Parameterized Compilation Lower Bounds for Restricted CNF-formulas

We show unconditional parameterized lower bounds in the area of knowledge compilation, more specifically on the size of circuits in decomposable negation normal form (DNNF) that encode CNF-formulas restricted by several graph width measures. In particular, we show that - there are CNF formulas of size $n$ and modular incidence treewidth $k$ whose smallest DNNF-encoding has size $n^{\Omega(k)}$, and - there are CNF formulas of size $n$ and incidence neighborhood diversity $k$ whose smallest DNNF-encoding has size $n^{\Omega(\sqrt{k})}$. These results complement recent upper bounds for compiling CNF into DNNF and strengthen---quantitatively and qualitatively---known conditional low\-er bounds for cliquewidth. Moreover, they show that, unlike for many graph problems, the parameters considered here behave significantly differently from treewidth

Algorithms for Coloring Reconfiguration Under Recolorability Constraints

Coloring reconfiguration is one of the most well-studied reconfiguration problems. In the problem, we are given two (vertex-)colorings of a graph using at most k colors, and asked to determine whether there exists a transformation between them by recoloring only a single vertex at a time, while maintaining a k-coloring throughout. It is known that this problem is solvable in linear time for any graph if k = 4. In this paper, we further investigate the problem from the viewpoint of recolorability constraints, which forbid some pairs of colors to be recolored directly. More specifically, the recolorability constraint is given in terms of an undirected graph R such that each node in R corresponds to a color, and each edge in R represents a pair of colors that can be recolored directly. In this paper, we give a linear-time algorithm to solve the problem under such a recolorability constraint if R is of maximum degree at most two. In addition, we show that the minimum number of recoloring steps required for a desired transformation can be computed in linear time for a yes-instance. We note that our results generalize the known positive ones for coloring reconfiguration

Sunflowers Meet Sparsity: A Linear-Vertex Kernel for Weighted Clique-Packing on Sparse Graphs

We study the kernelization complexity of the Weighted H-Packing problem on sparse graphs. For a fixed connected graph H, in the Weighted H-Packing problem the input is a graph G, a vertex-weight function w: V (G) → N, and positive integers k, t. The question is whether there exist k vertex-disjoint subgraphs H 1, ⋯, H k of G such that H i is isomorphic to H for each i ∈ [k] and the total weight of these k · |V (H)| vertices is at least t. It is known that the (unweighted) H-Packing problem admits a kernel with O(k |V (H)|-1) vertices on general graphs, and a linear kernel on planar graphs and graphs of bounded genus. In this work, we focus on case that H is a clique on h ≥ 3 vertices (which captures Triangle Packing) and present a linear-vertex kernel for Weighted Kh-Packing on graphs of bounded expansion, along with a kernel with O(k 1+ϵ) vertices on nowhere-dense graphs for all ϵ &gt; 0. To obtain these results, we combine two powerful ingredients in a novel way: the Erdos-Rado Sunflower lemma and the theory of sparsity.</p

Sunflowers Meet Sparsity: A Linear-Vertex Kernel for Weighted Clique-Packing on Sparse Graphs

We study the kernelization complexity of the Weighted H-Packing problem on sparse graphs. For a fixed connected graph H, in the Weighted H-Packing problem the input is a graph G, a vertex-weight function w: V (G) → N, and positive integers k, t. The question is whether there exist k vertex-disjoint subgraphs H 1, ⋯, H k of G such that H i is isomorphic to H for each i ∈ [k] and the total weight of these k · |V (H)| vertices is at least t. It is known that the (unweighted) H-Packing problem admits a kernel with O(k |V (H)|-1) vertices on general graphs, and a linear kernel on planar graphs and graphs of bounded genus. In this work, we focus on case that H is a clique on h ≥ 3 vertices (which captures Triangle Packing) and present a linear-vertex kernel for Weighted Kh-Packing on graphs of bounded expansion, along with a kernel with O(k 1+ϵ) vertices on nowhere-dense graphs for all ϵ &gt; 0. To obtain these results, we combine two powerful ingredients in a novel way: the Erdos-Rado Sunflower lemma and the theory of sparsity.</p

