Large Independent Sets Research Articles

The Approximability of Assortment Optimization Under Ranking Preferences

Assortment optimization has received significant attention in recent revenue management and combinatorial optimization literature. In “The Approximability of Assortment Optimization Under Ranking Preferences,” A. Aouad, V. Farias, R. Levi, and D. Segev provide best-possible approximability bounds for this problem under an almost general model specification, where preferences are expressed as a distribution over rankings. This paper shows how this optimization problem relates to the computational task of detecting large independent sets in graphs, allowing the establishment of strong complexity lower bounds with respect to various problem parameters. These findings are complemented by a number of algorithms that attain essentially best-possible approximation factors, proving that the hardness results are tight up to lower-order terms. Surprisingly, their results imply that a simple and widely studied policy, known as revenue-ordered assortments, achieves the best possible performance guarantee with respect to prices.

Research on algorithm for solving maximum independent set of semi-external data of large graph data

The maximum independent set algorithm for large-scale data semi-existing data is studied and the solving method of the largest independent set problem in large-map data is mainly analyzed in this paper. The specific research contents are mainly divided into semi-external map algorithm based on Greedy heuristic strategy, semi-external map algorithm based on swap and design, and implementation of semi-external graph algorithm processing function library. Experiments on a large number of real and artificially generated data sets show that the algorithm in this paper is very efficient both in time and in space. The largest independent set obtained by the algorithm can reach more than 96% of its theoretical upper bound for most of the data.

How far can the uncertainty on a Digital Soil Map be known?: A numerical experiment using pseudo values of clay content obtained from Vis-SWIR hyperspectral imagery

Digital Soil Map uncertainty is usually evaluated from a set of independent soil observations that are used to determine various uncertainty indicators. However, the number and locations of the sites that constitute these evaluations may impact the value of these indicators.In this paper, a numerical experiment on uncertainty indicators was performed using the pseudo values of topsoil clay content obtained from an airborne hyperspectral image in the Cap Bon region (Tunisia). These pseudo values form a soil pattern with a large extent (46% of 300 km2), high resolution (5 m) and good accuracy (R2val = 0.75) while being free of visible artefacts and pedologically plausible. Therefore, the dataset was considered a fair representation of reality while providing a quasi-unlimited choice of sites.The numerical experiment considered three Quantile Regression Forests as examples of DSM models by using inputs from relief soil covariates and geographical locations that were calibrated from 200, 2000 and 100,000 individuals respectively (low, medium and high quality models). Their uncertainty indicators were first evaluated by calculating four uncertainty indicators (ME, MSE, SSMSE and PICP) from a large independent validation set of 100,000 sites. These uncertainty indicators were then computed from independent evaluation sets of different sizes (from 50 to 500 sites) and from different locations (500 evaluation sets of each size). The independent evaluation sets were selected following a stratified random sampling using compact geographical strata.The numerical experiment showed that the values of the uncertainty indicators were highly variable across numbers and locations of sites. The largest variations were observed for evaluation sets with fewer than 100 sites, but non-negligible variations remained for larger evaluation datasets. This result suggested that evaluations from independent sets convey a non-negligible error on the uncertainty indicators, which increases as the number of sites decrease.Evaluations of DSM models from independent evaluation sets should be interpreted with care and uncertainty on validation results should be systematically estimated. For that, numerical experiments for benchmarking DSM models on known soil patterns across the world would be a valuable complement to the analytical expressions for the uncertainty indicators and the many DSM applications for which these analytical expressions are not valid. This would serve also to improve the sampling techniques for the calibration and evaluation datasets to reduce the error when estimating the uncertainty of a DSM product.

TopScore: Using Deep Neural Networks and Large Diverse Data Sets for Accurate Protein Model Quality Assessment.

The value of protein models obtained with automated protein structure prediction depends primarily on their accuracy. Protein model quality assessment is thus critical to select the model that can best answer biologically relevant questions from an ensemble of predictions. However, despite many advances in the field, different methods capture different types of errors, begging the question of which method to use. We introduce TopScore, a meta Model Quality Assessment Program (meta-MQAP) that uses deep neural networks to combine scores from 15 different primary predictors to predict accurate residue-wise and whole-protein error estimates. The predictions on six large independent data sets are highly correlated to superposition-independent errors in the model, achieving a Pearson's Rall2 of 0.93 and 0.78 for whole-protein and residue-wise error predictions, respectively. This is a significant improvement over any of the investigated primary MQAPs, demonstrating that much can be gained by optimally combining different methods and using different and very large data sets.

