For the vertex selection problem $(σ,ρ)$-DomSet one is given two fixed sets $σ$ and $ρ$ of integers and the task is to decide whether we can select vertices of the input graph such that, for every selected vertex, the number of selected neighbors is in $σ$ and, for every unselected vertex, the number of selected neighbors is in $ρ$ [Telle, Nord. J. Comp. 1994]. This framework covers many fundamental graph problems such as Independent Set and Dominating Set. We significantly extend the recent result by Focke et al. [SODA 2023] to investigate the case when $σ$ and $ρ$ are two (potentially different) residue classes modulo $m\ge 2$. We study the problem parameterized by treewidth and present an algorithm that solves in time $m^{tw} \cdot n^{O(1)}$ the decision, minimization and maximization version of the problem. This significantly improves upon the known algorithms where for the case $m \ge 3$ not even an explicit running time is known. We complement our algorithm by providing matching lower bounds which state that there is no $(m-ε)^{pw} \cdot n^{O(1)}$-time algorithm parameterized by pathwidth $pw$, unless SETH fails. For $m = 2$, we extend these bounds to the minimization version as the decision version is efficiently solvable.

Abstract In the general AntiFactor problem, a graph G and, for every vertex v of G, a set $$X_v\subseteq {\mathbb {N}}$$ X v ⊆ N of forbidden degrees is given. The task is to find a set S of edges such that the degree of v in S is not in the set $$X_v$$ X v . Standard techniques (dynamic programming plus fast convolution) can be used to show that if M is the largest forbidden degree, then the problem can be solved in time $$(M+2)^{{\operatorname {tw}}}\cdot n^{{\mathcal {O}}(1)}$$ ( M + 2 ) tw · n O ( 1 ) if a tree decomposition of width $${\operatorname {tw}}$$ tw is given. However, significantly faster algorithms are possible if the sets $$X_v$$ X v are sparse: our main algorithmic result shows that if every vertex has at most $$x$$ x forbidden degrees (we call this special case AntiFactor x), then the problem can be solved in time $$(x+1)^{{\mathcal {O}}({\operatorname {tw}})}\cdot n^{{\mathcal {O}}(1)}$$ ( x + 1 ) O ( tw ) · n O ( 1 ) . That is, AntiFactor x is fixed-parameter tractable parameterized by treewidth $${\operatorname {tw}}$$ tw and the maximum number $$x$$ x of excluded degrees. Our algorithm uses the technique of representative sets, which can be generalized to the optimization version, but (as expected) not to the counting version of the problem. In fact, we show that #AntiFactor 1 is already # $$[1]$$ [ 1 ] -hard parameterized by the width of the given decomposition. Moreover, we show that, unlike for the decision version, the standard dynamic programming algorithm is essentially optimal for the counting version. Formally, for a fixed nonempty set $$X$$ X , we denote by $$X$$ X -AntiFactor the special case where every vertex v has the same set $$X_v=X$$ X v = X of forbidden degrees. We show the following lower bound for every fixed set $$X$$ X : if there is an $$\epsilon >0$$ ϵ > 0 such that # $$X$$ X -AntiFactor can be

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets \(\sigma,\rho\) of non-negative integers, a \((\sigma,\rho)\) -set of a graph G is a set S of vertices such that \(|N(u)\cap S|\in\sigma\) for every \(u\in S\) , and \(|N(\!\textit{v})\cap S|\in\rho\) for every \(\textit{v}\not\in S\) . The problem of finding a \((\sigma,\rho)\) -set (of a certain size) unifies standard problems, such as Independent Set , Dominating Set , Independent Dominating Set , and many others. For all pairs of finite or cofinite sets \((\sigma,\rho)\) , we determine (under standard complexity assumptions) the best possible value \(c_{\sigma,\rho}\) such that there is an algorithm that counts \((\sigma,\rho)\) -sets in time \(c_{\sigma,\rho}^{\textsf{tw}}\cdot n^{O(1)}\) (if a tree decomposition of width \(\textsf{tw}\) is given in the input). Let \(s_{{\rm top}}\) denote the largest element of \(\sigma\) if \(\sigma\) is finite, or the largest missing integer \(+1\) if \(\sigma\) is cofinite; \(r_{{\rm top}}\) is defined analogously for \(\rho\) . Surprisingly, \(c_{\sigma,\rho}\) is often significantly smaller than the natural bound \(s_{{\rm top}}+r_{{\rm top}}+2\) achieved by existing algorithms. Toward defining \(c_{\sigma,\rho}\) , we say that \((\sigma,\rho)\) is \({\mathrm{m}}\) -structured if there is a pair \((\alpha,\beta)\) such that every integer in \(\sigma\) equals \(\alpha\) mod \({\mathrm{m}}\) , and every integer in \(\rho\) equals \(\beta\) mod \({\mathrm{m}}\) . Then, setting — \(c_{\sigma,\rho}=s_{{\rm top}}+r_{{\rm top}}+2\) if \((\sigma,\rho)\) is not \({\mathrm{m}}\) -structured for any \({\mathrm{m}}\geq 2\) , — \(c_{\sigma,\rho}=\max\{s_{{\rm top}},r_{{\rm top}}\}+2\) if \((\sigma,\rho)\) is 2-structured, but not \({\mathrm{m}}\) -structured for any \({\mathrm{m}}\geq 3\) , and