Search-Space Reduction via Essential Vertices

We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution

Resiliency Policies in Access Control Revisited

International audienceResiliency is a relatively new topic in the context of access control. Informally, it refers to the extent to which a multi-user computer system, subject to an authorization policy, is able to continue functioning if a number of authorized users are unavailable. Several interesting problems connected to resiliency were introduced by Li, Wang and Tripunitara [13], many of which were found to be intractable. In this paper, we show that these resiliency problems have unexpected connections with the workflow satisfiability problem (WSP). In particular, we show that an instance of the resiliency checking problem (RCP) may be reduced to an instance of WSP. We then demonstrate that recent advances in our understanding of WSP enable us to develop fixed-parameter tractable algorithms for RCP. Moreover, these algorithms are likely to be useful in practice, given recent experimental work demonstrating the advantages of bespoke algorithms to solve WSP. We also generalize RCP in several different ways, showing in each case how to adapt the reduction to WSP. Li et al also showed that the coexistence of resiliency policies and static separation-of-duty policies gives rise to further interesting questions. We show how our reduction of RCP to WSP may be extended to solve these problems as well and establish that they are also fixed-parameter tractable

Reconfigurations of Combinatorial Problems: Graph Colouring and Hamiltonian Cycle

We explore algorithmic aspects of two known combinatorial problems, Graph Colouring and Hamiltonian Cycle, by examining properties of their solution space. One can model the set of solutions of a combinatorial problem $P$ by the solution graph $R(P)$, where vertices are solutions of $P$ and there is an edge between two vertices, when the two corresponding solutions satisfy an adjacency reconfiguration rule. For example, we can define the reconfiguration rule for graph colouring to be that two solutions are adjacent when they differ in colour in exactly one vertex. The exploration of the properties of the solution graph $R(P)$ can give rise to interesting questions. The connectivity of $R(P)$ is the most prominent question in this research area. This is reasonable, since the main motivation for modelling combinatorial solutions as a graph is to be able to transform one into the other in a stepwise fashion, by following paths between solutions in the graph. Connectivity questions can be made binary, that is expressed as decision problems which accept a 'yes' or 'no' answer. For example, given two specific solutions, is there a path between them? Is the graph of solutions $R(P)$ connected? In this thesis, we first show that the diameter of the solution graph $R_{l}(G)$ of vertex $l$-colourings of k-colourable chordal and chordal bipartite graphs $G$ is $O(n^2)$, where $l > k$ and n is the number of vertices of $G$. Then, we formulate a decision problem on the connectivity of the graph colouring solution graph, where we allow extra colours to be used in order to enforce a path between two colourings with no path between them. We give some results for general instances and we also explore what kind of graphs pose a challenge to determine the complexity of the problem for general instances. Finally, we give a linear algorithm which decides whether there is a path between two solutions of the Hamiltonian Cycle Problem for graphs of maximum degree five, and thus providing insights towards the complexity classification of the decision problem

On the Complexity of Reconfiguration of Clique, Cluster Vertex Deletion, and Dominating Set

A graph problem P is a vertex-subset problem if feasible solutions for P consist of subsets of the vertices of a graph G. The st-connectivity problem for a vertex-subset problem P takes as input two feasible solutions S_s and S_t, and determines if there is a sequence of recon figuration steps that can be applied to transform S_s into S_t, such that each step results in a feasible solution of P of size bounded by k and each step is a vertex addition or deletion. For most NP-complete problems, this problem has been shown to be PSPACE-complete, while for some problems in P, this problem could be either in P or PSPACE-complete. However, knowing the complexity of a decision problem does not directly imply the complexity of its st-connectivity problem. Therefore, it is natural to ask whether we can fi nd a connection between the complexity of a decision problem and its st-connectivity problem when restricted to graph classes. This question motivated us to study the st-connectivity problems Clique Reconfiguration and Dominating Set Reconfiguration, whose decision problems' complexity for restricted graph classes is extensively studied, to get a better understanding of the boundary between polynomial-time solvable and intractable instances of these reconfi guration problems. Furthermore, we study the Cluster Vertex Deletion Reconfiguration problem, a problem whose decision problem is related to the Clique problem, to fi nd whether there is a connection between the complexity of this problem and the Clique Reconfiguration problem. Following are the main contributions of this thesis. First, we show that the Clique Re- configuration problem is linear-time solvable for paths, trees, bipartite graphs, chordal graphs, and cographs. Then, we prove that the Cluster Vertex Deletion Reconfiguration problem is linear-time solvable for paths and trees, and that it is NP-hard on bipartite graphs, and PSPACE-complete in general. Finally, we determine that the Dominating Set Reconfiguration problem is linear-time solvable for paths, cographs, trees, and interval graphs. Furthermore, we show that the problem is PSPACE-complete for general graphs, bipartite graphs, and split graphs