Are serum concentrations of vitamin B-12 causally related to cardiometabolic risk factors and disease? A Mendelian randomization study

Several observational studies have shown that low serum vitamin B-12 is associated with increased body mass index (BMI) and adverse cardiometabolic outcomes. However, it is unclear if these associations reflect a causal effect of vitamin B-12 on cardiometabolic risk factors and diseases, latent confounding, or reverse causality. The aims of this study were to investigate 1) the possible causal relation between vitamin B-12 and indicators of body fat, lipid, and glucose variables; type 2 diabetes (T2D); and cardiovascular disease by using a 2-sample Mendelian randomization (MR) method and 2) the possible pleiotropic role of fucosyltransferase 2 (FUT2). We selected 11 single nucleotide polymorphisms (SNPs) robustly associated with serum concentrations of vitamin B-12 in a previous genomewide association study (GWAS) in 45,576 individuals. We performed 2-sample MR analyses of the relation between vitamin B-12 and cardiometabolic risk factors and diseases with the use of publicly available GWAS summary statistics for 15 outcomes in ≤339,224 individuals. The robustness of results was tested with sensitivity analyses by using MR Egger regression and weighted-median estimation, and by performing additional analyses excluding a variant in the FUT2 gene, which may be pleiotropic. We found a suggestive causal relation between vitamin B-12 and fasting glucose and β cell function [homeostatic model assessment (HOMA) of β cell function (HOMA-B)]. However, we found no evidence that serum concentrations of vitamin B-12 were causally related to BMI, waist-to-hip ratio, plasma leptin, body fat, fasting insulin, insulin resistance (from HOMA of insulin resistance), glycated hemoglobin, triglycerides, T2D, coronary artery disease, or HDL, LDL, or total cholesterol. We found no evidence that serum concentrations of vitamin B-12 are causally related to body weight or the majority of cardiometabolic outcomes investigated. However, vitamin B-12 may have a causal effect on fasting glucose and HOMA-B, although these results will require replication in large independent data sets. This trialwas registered at http://www.isrctn.com/ISRCTN47414943 as ISRCTN47414943.

Abstract 4268: Magee Equations for predicting Oncotype DX recurrence risk: A further evaluation using a large independent data set

Abstract Purpose: Oncotype DX 21-gene assay is a costly clinical method to predict the risk of breast cancer recurrence in lymph node-negative and estrogen receptor (ER)-positive breast cancers. Magee Equations were linear regression models developed working on several standard clinically available histopathological markers for predicting the Oncotype DX risk score, with the intention to save expenses and time. Magee Equations were derived from 817 patients and initially evaluated on 255 separate patients, now publically accessible online to compute estimations of recurrence risk scores. The purpose of this study was to conduct a further validation of Magee Equations using a much larger independent dataset to examine its robustness in comparison to the initial evaluation. Methods and Materials: In a HIPAA-compliant and IRB-approved study, we collected 1,391 patients with invasive breast cancer at our institution and all those patients have Oncotype DX risk score available. This cohort is independent of all previous patient populations associated with the development and initial evaluation of Magee Equations. Out of the 1,391 patients, 1,210 have available the related histopathologic markers that had been used in Magee Equations, including ER status, progesterone receptor (PR) status, human epidermal growth factor receptor 2 (HER2) status, proliferation indices (Ki-67), Nottingham grade, and tumor size. We ran the three different forms of Magee Equations on all the 1,210 patients and compared the estimation output of each equation to the truth, i.e., the actual Oncotype DX risk score. The estimation results on the 1,210 patients were evaluated by two metrics: concordance and Pearson's correlation coefficients, and compared to the reported initial evaluation performance of Magee Equations on the 255 patients. Results: On the 1,210 cases, the concordance on the recurrence risk categories (i.e., low, intermediate, and high) between the actual Oncotype DX categories and the three Magee Equation-predicted categories was 61.9% (Eq. 1), 59.8% (Eq. 2), and 63.4% (Eq. 3), respectively, corresponding to the initial evaluation performance of 55.8%, 59.4%, and 54.4%. After eliminating the intermediate category, the concordance of the three equations increased to 99.0%, 98.3%, and 99.1%, respectively, corresponding to the initial performance of 100%, 98.6%, and 98.7%. Likewise, the Pearson's correlation on the continuous risk scores was 53.7%, 54.4%, and 56.2% for equations 1, 2, and 3, respectively, corresponding to the initial Pearson's correlation of 61.6%, 60.3%, and 59.4%. Conclusions: Magee Equations remain stable on 1,210 independent new patients, comparable to the initially reported evaluation performance. Clinical outcome studies are required to test the usefulness of Magee Equations as a stand-alone test. Citation Format: Dooman Arefan, Rohit Bhargava, David Dabbs, Rachel C. Jankowitz, Shandong Wu. Magee Equations for predicting Oncotype DX recurrence risk: A further evaluation using a large independent data set [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2018; 2018 Apr 14-18; Chicago, IL. Philadelphia (PA): AACR; Cancer Res 2018;78(13 Suppl):Abstract nr 4268.

