Independent set and matching permutations

Local algorithms on graphs are algorithms that run in parallel on the nodes of a graph to compute some global structural feature of the graph. Such algorithms use only local information available at nodes to determine local aspects of the global structure, while also potentially using some randomness. Research over the years has shown that such algorithms can be surprisingly powerful in terms of computing structures like large independent sets in graphs locally. These algorithms have also been implicitly considered in the work on graph limits, where a conjecture due to Hatami, Lovász and Szegedy [17] implied that local algorithms may be able to compute near-maximum independent sets in (sparse) random d-regular graphs. In this paper we refute this conjecture and show that every independent set produced by local algorithms is smaller that the largest one by a multiplicative factor of at least 1/2+1/(2√2) ≈ .853, asymptotically as d → ∞.

The maximum independent set problem is a classic and fundamental combinatorial challenge, where the objective is to find the largest subset of vertices in a graph such that no two vertices are adjacent. In this paper, we introduce a novel linear prioritized local algorithm tailored to address this problem on random d-regular graphs with a small and fixed degree d. Through exhaustive numerical simulations, we empirically investigated the independence ratio, i.e., the ratio between the cardinality of the independent set found and the order of the graph, which was achieved by our algorithm across random d-regular graphs with degree d ranging from 5 to 100. Remarkably, for every d within this range, our results surpassed the existing lower bounds determined by theoretical methods. Consequently, our findings suggest new conjectured lower bounds for the MIS problem on such graph structures. This finding has been obtained using a prioritized local algorithm. This algorithm is termed ‘prioritized’ because it strategically assigns priority in vertex selection, thereby iteratively adding them to the independent set.

Graphs with large minimum degree containing no copy of a clique on $r$ vertices ($K_r$) must contain relatively large independent sets. A classical result of Andrásfai, Erdős, and Sós implies that $K_r$-free graphs $G$ with degree larger than $((3r-7)/(3r-4))|V(G)|$ must be $(r-1)$-partite. An obvious consequence of this result is that the same degree threshold implies an independent set of order $(1/(r-1))|V(G)|$. The following paper provides improved bounds on the minimum degree which would imply the same conclusion. This problem was first considered by Brandt, and we provide improvements over these initial results for $r > 5$.

Three remarks on W2 graphs

A well-known conjecture of András Gyárfás and David Sumner states that for every positive integer m and every finite tree T there exists k such that all graphs that do not contain the clique Km or an induced copy of T have chromatic number at most k. The conjecture has been proved in many special cases, but the general case has been open for several decades. The main purpose of this paper is to consider a natural analogue of the conjecture for matroids, where it turns out, interestingly, to be false. Matroids are structures that result from abstracting the notion of independent sets in vector spaces: that is, a matroid is a set M together with a nonempty hereditary collection I of subsets deemed to be independent where all maximal independent subsets of every set are equicardinal. They can also be regarded as generalizations of graphs, since if G is any graph and I is the collection of all acyclic subsets of E(G), then the pair (E(G),I) is a matroid. In fact, it is a binary matroid, which means that it can be represented as a subset of a vector space over F2. To do this, we take the space of all formal sums of vertices and represent the edge vw by the sum v+w. A set of edges is easily seen to be acyclic if and only if the corresponding set of sums is linearly independent. There is a natural analogue of an induced subgraph for matroids: an induced restriction of a matroid M is a subset M′ of M with the property that adding any element of M−M′ to M′ produces a matroid with a larger independent set than M′. The natural analogue of a tree with m edges is the matroid Im, where one takes a set of size m and takes all its subsets to be independent. (Note, however, that unlike with graph-theoretic trees there is just one such matroid up to isomorphism for each m.) Every graph can be obtained by deleting edges from a complete graph. Analogously, every binary matroid can be obtained by deleting elements from a finite binary projective geometry, that is, the set of all one-dimensional subspaces in a finite-dimensional vector space over F2. Finally, the analogue of the chromatic number for binary matroids is a quantity known as the critical number introduced by Crapo and Rota, which in the case of a graph G turns out to be ⌈log2(χ(G))⌉ -- that is, roughly the logarithm of its chromatic number. One of the results of the paper is that a binary matroid can fail to contain I3 or the Fano plane F7 (which is the simplest projective geometry) as an induced restriction, but also have arbitrarily large critical number. By contrast, the critical number is at most two if one also excludes the matroid associated with K5 as an induced restriction. The main result of the paper is a structural description of all simple binary matroids that have neither I3 nor F7 as an induced restriction.

