22 research outputs found
Holographic Algorithm with Matchgates Is Universal for Planar CSP Over Boolean Domain
We prove a complexity classification theorem that classifies all counting
constraint satisfaction problems (CSP) over Boolean variables into exactly
three categories: (1) Polynomial-time tractable; (2) P-hard for general
instances, but solvable in polynomial-time over planar graphs; and (3)
P-hard over planar graphs. The classification applies to all sets of local,
not necessarily symmetric, constraint functions on Boolean variables that take
complex values. It is shown that Valiant's holographic algorithm with
matchgates is a universal strategy for all problems in category (2).Comment: 94 page
On the Complexity of Holant Problems
In this article we survey recent developments on the complexity of Holant problems. We discuss three different aspects of Holant problems: the decision version, exact counting, and approximate counting
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
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
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 as a powerful new tool in hardness proofs for
counting problems.Comment: 61 pages, 16 figures. An extended abstract appears in SODA 202
Clifford Gates in the Holant Framework
We show that the Clifford gates and stabilizer circuits in the quantum
computing literature, which admit efficient classical simulation, are
equivalent to affine signatures under a unitary condition. The latter is a
known class of tractable functions under the Holant framework.Comment: 14 page
Planar 3-way Edge Perfect Matching Leads to A Holant Dichotomy
We prove a complexity dichotomy theorem for a class of Holant problems on
planar 3-regular bipartite graphs. The complexity dichotomy states that for
every weighted constraint function defining the problem (the weights can
even be negative), the problem is either computable in polynomial time if
satisfies a tractability criterion, or \#P-hard otherwise. One particular
problem in this problem space is a long-standing open problem of Moore and
Robson on counting Cubic Planar X3C. The dichotomy resolves this problem by
showing that it is \numP-hard. Our proof relies on the machinery of signature
theory developed in the study of Holant problems. An essential ingredient in
our proof of the main dichotomy theorem is a pure graph-theoretic result:
Excepting some trivial cases, every 3-regular plane graph has a planar 3-way
edge perfect matching. The proof technique of this graph-theoretic result is a
combination of algebraic and combinatorial methods.
The P-time tractability criterion of the dichotomy is explicit. Other than
the known classes of tractable constraint functions (degenerate, affine,
product type, matchgates-transformable) we also identify a new infinite set of
P-time computable planar Holant problems; however, its tractability is not by a
direct holographic transformation to matchgates, but by a combination of this
method and a global argument. The complexity dichotomy states that everything
else in this Holant class is \#P-hard.Comment: arXiv admin note: text overlap with arXiv:2110.0117
The Complexity of Contracting Planar Tensor Network
Tensor networks have been an important concept and technique in many research
areas such as quantum computation and machine learning. We study the complexity
of evaluating the value of a tensor network. This is also called contracting
the tensor network. In this article, we focus on computing the value of a
planar tensor network where every tensor specified at a vertex is a Boolean
symmetric function. We design two planar gadgets to obtain a sub-exponential
time algorithm. The key is to remove high degree vertices while essentially not
changing the size of the tensor network. The algorithm runs in time
. Furthermore, we use a counting version of the
Sparsification Lemma to prove a matching lower bound
assuming \#ETH holds