Triangle-Tilings in Graphs Without Large Independent Sets

We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an n-vertex graph G with sublinear independence number. In this setting, we show that if δ(G) ≥ n/3 + o(n), then G has a triangle-tiling covering all but at most four vertices. Also, for every r ≥ 5, we asymptotically determine the minimum degree threshold for a perfect triangle-tiling under the additional assumptions that G is Kr-free and n is divisible by 3.

Non-coding RNA dysregulation in the amygdala region of schizophrenia patients contributes to the pathogenesis of the disease

Schizophrenia (SCZ) is a neuropsychiatric disorder with a complex genetic etiology. The redundancy of the gene networks underlying SCZ indicates that many gene combinations have the potential to cause a system dysfunction that can manifest as SCZ or a related neurodevelopmental disorder. Recent studies show that small non-coding microRNA (miRNA) and long non-coding RNA (lncRNA) are important factors in shaping these networks and are dynamically regulated by neuronal activation. We investigated the genome-wide transcription profiles of 46 human amygdala samples obtained from 22 SCZ patients and 24 healthy controls. Using RNA sequencing (RNA-seq), we determined lncRNA expression levels in all samples and generated miRNA profiles for 27 individuals (13 cases and 14 controls). Previous studies have identified differentially expressed miRNAs in SCZ, including miR-132, miR-212, and miR-34a/miR-34c. Here we report differential expression of a novel miRNA, miR1307, in SCZ. Notably, miR1307 maps to a locus previously associated with SCZ through GWAS. Additionally, one lncRNA that was overexpressed in SCZ, AC005009.2, also maps to a region previously associated with SCZ based on GWAS and overlapped SCZ-related genes. The results were replicated in a large independent data set of 254 dorsolateral prefrontal cortex samples from the CommonMind consortium. Taken together, these results suggest that miRNA and lncRNAs are important contributors to the pathogenesis of SCZ.

Extensive geographical and social structure in the paternal lineages of Saudi Arabia revealed by analysis of 27 Y-STRs

Saudi Arabia’s indigenous population is organized into patrilineal descent groups, but to date, little has been done to characterize its population structure, in particular with respect to the male-specific region of the Y chromosome. We have used the 27-STR Yfiler® Plus kit to generate haplotypes in 597 unrelated Saudi males, classified into five geographical regions (North, South, Central, East and West). Overall, Yfiler® Plus provides a good discrimination capacity of 95.3%, but this is greatly reduced (74.7%) when considering the reduced Yfiler® set of 17 Y-STRs, justifying the use of the expanded set of markers in this population. Comparison of the five geographical divisions reveals striking differences, with low diversity and similar haplotype spectra in the Central and Northern regions, and high diversity and similar haplotype spectra in the East and West. These patterns likely reflect the geographical isolation of the desert heartland of the peninsula, and the proximity to the sea of the Eastern and Western areas, and consequent historical immigration. We predicted haplogroups from Y-STR haplotypes, testing the performance of prediction by using a large independent set of Saudi Arabian Y-STR + Y-SNP data. Prediction indicated predominance (71%) of haplogroup J1, which was significantly more common in Central, Northern and Southern groups than in East and West, and formed a star-like expansion cluster in a median-joining network with an estimated age of ∼2800 years. Most of our 597 participants were sampled within Saudi Arabia itself, but ∼16% were sampled in the UK. Despite matching these two groups by home sub-region, we observed significant differences in haplotype and predicted haplogroup constitutions overall, and for most sub-regions individually. This suggests social structure influencing the probability of leaving Saudi Arabia, correlated with different Y-chromosome compositions. The UK-recruited sample is an inappropriate proxy for Saudi Arabia generally, and caution is needed when considering expatriate groups as representative of country of origin. Our study shows the importance of geographical and social structuring that may affect the utility of forensic databases and the interpretation of Y-STR profiles.

