On the power of symmetric linear programs

© 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.We consider families of symmetric linear programs (LPs) that decide a property of graphs (or other relational structures) in the sense that, for each size of graph, there is an LP defining a polyhedral lift that separates the integer points corresponding to graphs with the property from those corresponding to graphs without the property. We show that this is equivalent, with at most polynomial blow-up in size, to families of symmetric Boolean circuits with threshold gates. In particular, when we consider polynomial-size LPs, the model is equivalent to definability in a non-uniform version of fixed-point logic with counting (FPC). Known upper and lower bounds for FPC apply to the non-uniform version. In particular, this implies that the class of graphs with perfect matchings has polynomial-size symmetric LPs while we obtain an exponential lower bound for symmetric LPs for the class of Hamiltonian graphs. We compare and contrast this with previous results (Yannakakis 1991) showing that any symmetric LPs for the matching and TSP polytopes have exponential size. As an application, we establish that for random, uniformly distributed graphs, polynomial-size symmetric LPs are as powerful as general Boolean circuits. We illustrate the effect of this on the well-studied planted-clique problem.Peer ReviewedPostprint (author's final draft

Properties of random coverings of graphs

Wydział Matematyki i InformatykiPrzedmiotem rozprawy doktorskiej są asymptotyczne własności losowych nakryć grafów zdefiniowanych przez Amita i Liniala w 2002 roku, jako nowy model grafu losowego. Dla zadanego grafu bazowego G losowe nakrycie stopnia n, oznaczane jako Ḡ, otrzymujemy poprzez zastąpienie każdego wierzchołka v przez n-elementowy zbiór Ḡ_v oraz wybór, dla każdej krawędzi e={x,y} \in E(G), z jednostajnym prawdopodobieństwem, losowego skojarzenia pomiędzy zbiorami Ḡ_x i Ḡ_y. Pierwszym zagadnieniem poruszanym w pracy jest oszacowanie wielkości największej topologicznej kliki zawartej (jako podgraf) w losowym nakryciu danego grafu G. Udało się pokazać, że asymptotycznie prawie na pewno losowe nakrycie grafu G zawiera największą z możliwych topologiczną klikę. Drugim badanym zagadnieniem jest pytanie o istnienie w podniesieniu grafu cyklu Hamiltona. W pracy pokazujemy, że jeżeli graf G zawiera dwa krawędziowo rozłączne cykle Hamiltona, których suma nie jest grafem dwudzielnym i ma minimalny stopień co najmniej 5, to asymptotycznie prawie na pewno nakrycie Ḡ jest grafem hamiltonowskim.In the thesis we study selected properties of random coverings of graphs introduced by Amit and Linial in 2002. A random n-covering of a graph G, denoted by Ḡ, is obtained by replacing each vertex v of G by an n-element set Ḡ_v and then choosing, independently for every edge e = {x,y}\in E(G), uniformly at random a perfect matching between Ḡ_x and Ḡ_y. The first problem we consider is the typical size of the largest topological clique in random covering of given graph G. We show that asymptotically almost surely a random n-covering of a graph G contains the largest possible topological clique. The second property we examine is the existence of a Hamilton cycle in Ḡ. We show that if G contains two edge disjoint Hamilton cycles, which sum is not a bipartite graph and has minimum degree at least 5, then asymptotically almost surely Ḡ is Hamiltonian

On the power of symmetric linear programs

We consider families of symmetric linear programs (LPs) that decide a property of graphs (or other relational structures) in the sense that, for each size of graph, there is an LP defining a polyhedral lift that separates the integer points corresponding to graphs with the property from those corresponding to graphs without the property. We show that this is equivalent, with at most polynomial blow-up in size, to families of symmetric Boolean circuits with threshold gates. In particular, when we consider polynomial-size LPs, the model is equivalent to definability in a non-uniform version of fixed-point logic with counting (FPC). Known upper and lower bounds for FPC apply to the non-uniform version. In particular, this implies that the class of graphs with perfect matchings has polynomial-size symmetric LPs while we obtain an exponential lower bound for symmetric LPs for the class of Hamiltonian graphs. We compare and contrast this with previous results (Yannakakis 1991) showing that any symmetric LPs for the matching and TSP polytopes have exponential size. As an application, we establish that for random, uniformly distributed graphs, polynomial-size symmetric LPs are as powerful as general Boolean circuits. We illustrate the effect of this on the well-studied planted-clique problem

European Journal of Combinatorics Index, Volume 27

BACKGROUND: Diabetes is an inflammatory condition associated with iron abnormalities and increased oxidative damage. We aimed to investigate how diabetes affects the interrelationships between these pathogenic mechanisms. METHODS: Glycaemic control, serum iron, proteins involved in iron homeostasis, global antioxidant capacity and levels of antioxidants and peroxidation products were measured in 39 type 1 and 67 type 2 diabetic patients and 100 control subjects. RESULTS: Although serum iron was lower in diabetes, serum ferritin was elevated in type 2 diabetes (p = 0.02). This increase was not related to inflammation (C-reactive protein) but inversely correlated with soluble transferrin receptors (r = - 0.38, p = 0.002). Haptoglobin was higher in both type 1 and type 2 diabetes (p &lt; 0.001) and haemopexin was higher in type 2 diabetes (p &lt; 0.001). The relation between C-reactive protein and haemopexin was lost in type 2 diabetes (r = 0.15, p = 0.27 vs r = 0.63, p &lt; 0.001 in type 1 diabetes and r = 0.36, p = 0.001 in controls). Haemopexin levels were independently determined by triacylglycerol (R(2) = 0.43) and the diabetic state (R(2) = 0.13). Regarding oxidative stress status, lower antioxidant concentrations were found for retinol and uric acid in type 1 diabetes, alpha-tocopherol and ascorbate in type 2 diabetes and protein thiols in both types. These decreases were partially explained by metabolic-, inflammatory- and iron alterations. An additional independent effect of the diabetic state on the oxidative stress status could be identified (R(2) = 0.5-0.14). CONCLUSIONS: Circulating proteins, body iron stores, inflammation, oxidative stress and their interrelationships are abnormal in patients with diabetes and differ between type 1 and type 2 diabetes</p