arXiv subjects

David A. Cohen

Publications and source records attributed to David A. Cohen.

11 recordsLinked to original sources

arXiv2026

Binary constraints on one additional variable can create exponential ascents for local search

Local search in combinatorial optimisation can be viewed as an uphill climb on a corresponding fitness landscape, where the assignments visited by a strict local search follow an ascent. This hill-climbing is sometimes surprisingly efficient, but not always. Since fitness landscapes can be succinctly represented by Boolean valued constraint satisfaction problems (VCSPs), we ask: what properties of VCSPs ensure that all ascents are polynomial? Or alternatively, what are the "simplest" VCSPs with exponential ascents? Prior examples of VCSPs with exponential ascents were built up as a chain of gadgets of constraints. Here we investigate what happens for simpler starlike graphs. To establish lower bounds on the \emph{longest} ascent, we construct: (1) a star of binary constraints with a quadratic longest ascent, (2) a binary VCSP of treedepth 3 on 4n + 1 Boolean variables (made by gluing 2n triangles of constraints at a common centre variable) with an exponential longest ascent of length $10\cdot2^n - 9$, and (3) starlike VCSP of logarithmic vertex cover number with an exponential longest ascent. For steepest ascent, we prove (4) a tight bound of 2(n - 1) for VCSPs with star constraint graphs, and construct (5) a starlike tree VCSP with treedepth 3 with a quadratic steepest ascent. Finally, we prove upper bounds on the shortest ascent from any initial assignment: (6) linear for VCSPs of treedepth 3; and (7) $O(2^k (n - k))$ for VCSPs of vertex cover number k. Together (2) and (6) establish treedepth 3, and (3) and (7) establish logarithmic vertex cover number, as the first structural graph parameters for which the longest and shortest ascents can be exponentially separated. We discuss the consequences of our results for the parameterized complexity of local search.

cs.DM↗

arXiv2020

Representing fitness landscapes by valued constraints to understand the complexity of local search

Local search is widely used to solve combinatorial optimisation problems and to model biological evolution, but the performance of local search algorithms on different kinds of fitness landscapes is poorly understood. Here we consider how fitness landscapes can be represented using valued constraints, and investigate what the structure of such representations reveals about the complexity of local search. First, we show that for fitness landscapes representable by binary Boolean valued constraints there is a minimal necessary constraint graph that can be easily computed. Second, we consider landscapes as equivalent if they allow the same (improving) local search moves; we show that a minimal constraint graph still exists, but is NP-hard to compute. We then develop several techniques to bound the length of any sequence of local search moves. We show that such a bound can be obtained from the numerical values of the constraints in the representation, and show how this bound may be tightened by considering equivalent representations. In the binary Boolean case, we prove that a degree 2 or tree-structured constraint graph gives a quadratic bound on the number of improving moves made by any local search; hence, any landscape that can be represented by such a model will be tractable for any form of local search. Finally, we build two families of examples to show that the conditions in our tractability results are essential. With domain size three, even just a path of binary constraints can model a landscape with an exponentially long sequence of improving moves. With a treewidth-two constraint graph, even with a maximum degree of three, binary Boolean constraints can model a landscape with an exponentially long sequence of improving moves.

cs.DS↗

arXiv2019

Steepest ascent can be exponential in bounded treewidth problems

We investigate the complexity of local search based on steepest ascent. We show that even when all variables have domains of size two and the underlying constraint graph of variable interactions has bounded treewidth (in our construction, treewidth 7), there are fitness landscapes for which an exponential number of steps may be required to reach a local optimum. This is an improvement on prior recursive constructions of long steepest ascents, which we prove to need constraint graphs of unbounded treewidth.

cs.DM↗

arXiv2018

Binary Constraint Satisfaction Problems Defined by Excluded Topological Minors

The binary Constraint Satisfaction Problem (CSP) is to decide whether there exists an assignment to a set of variables which satisfies specified constraints between pairs of variables. A binary CSP instance can be presented as a labelled graph encoding both the forms of the constraints and where they are imposed. We consider subproblems defined by restricting the allowed form of this graph. One type of restriction that has previously been considered is to forbid certain specified substructures (patterns). This captures some tractable classes of the CSP, but does not capture classes defined by language restrictions, or the well-known structural property of acyclicity. In this paper we extend the notion of pattern and introduce the notion of a topological minor of a binary CSP instance. By forbidding a finite set of patterns from occurring as topological minors we obtain a compact mechanism for expressing novel tractable subproblems of the binary CSP, including new generalisations of the class of acyclic instances. Forbidding a finite set of patterns as topological minors also captures all other tractable structural restrictions of the binary CSP. Moreover, we show that several patterns give rise to tractable subproblems if forbidden as topological minors but not if forbidden as sub-patterns. Finally, we introduce the idea of augmented patterns that allows for the identification of more tractable classes, including all language restrictions of the binary CSP.