Anthropogenic landscapes support fewer rare bee species

The response of rare species to human land use is poorly known because rarity is difficult to study; however, it is also important because rare species compose most of biodiversity, and are disproportionately vulnerable. Regional bee pollinator faunas have not been assessed for rarity outside of Europe. Therefore, we do not know to what extent anthropogenic landscapes support rare North American bee biodiversity. We ask how richness and abundance of bee species respond to land use, within quartiles of species defined by their numerical, phenological, and geographical rarity. We conducted a field study to sample bee communities in forested, agricultural, and urban landscapes replicated across a large spatial extent of the northeastern United States. We used large independent data sets to classify observed bee species according to three forms of rarity: their numerical rarity (low regional frequency in a museum-based data set), phenological rarity (short flight season length) and geographical rarity (small range size). For all three forms of rarity, we found half as many rare bee species in agricultural landscapes compared to forest. We found half as many phenologically rare species in urban landscapes. Bees that had both shorter flight seasons and smaller range sizes were between one-third and one-half as rich in both types of anthropogenic landscapes, regardless of regional frequency. Although a minority of rare bee species were found in anthropogenic landscapes, our overall conclusion is that the native vegetation of our region, forest, is critical for supporting rare bee biodiversity.

A variant in PPP4R3A protects against alzheimer-related metabolic decline.

A reduction in glucose metabolism in the posterior cingulate cortex (PCC) predicts conversion to Alzheimer's disease (AD) and tracks disease progression, signifying its importance in AD. We aimed to use decline in PCC glucose metabolism as a proxy for the development and progression of AD to discover common genetic variants associated with disease vulnerability. We performed a genome-wide association study (GWAS) of decline in PCC fludeoxyglucose F 18 ([18 F] FDG) positron emission tomography measured in Alzheimer's Disease Neuroimaging Initiative participants (n = 606). We then performed follow-up analyses to assess the impact of significant single-nucleotide polymorphisms (SNPs) on disease risk and longitudinal cognitive performance in a large independent data set (n = 870). Last, we assessed whether significant SNP influence gene expression using two RNA sequencing data sets (n = 210 and n = 159). We demonstrate a novel genome-wide significant association between rs2273647-T in the gene, PPP4R3A, and reduced [18 F] FDG decline (p = 4.44 × 10-8 ). In a follow-up analysis using an independent data set, we demonstrate a protective effect of this variant against risk of conversion to MCI or AD (p = 0.038) and against cognitive decline in individuals who develop dementia (p = 3.41 × 10-15 ). Furthermore, this variant is associated with altered gene expression in peripheral blood and altered PPPP4R3A transcript expression in temporal cortex, suggesting a role at the molecular level. PPP4R3A is a gene involved in AD risk and progression. Given the protective effect of this variant, PPP4R3A should be further investigated as a gene of interest in neurodegenerative diseases and as a potential target for AD therapies. Ann Neurol 2017;82:900-911.

Herpes Zoster as a Risk Factor for Incident Giant Cell Arteritis.

Histopathologic studies have implicated herpes zoster (HZ) as a causative organism of giant cell arteritis (GCA). The purpose of this study was to assess the epidemiologic association of HZ events with incident GCA. We performed a retrospective cohort study in 2 large independent US administrative data sets: Medicare 5% and Truven Health Analytics MarketScan. Eligible subjects had 12 months of continuous coverage, were >50 years old, and had no history of GCA or polymyalgia rheumatica. HZ events (complicated and uncomplicated) and GCA were identified by the presence of International Classification of Diseases, Ninth Revision, Clinical Modification codes from physician visit or hospital discharge records. Antiviral therapies and vaccinations were identified from prescription claims and drug codes. Risk of incident GCA was calculated using multivariable Cox proportional hazards regression. Among 16,686,345 subjects, a total of 5,942 GCA cases occurred, with 3.1% (MarketScan) and 6.0% (Medicare) having preceding HZ events. Unadjusted GCA incidence rates were highest in the groups with complicated and uncomplicated HZ. After multivariable adjustment, complicated HZ was associated with an increased risk of GCA (hazard ratio [HR] 1.99 [95% confidence interval (95% CI) 1.32-3.02] in the Medicare cohort and 2.16 [95% CI 1.46-3.18] in the MarketScan cohort), as was uncomplicated HZ (HR 1.42 [95% CI 1.02-1.99] and HR 1.45 [95% CI 1.05-2.01] in the respective cohorts). Vaccination and antiviral treatment were not consistently associated with GCA risk, although antiviral treatment was marginally associated with a decreased risk of GCA in the Medicare cohort (HR 0.67 [95% CI 0.46-0.99]). HZ is associated with an increased risk of GCA. The infrequency of HZ in GCA patients suggests that it is only one potential trigger for GCA. Antivirals and vaccination did not consistently mitigate this risk.

