Victor Lagerkvist

Weak bases of Boolean co-clones

Abstract Universal algebra has proven to be a useful tool in the study of constraint satisfaction problems (CSP) since the complexity, up to logspace reductions, is determined by the clone of the constraint language. But two CSPs corresponding to the same clone may still differ substantially with respect to worst-case time complexity, which makes clones ill-suited when comparing running times of CSP problems. In this article we instead consider an algebra where each clone splits into an interval of strong partial clones such that a strong partial clone corresponds to the CSPs that are solvable within the same O ( c n ) bound. We investigate these intervals and give relational descriptions, weak bases, of the largest elements. They have a highly regular form and are in many cases easily relatable to the smallest members in the intervals, which suggests that the lattice of strong partial clones has a simpler structure than the lattice of partial clones.

Strong partial clones and the time complexity of SAT problems

Abstract Improving exact exponential-time algorithms for NP-complete problems is an expanding research area. Unfortunately, general methods for comparing the complexity of such problems are sorely lacking. In this article we study the complexity of SAT ( S ) with reductions increasing the amount of variables by a constant (CV-reductions) or a constant factor (LV-reductions). Using clone theory we obtain a partial order ≤ on languages such that SAT ( S ) is CV-reducible to SAT ( S ′ ) if S ≤ S ′ . With this ordering we identify the computationally easiest NP-complete SAT ( S ) problem ( SAT ( { R } ) ), which is strictly easier than 1-in-3-SAT. We determine many other languages in ≤ and bound their complexity in relation to SAT ( { R } ) . Using LV-reductions we prove that the exponential-time hypothesis is false if and only if all SAT ( S ) problems are subexponential. This is extended to cover degree-bounded SAT ( S ) problems. Hence, using clone theory, we obtain a solid understanding of the complexity of SAT ( S ) with CV- and LV-reductions.

Polynomially Closed Co-clones

Two well-studied closure operators for relations are based on primitive positive (p.p.) definitions and quantifier free p.p. definitions. The latter do however have limited expressiveness and the corresponding lattice of strong partial clones is uncountable. We consider implementations allowing polynomially many existentially quantified variables and obtain a dichotomy for co-clones where such implementations are enough to implement any relation and prove (1) that all remaining co-clones contain relations requiring a superpolynomial amount of quantified variables and (2) that the strong partial clones corresponding to two of these co-clones are of infinite order whenever the set of invariant relations can be finitely generated.

Blowing holes in various aspects of computational problems, with applications to constraint satisfaction

We consider methods for constructing NP-intermediate problems under the assumption that P i¾ź NP. We generalize Ladners original method for obtaining NP-intermediate problems by using parameters with various characteristics. In particular, this generalization allows us to obtain new insights concerning the complexity of CSP problems. We begin by fully characterizing the problems that admit NP-intermediate subproblems for a broad and natural class of parameterizations, and extend the result further such that structural CSP restrictions based on parameters that are hard to compute such as tree-width are covered. Hereby we generalize a result by Grohe on width parameters and NP-intermediate problems. For studying certain classes of problems, including CSPs parameterized by constraint languages, we consider more powerful parameterizations. First, we identify a new method for obtaining constraint languages Γ such that CSPΓ are NP-intermediate. The sets Γ can have very different properties compared to previous constructions by, for instance, Bodirsky & Grohe and provides insights into the algebraic approach for studying the complexity of infinite-domain CSPs. Second, we prove that the propositional abduction problem parameterized by constraint languages admits NP-intermediate problems. This settles an open question posed by Nordh & Zanuttini.

The power of primitive positive definitions with polynomially many variables

Two well-studied closure operators for relations are based on existentially quantified conjunctive formulas, primitive positive (p.p.) definitions, and primitive positive formulas without existenti ...

Upper and lower bounds on the time complexity of infinite-domain CSPs

The constraint satisfaction problem (CSP) is a widely studied problem with numerous applications in computer science. For infinite-domain CSPs, there are many results separating tractable and NP-hard cases while upper bounds on the time complexity of hard cases are virtually unexplored. Hence, we initiate a study of the worst-case time cmplexity of such CSPs. We analyse backtracking algorithms and show that they can be improved by exploiting sparsification. We present even faster algorithms based on enumerating finite structures. Last, we prove non-trivial lower bounds applicable to many interesting CSPs, under the assumption that the strong exponential-time hypothesis is true.

Relating the Time Complexity of Optimization Problems in Light of the Exponential-Time Hypothesis

Obtaining lower bounds for NP-hard problems has for a long time been an active area of research. Recent algebraic techniques introduced by Jonsson et al. (SODA 2013) show that the time complexity of the parameterized SAT(·) problem correlates to the lattice of strong partial clones. With this ordering they isolated a relation R such that SAT(R) can be solved at least as fast as any other NP-hard SAT(·) problem. In this paper we extend this method and show that such languages also exist for the max ones problem (Max-Ones(Γ)) and the Boolean valued constraint satisfaction problem over finite-valued constraint languages (VCSP(Δ)). With the help of these languages we relate Max-Ones and VCSP to the exponential time hypothesis in several different ways.

Kernelization of Constraint Satisfaction Problems: A Study Through Universal Algebra

A kernelization algorithm for a computational problem is a procedure which compresses an instance into an equivalent instance whose size is bounded with respect to a complexity parameter. For the constraint satisfaction problem (CSP), there exist many results concerning upper and lower bounds for kernelizability of specific problems, but it is safe to say that we lack general methods to determine whether a given problem admits a kernel of a particular size. In this paper, we take an algebraic approach to the problem of characterizing the kernelization limits of NP-hard CSP problems, parameterized by the number of variables. Our main focus is on problems admitting linear kernels, as has, somewhat surprisingly, previously been shown to exist. We show that a finite-domain CSP problem has a kernel with O(n) constraints if it can be embedded (via a domain extension) into a CSP which is preserved by a Maltsev operation. This result utilise a variant of the simple algorithm for Maltsev constraints. In the complementary direction, we give indication that the Maltsev condition might be a complete characterization for Boolean CSPs with linear kernels, by showing that an algebraic condition that is shared by all problems with a Maltsev embedding is also necessary for the existence of a linear kernel unless NP \(\subseteq \) co-NP/poly.

Time Complexity of Constraint Satisfaction via Universal Algebra

The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study the worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP complete and solvable in O(c^n) time.

A Dichotomy Theorem for the Inverse Satisfiability Problem

The inverse satisfiability problem over a set of Boolean relations Gamma (Inv-SAT(Gamma)) is the computational decision problem of, given a set of models R, deciding whether there exists a SAT(Gamma) instance with R as its set of models. This problem is co-NP-complete in general and a dichotomy theorem for finite Γ containing the constant Boolean relations was obtained by Kavvadias and Sideri. In this paper we remove the latter condition and prove that Inv-SAT(Gamma) is always either tractable or co-NP-complete for all finite sets of relations Gamma, thus solving a problem open since 1998. Very little of the techniques used by Kavvadias and Sideri are applicable and we have to turn to more recently developed algebraic approaches based on partial polymorphisms. We also consider the case when Γ is infinite, where the situation differs markedly from the case of SAT. More precisely, we show that there exists infinite Gamma such that Inv-SAT(Gamma) is tractable even though there exists finite Delta is subset of Gamma such that Inv-SAT(Delta) is co-NP-complete.