Collapsing Superstring Conjecture

In the Shortest Common Superstring (SCS) problem, one is given a collection of strings, and needs to find a shortest string containing each of them as a substring. SCS admits 2 11/23-approximation in polynomial time (Mucha, SODA\u2713). While this algorithm and its analysis are technically involved, the 30 years old Greedy Conjecture claims that the trivial and efficient Greedy Algorithm gives a 2-approximation for SCS. We develop a graph-theoretic framework for studying approximation algorithms for SCS. The framework is reminiscent of the classical 2-approximation for Traveling Salesman: take two copies of an optimal solution, apply a trivial edge-collapsing procedure, and get an approximate solution. In this framework, we observe two surprising properties of SCS solutions, and we conjecture that they hold for all input instances. The first conjecture, that we call Collapsing Superstring conjecture, claims that there is an elementary way to transform any solution repeated twice into the same graph G. This conjecture would give an elementary 2-approximate algorithm for SCS. The second conjecture claims that not only the resulting graph G is the same for all solutions, but that G can be computed by an elementary greedy procedure called Greedy Hierarchical Algorithm. While the second conjecture clearly implies the first one, perhaps surprisingly we prove their equivalence. We support these equivalent conjectures by giving a proof for the special case where all input strings have length at most 3 (which until recently had been the only case where the Greedy Conjecture was proven). We also tested our conjectures on millions of instances of SCS. We prove that the standard Greedy Conjecture implies Greedy Hierarchical Conjecture, while the latter is sufficient for an efficient greedy 2-approximate approximation of SCS. Except for its (conjectured) good approximation ratio, the Greedy Hierarchical Algorithm provably finds a 3.5-approximation, and finds exact solutions for the special cases where we know polynomial time (not greedy) exact algorithms: (1) when the input strings form a spectrum of a string (2) when all input strings have length at most 2

A 2-2/3 Approximation for the Shortest Superstring Problem

Given a collection of strings S={s_1, ..., s_n} over an alphabet \Sigma, a superstring \alpha of S is a string containing each s_i as a substring; that is, for each i, 1\u3c=i\u3c=n, \alpha contains a block of |s_i| consecutive characters that match s_i exactly. The shortest superstring problem is the problem of finding a superstring \alpha of minimum length. The shortest superstring problem has applications in both data compression and computational biology. In data compression, the problem is a part of a general model of string compression proposed by Gallant, Maier and Storer (JCSS \u2780). Much of the recent interest in the problem is due to its application to DNA sequence assembly. The problem has been shown to be NP-hard; in fact, it was shown by Blum et al.(JACM \u2794) to be MAX SNP-hard. The first O(1)-approximation was also due to Blum et al., who gave an algorithm that always returns a superstring no more than 3 times the length of an optimal solution. Several researchers have published results that improve on the approximation ratio; of these, the best previous result is our algorithm ShortString, which achieves a 2 3/4-approximation (WADS \u2795). We present our new algorithm, G-ShortString, which achieves a ratio of 2 2/3. It generalizes the ShortString algorithm, but the analysis differs substantially from that of ShortString. Our previous work identified classes of strings that have a nested periodic structure, and which must be present in the worst case for our algorithms. We introduced machinery to descibe these strings and proved strong structural properties about them. In this paper we extend this study to strings that exhibit a more relaxed form of the same structure, and we use this understanding to obtain our improved result

