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We develop a complexity theory for approximate real computations. We first produce a theory for exact computations but with condition numbers. The input size depends on a condition number, which is not assumed known by the machine. The…

Computational Complexity · Computer Science 2020-05-05 Gregorio Malajovich , Mike Shub

We demonstrate a polynomial approach to express the decision version of the directed Hamiltonian Cycle Problem (HCP), which is NP-Complete, as the Solvability of a Polynomial Equation with a constant number of variables, within a bounded…

Computational Complexity · Computer Science 2011-11-10 Deepak Chermakani

Contexts are terms with one `hole', i.e. a place in which we can substitute an argument. In context unification we are given an equation over terms with variables representing contexts and ask about the satisfiability of this equation.…

Logic in Computer Science · Computer Science 2013-11-11 Artur Jeż

Here we study the computational complexity of the constrained synchronization problem for the class of regular commutative constraint languages. Utilizing a vector representation of regular commutative constraint languages, we give a full…

Formal Languages and Automata Theory · Computer Science 2020-09-04 Stefan Hoffmann

We sketch a tentative proof of P-completeness for the $\beta$-convertibility problem on untyped planar (a.k.a. ordered or non-commutative) $\lambda$-terms.

Logic in Computer Science · Computer Science 2024-04-09 Anupam Das , Damiano Mazza , Lê Thành Dũng Nguyên , Noam Zeilberger

A set of stochastic matrices ${\cal P}$ is a consensus set if for every sequence of matrices $P(1), P(2), \ldots$ whose elements belong to ${\cal P}$ and every initial state $x(0)$, the sequence of states defined by $x(t) = P(t) P(t-1)…

Optimization and Control · Mathematics 2014-06-03 Vincent Blondel , Alex Olshevsky

Holzer and Holzer (Discrete Applied Mathematics 144(3):345--358, 2004) proved that the Tantrix(TM) rotation puzzle problem is NP-complete. They also showed that for infinite rotation puzzles, this problem becomes undecidable. We study the…

Computational Complexity · Computer Science 2008-06-09 Dorothea Baumeister , Joerg Rothe

In this paper we study the fine-grained complexity of finding exact and approximate solutions to problems in P. Our main contribution is showing reductions from exact to approximate solution for a host of such problems. As one (notable)…

Computational Complexity · Computer Science 2022-12-12 Lijie Chen , Shafi Goldwasser , Kaifeng Lyu , Guy N. Rothblum , Aviad Rubinstein

We extend the concept of polynomial time approximation algorithms to apply to problems for hierarchically specified graphs, many of which are PSPACE-complete. Assuming P != PSPACE, the existence or nonexistence of such efficient…

Combinatorics · Mathematics 2016-09-06 Madhav V. Marathe , Harry B. Hunt , S. S. Ravi

A discrete temporal constraint satisfaction problem is a constraint satisfaction problem (CSP) whose constraint language consists of relations that are first-order definable over $(\Bbb Z,<)$. Our main result says that every distance CSP is…

Logic · Mathematics 2016-04-27 Manuel Bodirsky , Barnaby Martin , Antoine Mottet

A partial complement of the graph $G$ is a graph obtained from $G$ by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph $G$ and graph class $\mathcal{G}$, is there a…

Computational Complexity · Computer Science 2020-06-11 Fedor V. Fomin , Petr A. Golovach , Torstein J. F. Strømme , Dimitrios M. Thilikos

The min-knapsack problem with compactness constraints extends the classical knapsack problem, in the case of ordered items, by introducing a restriction ensuring that they cannot be too far apart. This problem has applications in…

Optimization and Control · Mathematics 2025-04-28 Hubert Villuendas , Mathieu Besançon , Jérôme Malick

Determining whether two STRIPS planning instances are isomorphic is the simplest form of comparison between planning instances. It is also a particular case of the problem concerned with finding an isomorphism between a planning instance…

Artificial Intelligence · Computer Science 2024-06-25 Arnaud Lequen , Martin C. Cooper , Frédéric Maris

The problem of matrix completion and decomposition in the cone of positive semidefinite (PSD) matrices is a well-understood problem, with many important applications in areas such as linear algebra, optimization, and control theory. This…

Optimization and Control · Mathematics 2025-07-28 Ding Zhang , Axel Ringh , Li Qiu

In this note we show that pattern matching in permutations is polynomial time reducible to pattern matching in set partitions. In particular, pattern matching in set partitions is NP-Complete.

Combinatorics · Mathematics 2020-09-02 Thomas Grubb

The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces.…

Computational Geometry · Computer Science 2024-08-07 Mikkel Abrahamsen , Tillmann Miltzow , Nadja Seiferth

Equivalence checking is used to verify whether two programs produce equivalent outputs when given equivalent inputs. Research in this field mainly focused on improving equivalence checking accuracy and runtime performance. However, for…

Software Engineering · Computer Science 2024-09-04 Johann Glock , Josef Pichler , Martin Pinzger

Nonconvex optimization problems with an L1-constraint are ubiquitous, and are found in many application domains including: optimal control of hybrid systems, machine learning and statistics, and operations research. This paper shows that…

Optimization and Control · Mathematics 2017-09-27 Yonatan Mintz , Anil Aswani

Given $k$ collections of 2SAT clauses on the same set of variables $V$, can we find one assignment that satisfies a large fraction of clauses from each collection? We consider such simultaneous constraint satisfaction problems, and design…

Data Structures and Algorithms · Computer Science 2014-07-30 Amey Bhangale , Swastik Kopparty , Sushant Sachdeva

We study the complexity of PAC learning halfspaces in the presence of Massart noise. In this problem, we are given i.i.d. labeled examples $(\mathbf{x}, y) \in \mathbb{R}^N \times \{ \pm 1\}$, where the distribution of $\mathbf{x}$ is…

Machine Learning · Computer Science 2022-07-29 Ilias Diakonikolas , Daniel M. Kane , Pasin Manurangsi , Lisheng Ren