The following natural problem was raised independently by Erdős–Hajnal and Linial–Rabinovich in the early ’90 s. How large must the independence number alpha (G) of a graph G be whose every m vertices contain an independent set of size r? In this paper, we discuss new methods to attack this problem. The first new approach, based on bounding Ramsey numbers of certain graphs, allows us to improve the previously best lower bounds due to Linial–Rabinovich, Erdős–Hajnal and Alon–Sudakov. As an example, we prove that any n-vertex graph G having an independent set of size 3 among every 7 vertices has alpha (G) ge Omega (n^{5/12}). This confirms a conjecture of Erdős and Hajnal that alpha (G) should be at least n^{1/3+varepsilon } and brings the exponent halfway to the best possible value of 1/2. Our second approach deals with upper bounds. It relies on a reduction of the original question to the following natural extremal problem. What is the minimum possible value of the 2-density (The 2-density of a graph H is defined as m_2(H):=max _{H' subseteq H, |H'| ge 3} frac{e(H')-1}{|H'|-2}) of a graph on m vertices having no independent set of size r? This allows us to improve previous upper bounds due to Linial–Rabinovich, Krivelevich and Kostochka-Jancey. As part of our arguments, we link the problem of Erdős–Hajnal and Linial–Rabinovich and our new extremal 2-density problem to a number of other well-studied questions. This leads to many interesting directions for future research.

Large independent sets in random regular graphs

To develop a genotype-based ankylosing spondylitis (AS) risk prediction model that is more sensitive and specific than HLA-B27 typing. To develop the AS genetic risk scoring (AS-GRS) model, 648 individuals (285 cases and 363 controls) were examined for 5 copy number variants (CNV), 7 single-nucleotide polymorphisms (SNP), and an HLA-B27 marker by TaqMan assays. The AS-GRS model was developed using logistic regression and validated with a larger independent set (576 cases and 680 controls). Through logistic regression, we built the AS-GRS model consisting of 5 genetic components: HLA-B27, 3 CNV (1q32.2, 13q13.1, and 16p13.3), and 1 SNP (rs10865331). All significant associations of genetic factors in the model were replicated in the independent validation set. The discriminative ability of the AS-GRS model measured by the area under the curve was excellent: 0.976 (95% CI 0.96-0.99) in the model construction set and 0.951 (95% CI 0.94-0.96) in the validation set. The AS-GRS model showed higher specificity and accuracy than the HLA-B27-only model when the sensitivity was set to over 94%. When we categorized the individuals into quartiles based on the AS-GRS scores, OR of the 4 groups (low, intermediate-1, intermediate-2, and high risk) showed an increasing trend with the AS-GRS scores (r2 = 0.950) and the highest risk group showed a 494× higher risk of AS than the lowest risk group (95% CI 237.3-1029.1). Our AS-GRS could be used to identify individuals at high risk for AS before major symptoms appear, which may improve the prognosis for them through early treatment.

ABSTRACTAn independent set of size in a finite undirected graph is a set of vertices of the graph, no two of which are connected by an edge. Let be the number of independent sets of size in the graph and let . In 1987, Alavi, Malde, Schwenk, and Erdős asked if the independent set sequence of a tree is unimodal (the sequence goes up and then down). This problem is still open. In 2006, Levit and Mandrescu showed that the last third of the independent set sequence of a tree is decreasing. We show that the first 46.8% of the independent set sequence of a random tree is increasing with (exponentially) high probability as the number of vertices goes to infinity. So, the question of Alavi, Malde, Schwenk, and Erdős is “four‐fifths true,” with high probability. We also show unimodality of the independent set sequence of Erdős–Rényi random graphs, when the expected degree of a single vertex is large (with [exponentially] high probability as the number of vertices in the graph goes to infinity, except for a small region near the mode). A weaker result is shown for random regular graphs. The structure of independent sets of size as varies is of interest in probability, statistical physics, combinatorics, and computer science.

We study the following question raised by Erdős and Hajnal in the early 90’s. Over all n n -vertex graphs G G what is the smallest possible value of m m for which any m m vertices of G G contain both a clique and an independent set of size log ⁡ n \log n ? We construct examples showing that m m is at most 2 2 ( log ⁡ log ⁡ n ) 1 / 2 + o ( 1 ) 2^{2^{(\log \log n)^{1/2+o(1)}}} obtaining a twofold sub-polynomial improvement over the upper bound of about n \sqrt {n} coming from the natural guess, the random graph. Our (probabilistic) construction gives rise to new examples of Ramsey graphs, which while having no very large homogenous subsets contain both cliques and independent sets of size log ⁡ n \log n in any small subset of vertices. This is very far from being true in random graphs. Our proofs are based on an interplay between taking lexicographic products and using randomness.

