scholarly journals On the Kernelization of Global Constraints

2017 ◽  
Author(s):  
Clément Carbonnel ◽  
Emmanuel Hebrard
Keyword(s):  
Complexity Theory ◽  
Polynomial Time ◽  
Constraint Programming ◽  
Parameterized Complexity ◽  
Constraint Propagation ◽  
Decision Problems ◽  
Global Constraints ◽  
Theoretical Interest ◽  
Hard Decision

Kernelization is a powerful concept from parameterized complexity theory that captures (a certain idea of) efficient polynomial-time preprocessing for hard decision problems. However, exploiting this technique in the context of constraint programming is challenging. Building on recent results for the VertexCover constraint, we introduce novel "loss-less" kernelization variants that are tailored for constraint propagation. We showcase the theoretical interest of our ideas on two constraints, VertexCover and EdgeDominatingSet.

scholarly journals Constraint Games revisited

2017 ◽  
Author(s):  
Anthony Palmieri ◽  
Arnaud Lallouet
Keyword(s):  
Constraint Programming ◽  
Nash Equilibria ◽  
State Of The Art ◽  
Constraint Propagation ◽  
Global Constraints ◽  
Graphical Games ◽  
Previous State

Constraint Games are a recent framework proposed to model and solve static games where Constraint Programming is used to express players preferences. In this paper, we rethink their solving technique in terms of constraint propagation by considering players preferences as global constraints. It yields not only a more elegant but also a more efficient framework. Our new complete solver is faster than previous state-of-the-art and is able to find all pure Nash equilibria for some problems with 200 players. We also show that performances can greatly be improved for graphical games, allowing some games with 2000 players to be solved.

scholarly journals Polynomially Bounded Sequences and Polynomial Sequences

2015 ◽  
Vol 23 (3) ◽  
pp. 205-213
Author(s):  
Hiroyuki Okazaki ◽  
Yuichi Futa
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Turing Machine ◽  
Polynomial Time ◽  
Computation Time ◽  
Decision Problems ◽  
Computational Complexity Theory ◽  
Polynomial Sequences ◽  
Deterministic Turing Machine ◽  
Bounded Sequences

Abstract In this article, we formalize polynomially bounded sequences that plays an important role in computational complexity theory. Class P is a fundamental computational complexity class that contains all polynomial-time decision problems [11], [12]. It takes polynomially bounded amount of computation time to solve polynomial-time decision problems by the deterministic Turing machine. Moreover we formalize polynomial sequences [5].

scholarly journals Isomorphism, canonization, and definability for graphs of bounded rank width

2021 ◽  
Vol 64 (5) ◽  
pp. 98-105
Author(s):  
Martin Grohe ◽  
Daniel Neuen
Keyword(s):  
Fixed Point ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Descriptive Complexity ◽  
Graph Isomorphism Problem ◽  
Structural Graph ◽  
Graph Classes ◽  
Bounded Rank

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

scholarly journals Additive-error fine-grained quantum supremacy

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 329
Author(s):  
Tomoyuki Morimae ◽  
Suguru Tamaki
Keyword(s):  
Quantum Computing ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Boolean Circuits ◽  
Exponential Time ◽  
Fine Grained ◽  
Additive Error ◽  
Universal Quantum ◽  
The One ◽  
Computing Models

It is known that several sub-universal quantum computing models, such as the IQP model, the Boson sampling model, the one-clean qubit model, and the random circuit model, cannot be classically simulated in polynomial time under certain conjectures in classical complexity theory. Recently, these results have been improved to ``fine-grained" versions where even exponential-time classical simulations are excluded assuming certain classical fine-grained complexity conjectures. All these fine-grained results are, however, about the hardness of strong simulations or multiplicative-error sampling. It was open whether any fine-grained quantum supremacy result can be shown for a more realistic setup, namely, additive-error sampling. In this paper, we show the additive-error fine-grained quantum supremacy (under certain complexity assumptions). As examples, we consider the IQP model, a mixture of the IQP model and log-depth Boolean circuits, and Clifford+T circuits. Similar results should hold for other sub-universal models.

Chapter 17. Fixed-Parameter Tractability

2021 ◽  
Author(s):  
Marko Samer ◽  
Stefan Szeider
Keyword(s):  
Polynomial Time ◽  
Parameterized Complexity ◽  
Complexity Analysis ◽  
Fixed Parameter Tractability ◽  
Graph Representations ◽  
Fine Grained ◽  
Input Size ◽  
Backdoor Sets ◽  
Fixed Parameter ◽  
Problem Instances

Parameterized complexity is a new theoretical framework that considers, in addition to the overall input size, the effects on computational complexity of a secondary measurement, the parameter. This two-dimensional viewpoint allows a fine-grained complexity analysis that takes structural properties of problem instances into account. The central notion is “fixed-parameter tractability” which refers to solvability in polynomial time for each fixed value of the parameter such that the order of the polynomial time bound is independent of the parameter. This chapter presents main concepts and recent results on the parameterized complexity of the satisfiability problem and it outlines fundamental algorithmic ideas that arise in this context. Among the parameters considered are the size of backdoor sets with respect to various tractable base classes and the treewidth of graph representations of satisfiability instances.

