New Planar P-time Computable Six-Vertex Models and a Complete Complexity Classification

We discover new P-time computable six-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. We further prove that there are no more: Together, they exhaust all P-time computable six-vertex models on planar graphs, assuming #P is not P. This leads to the following exact complexity classification: For every parameter setting in ${\mathbb C}$ for the six-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) #P-hard for general graphs but computable in P-time for planar graphs, or (3) #P-hard even for planar graphs. The classification has an explicit criterion. The new P-time cases in (2) provably cannot be subsumed by Kasteleyn's algorithm. They are obtained by a non-local connection to #CSP, defined in terms of a "loop space". This is the first substantive advance toward a planar Holant classification with not necessarily symmetric constraints. We introduce M\"obius transformation on ${\mathbb C}$ as a powerful new tool in hardness proofs for counting problems.Comment: 61 pages, 16 figures. An extended abstract appears in SODA 202

The simple, little and slow things count : on parameterized counting complexity

In this thesis, we study the parameterized complexity of counting problems, as introduced by Flum and Grohe. This area mainly involves questions of the following kind: On inputs x with a parameter k, can we solve a given counting problem in time f(k)*|x|^c for a function f that depends only on k? In the positive case, we call the problem fixed-parameter tractable (fpt). Otherwise, we try to prove its #W[1]-hardness, which is the parameterized analogue of #P-hardness. We introduce a general technique that bridges parameterized counting complexity and the so-called Holant framework. We then apply this technique to the problem of counting perfect matchings (or equivalently, the permanent) subject to structural parameters of the input graph G: On the algorithmic side, we introduce a new tractable structural parameter, namely, the minimal size of an excluded single-crossing minor of G. We complement this by showing that counting perfect matchings is #W[1]-hard when parameterized by the size of an arbitrary excluded minor. Then we turn our attention to counting general subgraphs H other than perfect matchings in a host graph G. Instead of imposing structural parameters on G, we parameterize by the size of H, giving rise to the problems #Sub(C) for fixed graph classes C: For inputs H and G with H in C, we wish to count H-copies in G. Here, C could be the class of matchings, cycles, paths, or any other recursively enumerable class. We give a full dichotomy for these problems: Either #Sub(C) has a polynomial-time algorithm or it is #W[1]-complete. Assuming that FPT and #W[1] do not coincide, we can thus precisely identify the graph classes C for which the subgraph counting problem #Sub(C) admits polynomial-time algorithms. Furthermore, we obtain an unexpected application of our extensions to the Holant framework: We show that, given two unweighted graphs, it is C=P-complete to decide whether they have the same number of perfect matchings. Finally, we prove conditional lower bounds for counting problems under the counting exponential-time hypothesis #ETH. This hypothesis, introduced by Dell et al., asserts that the satisfying assignments to n-variable formulas in 3-CNF cannot be counted in time 2^o(n). Building upon this, we introduce a general technique that allows to derive tight lower bounds for other counting problems, such as counting perfect matchings, the Tutte polynomial, and the matching polynomial.Die vorliegende Arbeit befasst sich mit der parametrisierten Komplexität von Zählproblemen, einem von Flum und Grohe gegründeten Gebiet, in welchem Fragen der folgenden Art betrachtet werden: Können gegebene Probleme auf Eingaben x mit Parameter k in Zeit f(k)*|x|^c gelöst werden, wobei f eine Funktion ist, die nur von k abhängt? Im positiven Falle bezeichnen wir das Problem als parametrisierbar (FPT). Andernfalls versuchen wir typischerweise, dessen #W[1]-Härte zu beweisen - diese lässt sich vereinfachend als ein parametrisiertes Äquivalent der #P-Härte auffassen. Wir führen zunächst eine allgemeine Technik ein, welche die parametrisierte Zählkomplexität mit dem sogenannten Holant-Rahmenwerk verbindet. Anschließend setzen wir diese zum Zählen perfekter Paarungen (oder äquivalent, zur Auswertung der Permanente) unter strukturellen Parametern des Eingabegraphens G ein: Wir zeigen, dass das Zählen perfekter Paarungen parametrisierbar ist durch die minimale Größe eines ausgeschlossenen Minors von G, der höchstens eine Kreuzung besitzt. Dieses algorithmische Resultat komplementieren wir durch die #W[1]-Härte des Zählens perfekter Paarungen, wenn die minimale Größe eines beliebigen ausgeschlossenen Minors als Parameter betrachtet wird. Anschließend widmen wir uns dem Zählen beliebiger Subgraphen H in Graphen G. Anstelle von strukturellen Parametern betrachten wir die Größe von H als Parameter und erhalten hierdurch die Probleme #Sub(C) für feste Graphklassen C: Auf Eingaben H und G mit H in C gilt es, die H-Kopien in G zu zählen. Hierbei kann C die Klasse der Paarungen, Zyklen, Pfade, oder eine beliebige andere Klasse von Graphen darstellen. Wir zeigen eine vollständige Dichotomie für diese Probleme: Das Problem #Sub(C) ist entweder in P oder #W[1]-hart. Unter der gängigen Annahme, dass FPT und #W[1] nicht zusammenfallen, erhalten wir somit eine vollständige Klassifikation der Polynomialzeit-lösbaren Probleme #Sub(C). Weiterhin erhalten wir eine unerwartete Anwendung unserer Erweiterungen des Holant-Rahmenwerks: Wir zeigen die C=P-Vollständigkeit der Frage, ob die Anzahlen perfekter Paarungen in zwei gegebenen ungewichteten Graphen übereinstimmen. Schlussendlich zeigen wir bedingte untere Schranken für Zählprobleme unter der Zählversion der Exponentialzeithypothese #ETH, eingeführt durch Dell et al. Diese postuliert, dass die erfüllenden Belegungen in 3-KNF-Formeln mit n Variablen nicht in Zeit 2^o(n) gezählt werden können. Darauf aufbauend führen wir eine allgemeine Technik ein, die es ermöglicht, scharfe untere Schranken für andere Zählprobleme zu erhalten: Dies umfasst das Zählen perfekter Paarungen, das Tutte-Polynom und das Paarungs-Polynom