cs.CC↗

arXiv2018

On Singleton Arc Consistency for CSPs Defined by Monotone Patterns

Singleton arc consistency is an important type of local consistency which has been recently shown to solve all constraint satisfaction problems (CSPs) over constraint languages of bounded width. We aim to characterise all classes of CSPs defined by a forbidden pattern that are solved by singleton arc consistency and closed under removing constraints. We identify five new patterns whose absence ensures solvability by singleton arc consistency, four of which are provably maximal and three of which generalise 2-SAT. Combined with simple counter-examples for other patterns, we make significant progress towards a complete classification.

cs.CC↗

arXiv2017

Binarisation for Valued Constraint Satisfaction Problems

We study methods for transforming valued constraint satisfaction problems (VCSPs) to binary VCSPs. First, we show that the standard dual encoding preserves many aspects of the algebraic properties that capture the computational complexity of VCSPs. Second, we extend the reduction of CSPs to binary CSPs described by Bulin et al. [LMCS'15] to VCSPs. This reduction establishes that VCSPs over a fixed valued constraint language are polynomial-time equivalent to Minimum-Cost Homomorphism Problems over a fixed digraph.

cs.DM↗

arXiv2015

Variable and value elimination in binary constraint satisfaction via forbidden patterns

Variable or value elimination in a constraint satisfaction problem (CSP) can be used in preprocessing or during search to reduce search space size. A variable elimination rule (value elimination rule) allows the polynomial-time identification of certain variables (domain elements) whose elimination, without the introduction of extra compensatory constraints, does not affect the satisfiability of an instance. We show that there are essentially just four variable elimination rules and three value elimination rules defined by forbidding generic sub-instances, known as irreducible existential patterns, in arc-consistent CSP instances. One of the variable elimination rules is the already-known Broken Triangle Property, whereas the other three are novel. The three value elimination rules can all be seen as strict generalisations of neighbourhood substitution.

cs.CC↗

arXiv2014

The tractability of CSP classes defined by forbidden patterns

The constraint satisfaction problem (CSP) is a general problem central to computer science and artificial intelligence. Although the CSP is NP-hard in general, considerable effort has been spent on identifying tractable subclasses. The main two approaches consider structural properties (restrictions on the hypergraph of constraint scopes) and relational properties (restrictions on the language of constraint relations). Recently, some authors have considered hybrid properties that restrict the constraint hypergraph and the relations simultaneously. Our key contribution is the novel concept of a CSP pattern and classes of problems defined by forbidden patterns (which can be viewed as forbidding generic subproblems). We describe the theoretical framework which can be used to reason about classes of problems defined by forbidden patterns. We show that this framework generalises relational properties and allows us to capture known hybrid tractable classes. Although we are not close to obtaining a dichotomy concerning the tractability of general forbidden patterns, we are able to make some progress in a special case: classes of problems that arise when we can only forbid binary negative patterns (generic subproblems in which only inconsistent tuples are specified). In this case we are able to characterise very large classes of tractable and NP-hard forbidden patterns. This leaves the complexity of just one case unresolved and we conjecture that this last case is tractable.

cs.AI↗

arXiv2013

Tractable Combinations of Global Constraints

We study the complexity of constraint satisfaction problems involving global constraints, i.e., special-purpose constraints provided by a solver and represented implicitly by a parametrised algorithm. Such constraints are widely used; indeed, they are one of the key reasons for the success of constraint programming in solving real-world problems. Previous work has focused on the development of efficient propagators for individual constraints. In this paper, we identify a new tractable class of constraint problems involving global constraints of unbounded arity. To do so, we combine structural restrictions with the observation that some important types of global constraint do not distinguish between large classes of equivalent solutions.

cs.AI↗

arXiv2012

An Algebraic Theory of Complexity for Discrete Optimisation

Discrete optimisation problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimised can be expressed as a sum of functions of a restricted form. Here we present a unifying theory of complexity for problems of this kind. We show that the complexity of a finite-domain discrete optimisation problem is determined by certain algebraic properties of the objective function, which we call weighted polymorphisms. We define a Galois connection between sets of rational-valued functions and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised. These results provide a new approach to studying the complexity of discrete optimisation. We use this approach to identify certain maximal tractable subproblems of the general problem, and hence derive a complete classification of complexity for the Boolean case.

cs.CC↗

arXiv2008

The Expressive Power of Binary Submodular Functions

It has previously been an open problem whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This problem has been considered within several different contexts in computer science, including computer vision, artificial intelligence, and pseudo-Boolean optimisation. Using a connection between the expressive power of valued constraints and certain algebraic properties of functions, we answer this question negatively. Our results have several corollaries. First, we characterise precisely which submodular functions of arity 4 can be expressed by binary submodular functions. Next, we identify a novel class of submodular functions of arbitrary arities which can be expressed by binary submodular functions, and therefore minimised efficiently using a so-called expressibility reduction to the Min-Cut problem. More importantly, our results imply limitations on this kind of reduction and establish for the first time that it cannot be used in general to minimise arbitrary submodular functions. Finally, we refute a conjecture of Promislow and Young on the structure of the extreme rays of the cone of Boolean submodular functions.

cs.DM↗