Parameterized Rural Postman Problem

The Directed Rural Postman Problem (DRPP) can be formulated as follows: given a strongly connected directed multigraph $D=(V,A)$ with nonnegative integral weights on the arcs, a subset $R$ of $A$ and a nonnegative integer $\ell$, decide whether $D$ has a closed directed walk containing every arc of $R$ and of total weight at most $\ell$. Let $k$ be the number of weakly connected components in the the subgraph of $D$ induced by $R$. Sorge et al. (2012) ask whether the DRPP is fixed-parameter tractable (FPT) when parameterized by $k$, i.e., whether there is an algorithm of running time $O^*(f(k))$ where $f$ is a function of $k$ only and the $O^*$ notation suppresses polynomial factors. Sorge et al. (2012) note that this question is of significant practical relevance and has been open for more than thirty years. Using an algebraic approach, we prove that DRPP has a randomized algorithm of running time $O^*(2^k)$ when $\ell$ is bounded by a polynomial in the number of vertices in $D$. We also show that the same result holds for the undirected version of DRPP, where $D$ is a connected undirected multigraph

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 Boolean satisfiability problem (SAT), and 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 SAT problem admits a kernel of a particular size. This could be contrasted to the currently flourishing research program of determining the classical complexity of finite-domain CSP problems, where almost all non-trivial tractable classes have been identified with the help of algebraic properties. In this paper, we take an algebraic approach to the problem of characterizing the kernelization limits of NP-hard SAT and 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 CSP problem has a kernel with O(n) constraints if it can be embedded (via a domain extension) into a CSP problem which is preserved by a Maltsev operation. We also study extensions of this towards SAT and CSP problems with kernels with O(n^c) constraints, c>1, based on embeddings into CSP problems preserved by a k-edge operation, k > c. These results follow via a variant of the celebrated few subpowers algorithm. In the complementary direction, we give indication that the Maltsev condition might be a complete characterization of SAT problems 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 is included in co-NP/poly

Directed Multicut is W[1]-hard, Even for Four Terminal Pairs

We prove that Multicut in directed graphs, parameterized by the size of the cutset, is W[1]-hard and hence unlikely to be fixed-parameter tractable even if restricted to instances with only four terminal pairs. This negative result almost completely resolves one of the central open problems in the area of parameterized complexity of graph separation problems, posted originally by Marx and Razgon [SIAM J. Comput. 43(2):355-388 (2014)], leaving only the case of three terminal pairs open. Our gadget methodology allows us also to prove W[1]-hardness of the Steiner Orientation problem parameterized by the number of terminal pairs, resolving an open problem of Cygan, Kortsarz, and Nutov [SIAM J. Discrete Math. 27(3):1503-1513 (2013)].Comment: v2: Added almost tight ETH lower bound

Sparsification of SAT and CSP Problems via Tractable Extensions

Unlike polynomial kernelization in general, for which many non-trivial results and methods exist, only few non-trival algorithms are known for polynomial-time sparsification. Furthermore, excepting problems on restricted inputs (such as graph problems on planar graphs), most such results rely upon encoding the instance as a system of bounded-degree polynomial equations. In particular, for satisfiability (SAT) problems with a fixed constraint language Γ, every previously known result is captured by this approach; for several such problems, this is known to be tight. In this work, we investigate the limits of this approach—in particular, does it really cover all cases of non-trivial polynomial-time sparsification? We generalize the method using tools from the algebraic approach to constraint satisfaction problems (CSP). Every constraint that can be modelled via a system of linear equations, over some finite field F, also admits a finite domain extension to a tractable CSP with a Maltsev polymorphism; using known algorithms for Maltsev languages, we can show that every problem of the latter type admits a “basis” of O(n) constraints, which implies a linear sparsification for the original problem. This generalization appears to be strict; other special cases include constraints modelled via group equations over some finite group G. For sparsifications of polynomial but super-linear size, we consider two extensions of this. Most directly, we can capture systems of bounded-degree polynomial equations in a “lift-and-project” manner, by finding Maltsev extensions for constraints over c-tuples of variables, for a basis with O(nc) constraints. Additionally, we may use extensions with k-edge polymorphisms instead of requiring a Maltsev polymorphism. We also investigate characterizations of when such extensions exist. We give an infinite sequence of partial polymorphisms φ1, φ2, …which characterizes whether a language Γ has a Maltsev extension (of possibly infinite domain). In the complementary direction of proving lower bounds on kernelizability, we prove that for any language not preserved by φ1, the corresponding SAT problem does not admit a kernel of size O(n2−ε) for any ε > 0 unless the polynomial hierarchy collapses

Parameterized Two-Player Nash Equilibrium

We study the computation of Nash equilibria in a two-player normal form game from the perspective of parameterized complexity. Recent results proved hardness for a number of variants, when parameterized by the support size. We complement those results, by identifying three cases in which the problem becomes fixed-parameter tractable. These cases occur in the previously studied settings of sparse games and unbalanced games as well as in the newly considered case of locally bounded treewidth games that generalizes both these two cases