Analysis of heuristics

Heuristiken treten insbesondere im Zusammenhang mit Optimierungsproblemen in Erscheinung, bei solchen Problemen also, bei denen nicht nur eine Lösung zu finden ist, sondern unter mehreren möglichen Lösungen eine in einem objektiven Sinne beste Lösung ausfindig gemacht werden soll. Beim Problem kürzester Superstrings werden Heuristiken herangezogen, da mit exakten Algorithmen in Anbetracht der APX-Vollständigkeit des Problems nicht zu rechnen ist. Gegeben ist eine Menge S von Strings. Gesucht ist ein String s, so dass jeder String aus S Teilstring von s ist. Die Länge von s ist dabei zu minimieren. Die prominenteste Heuristik für das Problem kürzester Superstrings ist die Greedy-Heuristik, deren Approximationsfaktor derzeit jedoch nur unzureichend beschränkt werden kann. Es wird vermutet (die sogenannte Greedy-Conjecture), dass der Approximationsfaktor genau 2 beträgt, bewiesen werden kann aber nur, dass er nicht unter 2 und nicht über 3,5 liegt. Die Greedy-Conjecture ist das zentrale Thema des zweiten Kapitels. Die erzielten Ergebnisse sind im Wesentlichen: * Durch die Betrachtung von Greedyordnungen können bedingte lineare Ungleichungen nutzbar gemacht werden. Dieser Ansatz ermöglicht den Einsatz linearer Programmierung zum Auffinden interessanter Instanzen und eine Vertiefung des Verständnisses solcher schwerer Instanzen. Dieser Ansatz wird eingeführt und eine Interpretation des dualen Problems wird dargestellt. * Für die nichttriviale, große Teilklasse der bilinearen Greedyordnungen wird gezeigt, dass die Länge des von der Greedy-Heuristik gefundenen Superstrings und die des optimalen Superstrings sich höchstens um die Größe einer optimalen Kreisüberdeckung der Strings unterscheiden. Da eine optimale Kreisüberdeckung einer Menge von Strings stets höchstens so groß ist wie ein optimaler Superstring (man schließe einen Superstring zu einem einzelnen Kreis), ist das erzielte Ergebnis für die betrachtete Teilklasse der Greedyordnungen stärker als die klassische Greedy-Conjecture. * Es wird eine neue bedingte lineare Ungleichung auf Strings -- die Tripelungleichung -- gezeigt, die für das eben genannte Hauptergebnis wesentlich ist. * Schließlich wird gezeigt, dass die zum Nachweis der oberen Schranke von 3,5 für den Approximationsfaktor herangezogenen bedingten Ungleichungen (etwa die Monge-Ungleichung) inhärent zu schwach sind, um die Greedy-Conjecture selbst für lineare Greedyordnungen zu beweisen. Also ist die neue Tripelungleichung auch notwendig. Zuletzt wird gezeigt, dass das um die Tripelungleichung erweiterte System bedingter linearer Ungleichungen inhärent zu schwach ist, um die klassische Greedy-Conjecture für beliebige Greedyordnungen zu beweisen. Mit der Analyse von Queueing Strategien im Adversarial Queueing Modell wird auch ein Fall betrachtet, in dem Heuristiken auf Grund von anwendungsspezifischen Forderungen wie Online-Setup und Lokalität eingesetzt werden. Pakete sollen in einem Netzwerk verschickt werden, wobei jeder Rechner nur begrenzte Information über den Zustand des Netzwerks hat. Es werden Klassen von Queueing Strategien untersucht und insbesondere untersucht, wovon Queueing Strategien ihre lokalen Entscheidungen abhängig machen sollten, um ein gewisses Qualitätsmerkmal zu erreichen. Die hier erzielten Ergebnisse sind: * Jede Queueing Strategie, die ohne Zeitstempel arbeitet, kann zu einer exponentiell großen Queue und damit zu exponentiell großer Verzögerung (im Durchmesser und der Knotenzahl des Netzwerks) gezwungen werden. Dies war bisher nur für konkrete prominente Strategien bekannt. * Es wird eine neue Technik zur Feststellung der Stabilität von Queueing Strategien ohne Zeitnahme vorgestellt, die Aufschichtungskreise. Mit ihrer Hilfe können bekannte Stabilitätsbeweise prominenter Strategien vereinheitlicht werden und weitere Stabilitätsergebnisse erzielt werden. * Für die große Teilklasse distanzbasierter Queueing Strategien gelingt eine vollständige Klassifizierung aller 1-stabilen und universell stabilen Strategien

Foundations of Quantum Gravity : The Role of Principles Grounded in Empirical Reality

When attempting to assess the strengths and weaknesses of various principles in their potential role of guiding the formulation of a theory of quantum gravity, it is crucial to distinguish between principles which are strongly supported by empirical data - either directly or indirectly - and principles which instead (merely) rely heavily on theoretical arguments for their justification. These remarks are illustrated in terms of the current standard models of cosmology and particle physics, as well as their respective underlying theories, viz. general relativity and quantum (field) theory. It is argued that if history is to be of any guidance, the best chance to obtain the key structural features of a putative quantum gravity theory is by deducing them, in some form, from the appropriate empirical principles (analogous to the manner in which, say, the idea that gravitation is a curved spacetime phenomenon is arguably implied by the equivalence principle). It is subsequently argued that the appropriate empirical principles for quantum gravity should at least include (i) quantum nonlocality, (ii) irreducible indeterminacy, (iii) the thermodynamic arrow of time, (iv) homogeneity and isotropy of the observable universe on the largest scales. In each case, it is explained - when appropriate - how the principle in question could be implemented mathematically in a theory of quantum gravity, why it is considered to be of fundamental significance and also why contemporary accounts of it are insufficient.Comment: 21 pages. Some (mostly minor) corrections. Final published versio

