9,932 research outputs found
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
Counting Subgraphs in Somewhere Dense Graphs
We study the problems of counting copies and induced copies of a small pattern graph H in a large host graph G. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns H. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time f(H)?|G|^O(1) for some computable function f. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes ? as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis:
- Counting k-matchings in a graph G ? ? is fixed-parameter tractable if and only if ? is nowhere dense.
- Counting k-independent sets in a graph G ? ? is fixed-parameter tractable if and only if ? is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if ? is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting k-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in F-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting k-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19).
At the same time our proofs are much simpler: using structural characterisations of somewhere dense graphs, we show that a colourful version of a recent breakthrough technique for analysing pattern counting problems (Curticapean, Dell, Marx; STOC 17) applies to any subgraph-closed somewhere dense class of graphs, yielding a unified view of our current understanding of the complexity of subgraph counting
Counting Homomorphisms to Trees Modulo a Prime
Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article we study the complexity of #_pHomsToH, the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p. Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo 2, however, the influence of the structure of H on the tractability was shown to persist, which yields similar dichotomies.
Our main result states that for every tree H and every prime p the problem #_pHomsToH is either polynomial time computable or #_pP-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph H when counting modulo 2. In contrast to previous results on modular counting, the tractable cases of #_pHomsToH are essentially the same for all values of the modulo when H is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime p. These results are the first suggesting that such dichotomies hold not only for the one-bit functions of the modulo 2 case but also for the modular counting functions of all primes p
Counting Subgraphs in Somewhere Dense Graphs
We study the problems of counting copies and induced copies of a small
pattern graph in a large host graph . Recent work fully classified the
complexity of those problems according to structural restrictions on the
patterns . In this work, we address the more challenging task of analysing
the complexity for restricted patterns and restricted hosts. Specifically we
ask which families of allowed patterns and hosts imply fixed-parameter
tractability, i.e., the existence of an algorithm running in time for some computable function . Our main results present
exhaustive and explicit complexity classifications for families that satisfy
natural closure properties. Among others, we identify the problems of counting
small matchings and independent sets in subgraph-closed graph classes
as our central objects of study and establish the following crisp
dichotomies as consequences of the Exponential Time Hypothesis: (1) Counting
-matchings in a graph is fixed-parameter tractable if and
only if is nowhere dense. (2) Counting -independent sets in a
graph is fixed-parameter tractable if and only if
is nowhere dense. Moreover, we obtain almost tight conditional
lower bounds if is somewhere dense, i.e., not nowhere dense.
These base cases of our classifications subsume a wide variety of previous
results on the matching and independent set problem, such as counting
-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in
-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs
(Bressan, Roth; FOCS 21), as well as counting -independent sets in bipartite
graphs (Curticapean et al.; Algorithmica 19).Comment: 35 pages, 3 figures, 4 tables, abstract shortened due to ArXiv
requirement
Fine-grained dichotomies for the Tutte plane and Boolean #CSP
Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of
evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard
almost everywhere, and the remaining points admit polynomial-time algorithms.
Dell, Husfeldt, and Wahl\'en [9] and Husfeldt and Taslaman [12], in combination
with Curticapean [7], extended the #P-hardness results to tight lower bounds
under the counting exponential time hypothesis #ETH, with the exception of the
line , which was left open. We complete the dichotomy theorem for the
Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs
of a given -vertex graph cannot be determined in time unless
#ETH fails.
Another dichotomy theorem we strengthen is the one of Creignou and Hermann
[6] for counting the number of satisfying assignments to a constraint
satisfaction problem instance over the Boolean domain. We prove that all
#P-hard cases are also hard under #ETH. The main ingredient is to prove that
the number of independent sets in bipartite graphs with vertices cannot be
computed in time unless #ETH fails. In order to prove our results,
we use the block interpolation idea by Curticapean [7] and transfer it to
systems of linear equations that might not directly correspond to
interpolation.Comment: 16 pages, 1 figur
Answering Conjunctive Queries under Updates
We consider the task of enumerating and counting answers to -ary
conjunctive queries against relational databases that may be updated by
inserting or deleting tuples. We exhibit a new notion of q-hierarchical
conjunctive queries and show that these can be maintained efficiently in the
following sense. During a linear time preprocessing phase, we can build a data
structure that enables constant delay enumeration of the query results; and
when the database is updated, we can update the data structure and restart the
enumeration phase within constant time. For the special case of self-join free
conjunctive queries we obtain a dichotomy: if a query is not q-hierarchical,
then query enumeration with sublinear delay and sublinear update time
(and arbitrary preprocessing time) is impossible.
For answering Boolean conjunctive queries and for the more general problem of
counting the number of solutions of k-ary queries we obtain complete
dichotomies: if the query's homomorphic core is q-hierarchical, then size of
the the query result can be computed in linear time and maintained with
constant update time. Otherwise, the size of the query result cannot be
maintained with sublinear update time. All our lower bounds rely on the
OMv-conjecture, a conjecture on the hardness of online matrix-vector
multiplication that has recently emerged in the field of fine-grained
complexity to characterise the hardness of dynamic problems. The lower bound
for the counting problem additionally relies on the orthogonal vectors
conjecture, which in turn is implied by the strong exponential time hypothesis.
By sublinear we mean for some
, where is the size of the active domain of the current
database
Counting the learnable functions of structured data
Cover's function counting theorem is a milestone in the theory of artificial
neural networks. It provides an answer to the fundamental question of
determining how many binary assignments (dichotomies) of points in
dimensions can be linearly realized. Regrettably, it has proved hard to extend
the same approach to more advanced problems than the classification of points.
In particular, an emerging necessity is to find methods to deal with structured
data, and specifically with non-pointlike patterns. A prominent case is that of
invariant recognition, whereby identification of a stimulus is insensitive to
irrelevant transformations on the inputs (such as rotations or changes in
perspective in an image). An object is therefore represented by an extended
perceptual manifold, consisting of inputs that are classified similarly. Here,
we develop a function counting theory for structured data of this kind, by
extending Cover's combinatorial technique, and we derive analytical expressions
for the average number of dichotomies of generically correlated sets of
patterns. As an application, we obtain a closed formula for the capacity of a
binary classifier trained to distinguish general polytopes of any dimension.
These results may help extend our theoretical understanding of generalization,
feature extraction, and invariant object recognition by neural networks
A full dichotomy for Holant<sup>c</sup>, inspired by quantum computation
Holant problems are a family of counting problems parameterised by sets of
algebraic-complex valued constraint functions, and defined on graphs. They
arise from the theory of holographic algorithms, which was originally inspired
by concepts from quantum computation. Here, we employ quantum information
theory to explain existing results about holant problems in a concise way and
to derive two new dichotomies: one for a new family of problems, which we call
Holant, and, building on this, a full dichotomy for Holant. These two
families of holant problems assume the availability of certain unary constraint
functions -- the two pinning functions in the case of Holant, and four
functions in the case of Holant -- and allow arbitrary sets of
algebraic-complex valued constraint functions otherwise. The dichotomy for
Holant also applies when inputs are restricted to instances defined on
planar graphs. In proving these complexity classifications, we derive an
original result about entangled quantum states.Comment: 57 pages, combines edited versions of arXiv:1702.00767 and
arXiv:1704.05798 with some new result
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