A Holant Dichotomy: Is the FKT Algorithm Universal?

We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. This dichotomy is specifically to answer the question: Is the FKT algorithm under a holographic transformation a \emph{universal} strategy to obtain polynomial-time algorithms for problems over planar graphs that are intractable in general? This dichotomy is a culmination of previous ones, including those for Spin Systems, Holant, and #CSP. A recurring theme has been that a holographic reduction to FKT is a universal strategy. Surprisingly, for planar Holant, we discover new planar tractable problems that are not expressible by a holographic reduction to FKT. In previous work, an important tool was a dichotomy for #CSP^d, which denotes #CSP where every variable appears a multiple of d times. However its proof violates planarity. We prove a dichotomy for planar #CSP^2. We apply this planar #CSP^2 dichotomy in the proof of the planar Holant dichotomy. As a special case of our new planar tractable problems, counting perfect matchings (#PM) over k-uniform hypergraphs is polynomial-time computable when the incidence graph is planar and k >= 5. The same problem is #P-hard when k=3 or k=4, which is also a consequence of our dichotomy. When k=2, it becomes #PM over planar graphs and is tractable again. More generally, over hypergraphs with specified hyperedge sizes and the same planarity assumption, #PM is polynomial-time computable if the greatest common divisor of all hyperedge sizes is at least 5.Comment: 128 pages, 36 figure

The Computational Power of Non-interacting Particles

Shortened abstract: In this thesis, I study two restricted models of quantum computing related to free identical particles. Free fermions correspond to a set of two-qubit gates known as matchgates. Matchgates are classically simulable when acting on nearest neighbors on a path, but universal for quantum computing when acting on distant qubits or when SWAP gates are available. I generalize these results in two ways. First, I show that SWAP is only one in a large family of gates that uplift matchgates to quantum universality. In fact, I show that the set of all matchgates plus any nonmatchgate parity-preserving two-qubit gate is universal, and interpret this fact in terms of local invariants of two-qubit gates. Second, I investigate the power of matchgates in arbitrary connectivity graphs, showing they are universal on any connected graph other than a path or a cycle, and classically simulable on a cycle. I also prove the same dichotomy for the XY interaction. Free bosons give rise to a model known as BosonSampling. BosonSampling consists of (i) preparing a Fock state of n photons, (ii) interfering these photons in an m-mode linear interferometer, and (iii) measuring the output in the Fock basis. Sampling approximately from the resulting distribution should be classically hard, under reasonable complexity assumptions. Here I show that exact BosonSampling remains hard even if the linear-optical circuit has constant depth. I also report several experiments where three-photon interference was observed in integrated interferometers of various sizes, providing some of the first implementations of BosonSampling in this regime. The experiments also focus on the bosonic bunching behavior and on validation of BosonSampling devices. This thesis contains descriptions of the numerical analyses done on the experimental data, omitted from the corresponding publications.Comment: PhD Thesis, defended at Universidade Federal Fluminense on March 2014. Final version, 208 pages. New results in Chapter 5 correspond to arXiv:1106.1863, arXiv:1207.2126, and arXiv:1308.1463. New results in Chapter 6 correspond to arXiv:1212.2783, arXiv:1305.3188, arXiv:1311.1622 and arXiv:1412.678

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