Holographic Algorithms Beyond Matchgates

Holographic algorithms introduced by Valiant are composed of two ingredients: matchgates, which are gadgets realizing local constraint functions by weighted planar perfect matchings, and holographic reductions, which show equivalences among problems with different descriptions via certain basis transformations. In this paper, we replace matchgates in the paradigm above by the affine type and the product type constraint functions, which are known to be tractable in general (not necessarily planar) graphs. More specifically, we present polynomial-time algorithms to decide if a given counting problem has a holographic reduction to another problem defined by the affine or product-type functions. Our algorithms also find a holographic transformation when one exists. We further present polynomial-time algorithms of the same decision and search problems for symmetric functions, where the complexity is measured in terms of the (exponentially more) succinct representations. The algorithm for the symmetric case also shows that the recent dichotomy theorem for Holant problems with symmetric constraints is efficiently decidable. Our proof techniques are mainly algebraic, e.g., using stabilizers and orbits of group actions.Comment: Inf. Comput., to appear. Author accepted manuscrip

Complexity dichotomies for approximations of counting problems

Αυτή η διπλωματική αποτελεί μια επισκόπηση θεωρημάτων διχοτομίας για υπολογιστικά προβλήματα, και ειδικότερα προβλήματα μέτρησης. Θεώρημα διχοτομίας στην υπολογιστική πολυπλοκότητα είναι ένας πλήρης χαρασκτηρισμός των μελών μιας κλάσης προβλημάτων, σε υπολογιστικά δύσκολα και υπολογιστικά εύκολα, χωρίς να υπάρχουν προβλήματα ενδιάμεσης πολυπλοκότητας στην κλάση αυτή. Λόγω του θεωρήματος του Ladner, δεν μπορούμε να έχουμε διχοτομία για ολόκληρες τις κλάσεις NP και #P, παρόλα αυτά υπάρχουν μεγάλες υποκλάσεις της NP (#P) για τις οποίες ισχύουν θεωρήματα διχοτομίας. Συνεχίζουμε με την εκδοχή απόφασης του προβλήματος ικανοποίησης περιορισμών (CSP), μία κλάση προβλήμάτων της NP στην οποία δεν εφαρμόζεται το θεώρημα του Ladner. Δείχνουμε τα θεωρήματα διχοτομίας που υπάρχουν για ειδικές περιπτώσεις του CSP. Στη συνέχεια επικεντρωνόμαστε στα προβλήματα μέτρησης παρουσιάζοντας τα παρακάτω μοντέλα: Ομομορφισμοί γράφων, μετρητικό πρόβλημα ικανοποίησης περιορισμών (#CSP), και προβλήματα Holant. Αναφέρουμε τα θεωρήματα διχοτομίας που γνωρίζουμε γι' αυτά. Στο τελευταίο και κύριο κεφάλαιο, χαλαρώνουμε την απαίτηση ακριβών υπολογισμών, και αρκούμαστε στην προσέγγιση των προβλημάτων. Παρουσιάζουμε τα μέχρι σήμερα γνωστά θεωρήματα κατάταξης για το #CSP. Πολλά ερωτήματα στην περιοχή παραμένουν ανοιχτά. Το παράρτημα είναι μια εισαγωγή στους ολογραφικούς αλγορίθμους, μία πρόσφατη αλγοριθμική τεχνική για την εύρεση πολυωνυμικών αλγορίθμων (ακριβείς υπολογισμοί) σε προβλήματα μέτρησης.This thesis is a survey of dichotomy theorems for computational problems, focusing in counting problems. A dichotomy theorem in computational complexity, is a complete classification of the members of a class of problems, in computationally easy and computationally hard, with the set of problems of intermediate complexity being empty. Due to Ladner's theorem we cannot find a dichotomy theorem for the whole classes NP and #P, however there are large subclasses of NP (#P), that model many "natural" problems, for which dichotomy theorems exist. We continue with the decision version of constraint satisfaction problems (CSP), a class of problems in NP, for which Ladner's theorem doesn't apply. We obtain a dichotomy theorem for some special cases of CSP. We then focus on counting problems presenting the following frameworks: graph homomorphisms, counting constraint satisfaction (#CSP) and Holant problems; we provide the known dichotomies for these frameworks. In the last and main chapter of this thesis we relax the requirement of exact computation, and settle in approximating the problems. We present the known cassification theorems for cases of #CSP. Many questions in terms of approximate counting problems remain open. The appendix introduces a recent technique for obtaining exact polynomial-time algorithms for counting problems, namely the holographic algorithms

Holographic Algorithms Beyond Matchgates

Holographic algorithms based on matchgates were introduced by Valiant. These algorithms run in polynomial-time and are intrinsically for planar problems. We introduce two new families of holographic algorithms, which work over general, i.e., not necessarily planar, graphs. The two underlying families of constraint functions are of the affine and product types. These play the role of Kasteleyn’s algorithm for counting planar perfect matchings. The new algorithms are obtained by transforming a problem to one of these two families by holographic reductions. We present a polynomial-time algorithm to decide if a given counting problem has a holographic algorithm using these constraint families. When the constraints are symmetric, we give a polynomial-time decision procedure in the size of the succinct presentation of symmetric constraint functions. This procedure shows that the recent dichotomy theorem for Holant problems with symmetric constraints is polynomial-time decidable

Aeronautical Engineering: A special bibliography with indexes, supplement 46, July 1974

This special bibliography lists 374 reports, articles, and other documents introduced into the NASA scientific and technical information system in June 1974

Technology 2000, volume 1

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Aeronautical engineering: A continuing bibliography with indexes (supplement 315)

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Research and technology highlights, 1993

This report contains highlights of the major accomplishments and applications that have been made by Langley researchers and by our university and industry colleagues during the past year. The highlights illustrate both the broad range of the research and technology activities supported by NASA Langley Research Center and the contributions of this work toward maintaining United States leadership in aeronautics and space research. This report also describes some of the Center's most important research and testing facilities

Aeronautical Engineering: A continuing bibliography, supplement 96

This bibliography lists 448 reports, articles, and other documents introduced into the NASA scientific and technical information system in April 1978