. In large graphs finding the minimum dominating set is difficult. The minimum dominating set problem in graphs seek a set D of minimum number of vertices such that each vertex of the graph is either in D or adjacent to a vertex in D. In a graph on n nodes if there is a single node of degree n-1 then that single node forms a minimum dominating set. In the proposed work, a designed network called Graph Neural Network (GNN) is used to identify the node of degree N-1 in a graph having a single node of degree N-1 as it forms the minimum dominating set in the graph.. The network is simulated for graphs with nodes varying from 5 to 15. The state dimension of the input vectors are analyzed for better convergence. It has been found that the minimum dominating set was correctly identified from 80% to 90% of the graphs when the state dimension was 2 and 3. It has been observed that when the state dimension was 2, the convergence was fast as it requires minimum hidden neurons than other state dimensions. It has also been observed that when the state dimension was greater than 3, convergence requires hours of time and more number of hidden neurons. GNN was able to identify the minimum dominating set in a graph on n vertices which has a single node of degree n-1.

Variable objective search

This paper introduces the variable objective search framework for combinatorial optimization. The method utilizes different objective functions used in alternative mathematical programming formulations of the same combinatorial optimization problem in an attempt to improve the solutions obtained using each of these formulations individually. The proposed technique is illustrated using alternative quadratic unconstrained binary formulations of the classical maximum independent set problem in graphs.

On characterization of maximal independent sets via quadratic optimization

This article investigates the local maxima properties of a box-constrained quadratic optimization formulation of the maximum independent set problem in graphs. Theoretical results characterizing bi...

The complexity of dissociation set problems in graphs

A subset of vertices in a graph is called a dissociation set if it induces a subgraph with a vertex degree of at most 1. The maximum dissociation set problem, i.e., the problem of finding a dissociation set of maximum size in a given graph is known to be NP-hard for bipartite graphs. We show that the maximum dissociation set problem is NP-hard for planar line graphs of planar bipartite graphs. In addition, we describe several polynomially solvable cases for the problem under consideration. One of them deals with the subclass of the so-called chair-free graphs. Furthermore, the related problem of finding a maximal (by inclusion) dissociation set of minimum size in a given graph is studied, and NP-hardness results for this problem, namely for weakly chordal and bipartite graphs, are derived. Finally, we provide inapproximability results for the dissociation set problems mentioned above.

Solving the minimum M-dominating set problem by a continuous optimization approach based on DC programming and DCA

We propose a new optimization approach based on DC (Difference of Convex functions) programming and DCA (DC Algorithm) to the so-called Minimum M-Dominating Set problem in graphs. This problem is beforehand re-casted as a polyhedral DC program with the help of exact penalty in DC programming. The related DCA is original and computer efficient because it consists of solving a few linear programs and converges after a finite number of iterations to an integer solution while working in a continuous domain. Numerical simulations show the efficiency and robustness of DCA and its superiority with respect to standard methods.

On the Complexity of Dissociation Set Problems in Graphs

A subset of vertices in a graph is called a dissociation set if it induces a subgraph with vertex degree at most 1. A dissociation set D is maximal if no other dissociation set contains D. The complexity of finding a dissociation set of maximum size in line graphs and finding a maximal dissociation set of minimum size in general graphs is considered.

Gowers Uniformity, Influence of Variables, and PCPs

We study the relation of query complexity and soundness in probabilistically checkable proofs (PCPs). We present a PCP verifier for languages that are Unique-Games-Hard and such that the verifier makes q queries, has almost perfect completeness, and has soundness error at most $2q/2^q+\varepsilon$ for arbitrarily small $\varepsilon>0$. For values of q of the form $2^t-1$, the soundness error is $(q+1)/2^q+\varepsilon$. Charikar, Makarychev, and Makarychev show that there is a constant $\beta$ such that every language that has a verifier of query complexity q and a ratio of soundness error to completeness smaller than $\beta q/2^q$ is decidable in polynomial time. Up to the value of the multiplicative constant and to the validity of the Unique Games Conjecture, our result is therefore tight. As a corollary, we show that approximating the Maximum Independent Set problem in graphs of degree $\Delta$ within a factor better than $\Delta/(\log\Delta)^\alpha$ is Unique-Games-Hard for a certain constant $\alpha>0$. Our main technical results are (i) a connection between the Gowers uniformity of a boolean function and the influence of its variables and (ii) the proof that “Gowers uniform” functions pass the “hypergraph linearity test” approximately with the same probability of a random function. The connection between Gowers uniformity and influence might have other applications.

Approximating minimum independent dominating sets in wireless networks

We present the first polynomial-time approximation scheme (PTAS) for the Minimum Independent Dominating Set problem in graphs of polynomially bounded growth which are used to model wireless communication networks. The approach presented yields a robust algorithm, that is, it accepts any undirected graph as input, and returns a ( 1 + ε ) -approximate minimum independent dominating set, or a certificate showing that the input graph does not satisfy the bounded growth property.

Approximation Algorithms for Connected Dominating Sets

The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(Δ)+2 and H(Δ)+2 are presented, where Δ is the maximum degree and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited or has at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (cn+1) \ln n where cn ln k is the approximation factor for the node weighted Steiner tree problem (currently cn = 1.6103 ). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c+1) H(Δ) +c-1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644 ).

Feedback vertex sets in mesh-based networks

In this paper, we consider the minimum feedback vertex set problem in graphs, i.e., the problem of finding a minimal cardinality subset of the vertices, whose removal makes a graph acyclic. The problem is N P -hard for general topologies, but optimal and near-optimal solutions have been provided for particular networks. In this paper, the problem is considered for undirected graphs with the following topologies: two- and higher-dimensional meshes of trees, trees of meshes, and pyramid networks. For the two-dimensional meshes of trees the results are optimal; for the higher-dimensional meshes of trees and the tree of meshes the results are asymptotically optimal. For the pyramid networks, there remains a small factor between the upper and the lower bounds.