830 research outputs found
Expanding the expressive power of Monadic Second-Order logic on restricted graph classes
We combine integer linear programming and recent advances in Monadic
Second-Order model checking to obtain two new algorithmic meta-theorems for
graphs of bounded vertex-cover. The first shows that cardMSO1, an extension of
the well-known Monadic Second-Order logic by the addition of cardinality
constraints, can be solved in FPT time parameterized by vertex cover. The
second meta-theorem shows that the MSO partitioning problems introduced by Rao
can also be solved in FPT time with the same parameter. The significance of our
contribution stems from the fact that these formalisms can describe problems
which are W[1]-hard and even NP-hard on graphs of bounded tree-width.
Additionally, our algorithms have only an elementary dependence on the
parameter and formula. We also show that both results are easily extended from
vertex cover to neighborhood diversity.Comment: Accepted for IWOCA 201
Solving Hard Stable Matching Problems Involving Groups of Similar Agents
Many important stable matching problems are known to be NP-hard, even when
strong restrictions are placed on the input. In this paper we seek to identify
structural properties of instances of stable matching problems which will allow
us to design efficient algorithms using elementary techniques. We focus on the
setting in which all agents involved in some matching problem can be
partitioned into k different types, where the type of an agent determines his
or her preferences, and agents have preferences over types (which may be
refined by more detailed preferences within a single type). This situation
would arise in practice if agents form preferences solely based on some small
collection of agents' attributes. We also consider a generalisation in which
each agent may consider some small collection of other agents to be
exceptional, and rank these in a way that is not consistent with their types;
this could happen in practice if agents have prior contact with a small number
of candidates. We show that (for the case without exceptions), several
well-studied NP-hard stable matching problems including Max SMTI (that of
finding the maximum cardinality stable matching in an instance of stable
marriage with ties and incomplete lists) belong to the parameterised complexity
class FPT when parameterised by the number of different types of agents needed
to describe the instance. For Max SMTI this tractability result can be extended
to the setting in which each agent promotes at most one `exceptional' candidate
to the top of his/her list (when preferences within types are not refined), but
the problem remains NP-hard if preference lists can contain two or more
exceptions and the exceptional candidates can be placed anywhere in the
preference lists, even if the number of types is bounded by a constant.Comment: Results on SMTI appear in proceedings of WINE 2018; Section 6
contains work in progres
The Algebraic Intersection Type Unification Problem
The algebraic intersection type unification problem is an important component
in proof search related to several natural decision problems in intersection
type systems. It is unknown and remains open whether the algebraic intersection
type unification problem is decidable. We give the first nontrivial lower bound
for the problem by showing (our main result) that it is exponential time hard.
Furthermore, we show that this holds even under rank 1 solutions (substitutions
whose codomains are restricted to contain rank 1 types). In addition, we
provide a fixed-parameter intractability result for intersection type matching
(one-sided unification), which is known to be NP-complete.
We place the algebraic intersection type unification problem in the context
of unification theory. The equational theory of intersection types can be
presented as an algebraic theory with an ACI (associative, commutative, and
idempotent) operator (intersection type) combined with distributivity
properties with respect to a second operator (function type). Although the
problem is algebraically natural and interesting, it appears to occupy a
hitherto unstudied place in the theory of unification, and our investigation of
the problem suggests that new methods are required to understand the problem.
Thus, for the lower bound proof, we were not able to reduce from known results
in ACI-unification theory and use game-theoretic methods for two-player tiling
games
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A modular, efficient formalisation of real algebraic numbers
This paper presents a construction of the real algebraic numbers with executable arithmetic operations in Isabelle/HOL. Instead of verified resultants, arithmetic operations on real algebraic numbers are based on a decision procedure to decide the sign of a bivariate polynomial (with rational coefficients) at a real algebraic point. The modular design allows the safe use of fast external code. This work can be the basis for decision procedures that rely on real algebraic numbers.The CSC Cambridge International Scholarship is generously funding Wenda Li’s Ph.D. course.This is the author accepted manuscript. The final version is available from the Association for Computing Machinery via http://dx.doi.org/10.1145/2854065.285407
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