Characterizing 4-critical graphs with Ore-degree at most seven

Dirac introduced the notion of a k-critical graph, a graph that is not (k−1)-colorable but whose every proper subgraph is (k−1)-colorable. Brook's Theorem states that every graph with maximum degree k is k-colorable unless it contains a subgraph isomorphic to Kk+1 or an odd cycle (for k=2). Equivalently, for all k≥4, the only k-critical graph of maximum degree k−1 is Kk. A natural generalization of Brook's theorem is to consider the Ore-degree of a graph, which is the maximum of d(u)+d(v) over all uv∈E(G). Kierstead and Kostochka proved that for all k≥6 the only k-critical graph with Ore-degree at most 2k−1 is Kk. Kostochka, Rabern and Stiebitz proved that the only 5-critical graphs with Ore-degree at most 9 are K5 and a graph they called O5.A different generalization of Brook's theorem, motivated by Hajos' construction, is Ore's conjectured bound on the minimum density of a k-critical graph. Recently, Kostochka and Yancey proved Gallai's conjecture. Their proof for k≥5 implies the above results on Ore-degree. However, the case for k=4 remains open, which is the subject of this paper.Kostochka and Yancey's short but beautiful proof for the case k=4 says that if G is a 4-critical graph, then |E(G)|≥(5|V(G)|−2)/3. We prove the following bound which is better when there exists a large independent set of degree three vertices: if G is a 4-critical graph G, then |E(G)|≥1.6|V(G)|+.2α(D3(G))−.6, where D3(G) is the graph induced by the degree three vertices of G. As a corollary, we characterize the 4-critical graphs with Ore-degree at most seven as precisely the graphs of Ore-degree seven in the family of graphs obtained from K4 and Ore compositions.

Predicting perceived visual complexity of abstract patterns using computational measures: The influence of mirror symmetry on complexity perception.

Visual complexity is relevant for many areas ranging from improving usability of technical displays or websites up to understanding aesthetic experiences. Therefore, many attempts have been made to relate objective properties of images to perceived complexity in artworks and other images. It has been argued that visual complexity is a multidimensional construct mainly consisting of two dimensions: A quantitative dimension that increases complexity through number of elements, and a structural dimension representing order negatively related to complexity. The objective of this work is to study human perception of visual complexity utilizing two large independent sets of abstract patterns. A wide range of computational measures of complexity was calculated, further combined using linear models as well as machine learning (random forests), and compared with data from human evaluations. Our results confirm the adequacy of existing two-factor models of perceived visual complexity consisting of a quantitative and a structural factor (in our case mirror symmetry) for both of our stimulus sets. In addition, a non-linear transformation of mirror symmetry giving more influence to small deviations from symmetry greatly increased explained variance. Thus, we again demonstrate the multidimensional nature of human complexity perception and present comprehensive quantitative models of the visual complexity of abstract patterns, which might be useful for future experiments and applications.

Minimum sum coloring for large graphs with extraction and backward expansion search

The Minimum Sum Coloring Problem (MSCP) is a relevant model tightly related to the classical vertex coloring problem (VCP). MSCP is known to be NP-hard, thus solving the problem for large graphs is particular challenging. Based on the general “reduce-and-solve” principle and inspired by the work for the VCP, we present an extraction and backward expansion search approach (EBES) to compute the upper and lower bounds for the MSCP on large graphs. The extraction phase reduces the given graph by extracting large collections of pairwise disjoint large independent sets (or color classes). The backward extension phase adds the extracted independent sets to recover the intermediate graphs while optimizing the sum coloring of each intermediate graph. We assess the proposed approach on a set of 35 large benchmark graphs with 450–4000 vertices from the DIMACS and COLOR graph coloring competitions. Computational results show that EBES is able to find improved upper bounds for 19 graphs and improved lower bounds for 12 graphs.

Comprehensive Evaluation of TFF3 Promoter Hypomethylation and Molecular Biomarker Potential for Prostate Cancer Diagnosis and Prognosis.