Independent sets play a central role in distributed algorithmics. We examine here the minimal requirements for computing non-trivial independent sets. In particular, we focus on algorithms that operate in a single communication round. A classic result of Linial shows that a constant number of rounds does not suffice to compute a maximal independent set. We are therefore interested in the size of the solution that can be computed, especially in comparison to the optimal. Our main result is a randomized one-round algorithm that achieves poly-logarithmic approximation on graphs of polynomially bounded-independence. Specifically, we show that the algorithm achieves the Caro-Wei bound (an extension of the Turán bound for independent sets) in general graphs up to a constant factor, and that the Caro-Wei bound yields a poly-logarithmic approximation on bounded-independence graphs. The algorithm uses only a single bit message and operates in a beeping model, where a node receives only the disjunction of the bits transmitted by its neighbors. We give limitation results that show that these are the minimal requirements for obtaining non-trivial solutions. In particular, a sublinear approximation cannot be obtained in a single round on general graphs, nor when nodes cannot both transmit and receive messages. We also show that our analysis of the Caro-Wei bound on polynomially bounded-independence graphs is tight, and that the poly-logarithmic approximation factor does not extend to mathrm {O}(1)-claw free graphs.

The independent set sequence of regular bipartite graphs

The kth p‐power of a graph G is the graph on the vertex set $V(G)^k$, where two k‐tuples are adjacent iff the number of their coordinates which are adjacent in G is not congruent to 0 modulo p. The clique number of powers of G is polylogarithmic in the number of vertices; thus graphs with small independence numbers in their p‐powers do not contain large homogeneous subsets. We provide algebraic upper bounds for the asymptotic behavior of independence numbers of such powers, settling a conjecture of [N. Alon and E. Lubetzky, Combinatorica, 27 (2007), pp. 13–33] up to a factor of 2. For precise bounds on some graphs, we apply Delsarte’s linear programming bound and Hoffman’s eigenvalue bound. Finally, we show that for any nontrivial graph G, one can point out specific induced subgraphs of large p‐powers of G with neither a large clique nor a large independent set. We prove that the larger the Shannon capacity of $\overline{G}$ is, the larger these subgraphs are, and if G is the complete graph, then some p‐power of G matches the bounds of the Frankl–Wilson Ramsey construction, and is in fact a subgraph of a variant of that construction.

We study the problem of selecting a subset of k random variables to observe that will yield the best linear prediction of another variable of interest, given the pairwise correlations between the observation variables and the predictor variable. Under preserving reductions, this problem is equivalent to the sparse approximation problem of approximating signals concisely. The subset selection problem is NP-hard in general; in this paper, we propose and analyze exact and algorithms for several special cases of practical interest. Specifically, we give an FPTAS when the covariance matrix has constant bandwidth, and exact algorithms when the associated covariance graph, consisting of edges for pairs of variables with non-zero correlation, forms a tree or has a large (known) independent set. Furthermore, we give an exact algorithm when the variables can be embedded into a line such that the covariance decreases exponentially in the distance, and a constant-factor when the variables have no conditional suppressor variables. Much of our reasoning is based on perturbation results for the R2 multiple correlation measure, which is frequently used as a natural measure for goodness-of-fit statistics. It lies at the core of our FPTAS, and also allows us to extend our exact algorithms to algorithms when the matrix falls into one of the above classes. We also use our perturbation analysis to prove guarantees for the widely used Forward Regression heuristic under the assumption that the observation variables are nearly independent.

Assume that the world is a graph. There are two cases to be examined: either this graph is finite or it is infinite. In this chapter, the authors explore the two possibilities, knowing that, in each case, the philosopher's quest is that of a stable subset whose name has always been called “the Absolute”. They appeal to the Erdos–Hajnal conjecture, which states that families of graphs defined by forbidden induced subgraphs have either large cliques or large independent (i.e. stable) sets. Philosophers who seek to reduce the graph of their philosophical architectonics to an independent set should therefore avoid this kind of configuration, except if they wanted, in the Hegelian way, to identify the whole of their system with the absolute itself. Infinite attributes or infinite modes are their examples in Spinoza's Ethics.