In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing $W[1]$-hardness proofs for these problems, since XNLP-hardness implies $W[t]$-hardness for all $t$. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space.
In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, ($q$-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth.
We show that the $b$-Coloring problem is complete for the class XNLP when parameterized by the pathwidth of the input graph. Besides determining the precise parameterized complexity of this problem, this implies that b-Coloring parameterized by pathwidth is $W[t]$-hard for all $t$, and resolves the parameterized complexity of $b$-Coloring parameterized by treewidth.
We present completeness results for inference in Bayesian networks with respect to two different parameterizations, namely the number of variables and the topological vertex separation number. For this we introduce the parameterized complexity classes $\mathsf{W[1]PP}$ and $\mathsf{XLPP}$, which relate to $\mathsf{W[1]}$ and $\mathsf{XNLP}$ respectively as $\mathsf{PP}$ does to $\mathsf{NP}$. The second parameter is intended as a natural translation of the notion of pathwidth to the case of directed acyclic graphs, and as such it is a stronger parameter than the more commonly considered treewi…
We show that some natural problems that are XNLP-hard (which implies W[t]-hardness for all t) when parameterized by pathwidth or treewidth, become FPT when parameterized by stable gonality, a novel graph parameter based on optimal maps from graphs to trees. The problems we consider are classical flow and orientation problems, such as Undirected Flow with Lower Bounds (which is strongly NP-complete, as shown by Itai), Minimum Maximum Outdegree (for which W[1]-hardness for treewidth was proven by Szeider), and capacitated optimization problems such as Capacitated (Red-Blue) Dominating Set (for w…
The concept of space-bounded computability has become significantly important in handling vast data sets on memory-limited computing devices. To replenish the existing short list of NL-complete problems whose instance sizes are dictated by log-space size parameters, we propose new additions obtained directly from natural parameterizations of three typical NP-complete problems -- the vertex cover problem, the exact cover by 3-sets problem, and the 3-dimensional matching problem. With appropriate restrictions imposed on their instances, the proposed decision problems parameterized by appropriate…
We define a theory of parameterized algebraic complexity classes in analogy to parameterized Boolean counting classes. We define the classes VFPT and VW[t], which mirror the Boolean counting classes #FPT and #W[t], and define appropriate reductions and completeness notions. Our main contribution is the VW[1]-completeness proof of the parameterized clique family. This proof is far more complicated than in the Boolean world. It requires some new concepts like composition theorems for bounded exponential sums and Boolean-arithmetic formulas. In addition, we also look at two polynomials linked to …
We examine a parameterized complexity class for randomized computation where only the error bound and not the full runtime is allowed to depend more than polynomially on the parameter, based on a proposal by Kwisthout in [15,16]. We prove that this class, for which we propose the shorthand name PPPT, has a robust definition and is in fact equal to the intersection of the classes paraBPP and PP. This result is accompanied by a Cook-style proof of completeness for the corresponding promise class (under a suitable notion of reduction) for parameterized approximation versions of the inference prob…
We study a variant of Min Cost Flow in which the flow needs to be connected. Specifically, in the Connected Flow problem one is given a directed graph $G$, along with a set of demand vertices $D \subseteq V(G)$ with demands $\mathsf{dem}: D \rightarrow \mathbb{N}$, and costs and capacities for each edge. The goal is to find a minimum cost flow that satisfies the demands, respects the capacities and induces a (strongly) connected subgraph. This generalizes previously studied problems like the (Many Visits) TSP. We study the parameterized complexity of Connected Flow parameterized by $|D|$, the …
A strong geodetic set of a graph~$G=(V,E)$ is a vertex set~$S \subseteq V(G)$ in which it is possible to cover all the remaining vertices of~$V(G) \setminus S$ by assigning a unique shortest path between each vertex pair of~$S$. In the Strong Geodetic problem (SG) a graph~$G$ and a positive integer~$k$ are given as input and one has to decide whether~$G$ has a strong geodetic set of cardinality at most~$k$. This problem is known to be NP-hard for general graphs. In this work we introduce the Strong Geodetic Recognition problem (SGR), which consists in determining whether even a given vertex se…
The parameterized complexity of counting minimum cuts stands as a natural question because Ball and Provan showed its #P-completeness. For any undirected graph $G=(V,E)$ and two disjoint sets of its vertices $S,T$, we design a fixed-parameter tractable algorithm which counts minimum edge $(S,T)$-cuts parameterized by their size $p$. Our algorithm operates on a transformed graph instance. This transformation, called drainage, reveals a collection of at most $n=\left| V \right|$ successive minimum $(S,T)$-cuts $Z_i$. We prove that any minimum $(S,T)$-cut $X$ contains edges of at least one cut $Z…
Given a graph property $Φ$, the problem $\#\mathsf{IndSub}(Φ)$ asks, on input a graph $G$ and a positive integer $k$, to compute the number of induced subgraphs of size $k$ in $G$ that satisfy $Φ$. The search for explicit criteria on $Φ$ ensuring that $\#\mathsf{IndSub}(Φ)$ is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some par…
We investigate the parameterized complexity of the recognition problem for the proper $H$-graphs. The $H$-graphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph $H$, and the properness means that the containment relationship between the representations of the vertices is forbidden. The class of $H$-graphs was introduced as a natural (parameterized) generalization of interval and circular-arc graphs by Biró, Hujter, and Tuza in 1992, and the proper $H$-graphs were introduced by Chaplick et al. in WADS 2019 as a generalization of proper interval and circular-a…
The splitting number of a graph $G=(V,E)$ is the minimum number of vertex splits required to turn $G$ into a planar graph, where a vertex split removes a vertex $v \in V$, introduces two new vertices $v_1, v_2$, and distributes the edges formerly incident to $v$ among its two split copies $v_1, v_2$. The splitting number problem is known to be NP-complete. In this paper we shift focus to the splitting number of graph drawings in $\mathbb R^2$, where the new vertices resulting from vertex splits can be re-embedded into the existing drawing of the remaining graph. We first provide a non-uniform …
On the Intractability of the Minimum Distance Problem for Regular LDPC Codes
School of Mathematics, Shandong University（山东大学数学学院）；Academy of Mathematics and Systems Science, CAS, University of Chinese Academy of Sciences（中国科学院数学与系统科学研究院，中国科学院大学）；Hangzhou Research Center, Huawei Technologies Co. Ltd.（华为技术有限公司杭州研究中心）
The problem of matching a query string to a directed graph, whose vertices are labeled by strings, has application in different fields, from data mining to computational biology. Several variants of the problem have been considered, depending on the fact that the match is exact or approximate and, in this latter case, which edit operations are considered and where are allowed. In this paper we present results on the complexity of the approximate matching problem, where edit operations are symbol substitutions and are allowed only on the graph labels or both on the graph labels and the query st…