Overdiagnosis and overtreatment of clinically insignificant tumors remains a major problem in prostate cancer (PC) due to suboptimal diagnostic and prognostic tools. Thus, novel biomarkers are urgently needed. In this study, we investigated the biomarker potential of Trefoil factor 3 (TFF3) promoter methylation and RNA expression levels for PC. Initially, by quantitative methylation specific PCR (qMSP) analysis of a large radical prostatectomy (RP) cohort (n = 292), we found that the TFF3 promoter was significantly hypomethylated in PC compared to non-malignant (NM) prostate tissue samples (p < 0.001) with an AUC (area under the curve) of 0.908 by receiver operating characteristics (ROC) curve analysis. Moreover, significant TFF3 promoter hypomethylation (p ≤ 0.010) as well as overexpression (p < 0.001) was found in PC samples from another large independent patient sample set (498 PC vs. 67 NM) analyzed by Illumina 450K DNA methylation arrays and/or RNA sequencing. TFF3 promoter methylation and transcriptional expression levels were inversely correlated, suggesting that epigenetic mechanisms contribute to the regulation of gene activity. Furthermore, low TFF3 expression was significantly associated with high ERG, ETS transcription factor (ERG) expression (p < 0.001), as well as with high Gleason score (p < 0.001), advanced pathological T-stage (p < 0.001), and prostate-specific antigen (PSA) recurrence after RP (p = 0.013; univariate Cox regression analysis). There were no significant associations between TFF3 promoter methylation levels, ERG status, or PSA recurrence in these RP cohorts. In conclusion, our results demonstrated diagnostic biomarker potential of TFF3 promoter hypomethylation for PC as well as prognostic biomarker potential of TFF3 RNA expression. To the best of our knowledge, this is the most comprehensive study of TFF3 promoter methylation and transcriptional expression in PC to date.

Induced Subgraphs With Many Distinct Degrees

Let hom(G) denote the size of the largest clique or independent set of a graphG. In 2007, Bukh and Sudakov proved that everyn-vertex graphGwith hom(G) =O(logn) contains an induced subgraph with Ω(n1/2) distinct degrees, and raised the question of deciding whether an analogous result holds for everyn-vertex graphGwith hom(G) =O(nϵ), whereϵ&gt; 0 is a fixed constant. Here, we answer their question in the affirmative and show that every graphGonnvertices contains an induced subgraph with Ω((n/hom(G))1/2) distinct degrees. We also prove a stronger result for graphs with large cliques or independent sets and show, for any fixedk∈ ℕ, that if ann-vertex graphGcontains no induced subgraph withkdistinct degrees, then hom(G)⩾n/(k− 1) −o(n); this bound is essentially best possible.

Limits of local algorithms over sparse random graphs

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. Recent research has shown that such algorithms show significant promise in computing structures like large independent sets in graphs locally. Indeed the promise led to a conjecture by Hatami, Lovász and Szegedy [Geom. Funct. Anal. 24 (2014) 269–296] that local algorithms defined specifically as so-called i.i.d. factors may be able to find approximately largest independent sets in random $d$-regular graphs. In this paper, we refute this conjecture and show that every independent set produced by local algorithms is multiplicative factor $1/2+1/(2\sqrt{2})$ smaller than the largest, asymptotically as $d\rightarrow\infty$. Our result is based on an important clustering phenomena predicted first in the literature on spin glasses, and recently proved rigorously for a variety of constraint satisfaction problems on random graphs. Such properties suggest that the geometry of the solution space can be quite intricate. The specific clustering property that we prove and apply in this paper shows that typically every two large independent sets in a random graph either have a significant intersection, or have a very small intersection. As a result, large independent sets are clustered according to the proximity to each other. While the clustering property was postulated earlier as an obstruction for the success of local algorithms, our result is the first one where the clustering property is used to formally prove limits on local algorithms.

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.

Extracting List Colorings from Large Independent Sets

AbstractWe take an application of the Kernel Lemma by Kostochka and Yancey [11] to its logical conclusion. The consequence is a sort of magical way to draw conclusions about list coloring (and online list coloring) just from the existence of an independent set incident to many edges. We use this to prove an Ore‐degree version of Brooks' Theorem for online list‐coloring. The Ore‐degree of an edge in a graph G is . The Ore‐degree of G is . We show that every graph with and is online ‐choosable. In addition, we prove an upper bound for online list‐coloring triangle‐free graphs: . Finally, we characterize Gallai trees as the connected graphs G with no independent set incident to at least edges.