The Meaning of System

2014 ◽  
Vol 1 (1) ◽  
pp. 21-34 ◽  
Author(s):  
Steen Leleur
Keyword(s):  
Systems Theory ◽  
Complexity Theory ◽  
Explanatory Power ◽  
Theoretical Interest ◽  
Core Concept ◽  
The Core ◽  
Systems Practice ◽  
Actual Meaning ◽  
Current Systems ◽  
Real World Problems

This article reviews the generic meaning of ‘system’ and complements more conventional system notions with a system perception based on recent complexity theory. With system as the core concept of systems theory, its actual meaning is not just of theoretical interest but is highly relevant also for systems practice. It is argued that complexity theory and thinking with reference to Luhmann a.o. ought to be recognised and paid attention to by the systems community. Overall, it is found that a complexity orientation may contribute to extend and enrich the explanatory power of current systems theory when used to complex real-world problems. As regards systems practice it is found that selective use and combination of five presented research approaches (functionalist, interpretive, emancipatory, postmodern and complexity) which function as different but complementing ‘epistemic lenses’ in a process described as constructive circularity, may strengthen the exploration and learning efforts in systems-based intervention.

scholarly journals Complexity-theoretic aspects of expanding cellular automata

2020 ◽  
Author(s):  
Augusto Modanese
Keyword(s):  
Cellular Automata ◽  
Polynomial Time ◽  
Time Complexity ◽  
Truth Table ◽  
Decision Problems ◽  
Turing Machines ◽  
One Dimensional ◽  
Alternative Characterization ◽  
Time Class ◽  
Theoretical Standpoint

Abstract The expanding cellular automata (XCA) variant of cellular automata is investigated and characterized from a complexity-theoretical standpoint. An XCA is a one-dimensional cellular automaton which can dynamically create new cells between existing ones. The respective polynomial-time complexity class is shown to coincide with $${\le _{tt}^p}(\textsf {NP})$$ ≤ tt p ( NP ) , that is, the class of decision problems polynomial-time truth-table reducible to problems in $$\textsf {NP}$$ NP . An alternative characterization based on a variant of non-deterministic Turing machines is also given. In addition, corollaries on select XCA variants are proven: XCAs with multiple accept and reject states are shown to be polynomial-time equivalent to the original XCA model. Finally, XCAs with alternative acceptance conditions are considered and classified in terms of $${\le _{tt}^p}(\textsf {NP})$$ ≤ tt p ( NP ) and the Turing machine polynomial-time class $$\textsf {P}$$ P .

scholarly journals Schedulers and redundancy for a class of constraint propagation rules

2005 ◽  
Vol 5 (4-5) ◽  
pp. 441-465 ◽  
Author(s):  
SEBASTIAN BRAND ◽  
KRZYSZTOF R. APT
Keyword(s):  
Constraint Programming ◽  
Constraint Propagation ◽  
Iteration Algorithm ◽  
Rule Based ◽  
Propagation Rules

We study here schedulers for a class of rules that naturally arise in the context of rule-based constraint programming. We systematically derive a scheduler for them from a generic iteration algorithm of Apt (2000). We apply this study to so-called membership rules of Apt and Monfroy (2001). This leads to an implementation that yields a considerably better performance for these rules than their execution as standard CHR rules. Finally, we show how redundant rules can be identified and how appropriately reduced sets of rules can be computed.

FUNCTIONAL PEARL Linear lambda calculus and PTIME-completeness

2004 ◽  
Vol 14 (6) ◽  
pp. 623-633 ◽  
Author(s):  
HARRY G. MAIRSON
Keyword(s):  
Polynomial Time ◽  
Linear Logic ◽  
Lambda Calculus ◽  
Type Inference ◽  
Decision Problems ◽  
Logical Interpretation

We give transparent proofs of the PTIME-completeness of two decision problems for terms in the λ-calculus. The first is a reproof of the theorem that type inference for the simply-typed λ-calculus is PTIME-complete. Our proof is interesting because it uses no more than the standard combinators Church knew of some 70 years ago, in which the terms are linear affine – each bound variable occurs at most once. We then derive a modification of Church's coding of Booleans that is linear, where each bound variable occurs exactly once. A consequence of this construction is that any interpreter for linear λ-calculus requires polynomial time. The logical interpretation of this consequence is that the problem of normalizing proofnets for multiplicative linear logic (MLL) is also PTIME-complete.

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