A New Randomized Algorithm to Approximate the Star Discrepancy Based on Threshold Accepting

We present a new algorithm for estimating the star discrepancy of arbitrary point sets. Similar to the algorithm for discrepancy approximation of Winker and Fang [SIAM J. Numer. Anal. 34 (1997), 2028{2042] it is based on the optimization algorithm threshold accepting. Our improvements include, amongst others, a non-uniform sampling strategy which is more suited for higher-dimensional inputs and additionally takes into account the topological characteristics of given point sets, and rounding steps which transform axis-parallel boxes, on which the discrepancy is to be tested, into critical test boxes. These critical test boxes provably yield higher discrepancy values, and contain the box that exhibits the maximum value of the local discrepancy. We provide comprehensive experiments to test the new algorithm. Our randomized algorithm computes the exact discrepancy frequently in all cases where this can be checked (i.e., where the exact discrepancy of the point set can be computed in feasible time). Most importantly, in higher dimension the new method behaves clearly better than all previously known methods

Alternative parameterizations of Metric Dimension

A set of vertices $W$ in a graph $G$ is called resolving if for any two distinct $x,y\in V(G)$, there is $v\in W$ such that ${\rm dist}_G(v,x)\neq{\rm dist}_G(v,y)$, where ${\rm dist}_G(u,v)$ denotes the length of a shortest path between $u$ and $v$ in the graph $G$. The metric dimension ${\rm md}(G)$ of $G$ is the minimum cardinality of a resolving set. The Metric Dimension problem, i.e. deciding whether ${\rm md}(G)\le k$, is NP-complete even for interval graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs) from the lens of parameterized complexity. The problem parameterized by $k$ was proved to be $W[2]$-hard by Hartung and Nichterlein (2013) and we study the dual parameterization, i.e., the problem of whether ${\rm md}(G)\le n- k,$ where $n$ is the order of $G$. We prove that the dual parameterization admits (a) a kernel with at most $3k^4$ vertices and (b) an algorithm of runtime $O^*(4^{k+o(k)}).$ Hartung and Nichterlein (2013) also observed that Metric Dimension is fixed-parameter tractable when parameterized by the vertex cover number $vc(G)$ of the input graph. We complement this observation by showing that it does not admit a polynomial kernel even when parameterized by $vc(G) + k$. Our reduction also gives evidence for non-existence of polynomial Turing kernels

Path-Contractions, Edge Deletions and Connectivity Preservation

We study several problems related to graph modification problems under connectivity constraints from the perspective of parameterized complexity: {\sc (Weighted) Biconnectivity Deletion}, where we are tasked with deleting~$k$ edges while preserving biconnectivity in an undirected graph, {\sc Vertex-deletion Preserving Strong Connectivity}, where we want to maintain strong connectivity of a digraph while deleting exactly~$k$ vertices, and {\sc Path-contraction Preserving Strong Connectivity}, in which the operation of path contraction on arcs is used instead. The parameterized tractability of this last problem was posed by Bang-Jensen and Yeo [DAM 2008] as an open question and we answer it here in the negative: both variants of preserving strong connectivity are $\sf W[1]$-hard. Preserving biconnectivity, on the other hand, turns out to be fixed parameter tractable and we provide a $2^{O(k\log k)} n^{O(1)}$-algorithm that solves {\sc Weighted Biconnectivity Deletion}. Further, we show that the unweighted case even admits a randomized polynomial kernel. All our results provide further interesting data points for the systematic study of connectivity-preservation constraints in the parameterized setting

On the Workflow Satisfiability Problem with Class-Independent Constraints for Hierarchical Organizations

A workflow specification defines a set of steps, a set of users, and an access control policy. The policy determines which steps a user is authorized to perform and imposes constraints on which sets of users can perform which sets of steps. The workflow satisfiability problem (WSP) is the problem of determining whether there exists an assignment of users to workflow steps that satisfies the policy. Given the computational hardness of WSP and its importance in the context of workflow management systems, it is important to develop algorithms that are as efficient as possible to solve WSP. In this article, we study the fixed-parameter tractability of WSP in the presence of class-independent constraints, which enable us to (1) model security requirements based on the groups to which users belong and (2) generalize the notion of a user-independent constraint. Class-independent constraints are defined in terms of equivalence relations over the set of users. We consider sets of nested equivalence relations because this enables us to model security requirements in hierarchical organizations. We prove that WSP is fixed-parameter tractable (FPT) for class-independent constraints defined over nested equivalence relations and develop an FPT algorithm to solve WSP instances incorporating such constraints. We perform experiments to evaluate the performance of our algorithm and compare it with that of SAT4J, an off-the-shelf pseudo-Boolean SAT solver. The results of these experiments demonstrate that our algorithm significantly outperforms SAT4J for many instances of WSP