Hidden Citations Obscure True Impact in Science

References, the mechanism scientists rely on to signal previous knowledge, lately have turned into widely used and misused measures of scientific impact. Yet, when a discovery becomes common knowledge, citations suffer from obliteration by incorporation. This leads to the concept of hidden citation, representing a clear textual credit to a discovery without a reference to the publication embodying it. Here, we rely on unsupervised interpretable machine learning applied to the full text of each paper to systematically identify hidden citations. We find that for influential discoveries hidden citations outnumber citation counts, emerging regardless of publishing venue and discipline. We show that the prevalence of hidden citations is not driven by citation counts, but rather by the degree of the discourse on the topic within the text of the manuscripts, indicating that the more discussed is a discovery, the less visible it is to standard bibliometric analysis. Hidden citations indicate that bibliometric measures offer a limited perspective on quantifying the true impact of a discovery, raising the need to extract knowledge from the full text of the scientific corpus

Zenonovi paradoksi i kvantni mikrosvet: šta govore aporije

The article considers new approaches to four of Zeno’s paradoxes: the Arrow, Achilles and the Tortoise, the Dichotomy, and the Stadium. The paradoxes are analyzed in the light of current research in the field of elementary particle physics and some promising directions in the development of the quantum gravity. Physical theories, provided with the necessary philosophical interpretation, are used in order to clarify Zeno’s paradoxes and to search for answers to them. The text shows that using modern approaches to solve the paradoxes is not effective, because the paradoxes become irrelevant when analyzed in the context of microworld physics, at very small scales. The main part of the paper is devoted to demonstrating this circum- stance – that the questions posed by the paradoxes are impossible to answer (at least in their classical interpretation). As a possible explanation, the article puts forward that in the formulation of the paradoxes, the properties of the macroworld and the microworld are mixed (which is historically justified, given the intuitive homogeneity of the large and the small, and the fact that non-classical physics – quantum mechanics – did not emerge until the twentieth century); that is, from the observation of large physical objects, a transition is made to the infinitely small in terms of discreteness and continuity. However, the principles of organization of space at very small scales are beginning to be clarified in general terms only now, and, perhaps, these principles may turn out to be quite far from the classical ideas about fundamental physical reality.U članku se razmatraju novi pristupi pitanju četiri Zenonova paradoksa: strela, Ahil i kornjača, Dihotomija i Stadion. Paradoksi su analizirani u svetlu aktuelnih istraživanja u oblasti fizike elementarnih čestica i nekih obećavajućih pravaca u razvoju kvantne gravitacije. Fizičke teorije, opremljene neophodnim filozofskim tumačenjem, koriste se kako bi se razjasnili Zenonovi paradoksi i tražili odgovori na njih. Tekst pokazuje da korišćenje savremenih pristupa za rešavanje paradoksa nije efikasno, jer paradoksi postaju irelevantni kada se analiziraju u kontekstu fizike mikrosveta, na veoma malim razmerama. Glavni deo rada posvećen je demonstraciji ove okolnosti – da je na pitanja koja postavljaju paradoksi nemoguće odgovoriti (barem u njihovoj klasičnoj interpretaciji). Kao moguće objašnjenje, u članku se navodi da se u formulisanju paradoksa mešaju svojstva makrosveta i mikrosveta (što je istorijski opravdano, s obzirom na intuitivnu homogenost velikog i malog i činjenicu da neklasična fizika – kvantna mehanika – pojavila se tek u dvadesetom veku); odnosno od posmatranja velikih fizičkih objekata prelazi se na beskonačno male u smislu diskretnosti i kontinuiteta. Međutim, principi organizacije prostora u veoma malim razmerama počinju da se uopšteno razjašnjavaju tek sada i, možda, ovi principi se mogu ispostaviti kao prilično strani klasičnim idejama o fundamentalnoj fizičkoj stvarnosti