We study a general family of problems that form a common generalization of classic hitting (also referred to as covering or transversal) and packing problems. An instance of X-HitPack asks: Can removing k (deletable) vertices of a graph G prevent us from packing $\ell$ vertex-disjoint objects of type X? This problem captures a spectrum of problems with standard hitting and packing on opposite ends. Our main motivating question is whether the combination X-HitPack can be significantly harder than these two base problems. Already for a particular choice of X, this question can be posed for many different complexity notions, leading to a large, so-far unexplored domain in the intersection of the areas of hitting and packing problems. On a high-level, we present two case studies: (1) X being all cycles, and (2) X being all copies of a fixed graph H. In each, we explore the classical complexity, as well as the parameterized complexity with the natural parameters k+l and treewidth. We observe that the combined problem can be drastically harder than the base problems: for cycles or for H being a connected graph with at least 3 vertices, the problem is Σ_2^P-complete and requires double-exponential dependence on the treewidth of the graph (assuming the Exponential-Time Hypothesis). In contrast, the combined problem admits qualitatively similar running times as the base problems in some cases, although significant novel ideas are required. For example, for X being all cycles, we establish a 2^poly(k+l)n^O(1) algorithm using an involved branching method. Also, for X being all edges (i.e., H = K_2; this combines Vertex Cover and Maximum Matching) the problem can be solved in time 2^\poly(tw)n^O(1) on graphs of treewidth tw. The key step enabling this running time relies on a combinatorial bound obtained from an algebraic (linear delta-matroid) representation of possible matchings.

Abstract In this paper, we consider a general notion of convolution. Let $$D$$ D be a finite domain and let $$D^n$$ D n be the set of n-length vectors (tuples) of $$D$$ D . Let $$f :D\times D\rightarrow D$$ f : D × D → D be a function and let $$\oplus _f$$ ⊕ f be a coordinate-wise application of f. The $$f$$ f -Convolution of two functions $$g,h :D^n \rightarrow \{-M,\ldots ,M\}$$ g , h : D n → { - M , … , M } is $$\begin{aligned} (g \mathbin {\circledast _{f}}h)(\textbf{v}) {:}{=}\sum _{\begin{array}{c} \textbf{v}_g,\textbf{v}_h \in D^n\\ \text {s.t. } \textbf{v}= \textbf{v}_g \oplus _f \textbf{v}_h \end{array}} g(\textbf{v}_g) \cdot h(\textbf{v}_h) \end{aligned}$$ ( g ⊛ f h ) ( v ) : = ∑ v g , v h ∈ D n s.t. v = v g ⊕ f v h g ( v g ) · h ( v h ) for every $$\textbf{v}\in D^n$$ v ∈ D n . This problem generalizes many fundamental convolutions such as Subset Convolution, XOR Product, Covering Product or Packing Product, etc. For arbitrary function f and domain $$D$$ D we can compute $$f$$ f -Convolution via brute-force enumeration in $$\widetilde{{\mathcal {O}}}(|D|^{2n} \cdot \textrm{polylog}(M))$$ O

Abstract The leafage of a chordal graph G is the minimum integer $$\ell $$ ℓ such that G can be realized as an intersection graph of subtrees of a tree with $$\ell $$ ℓ leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time $$2^{\mathcal {O}(\ell ^2)} \cdot n^{\mathcal {O}(1)}$$ 2 O ( ℓ 2 ) · n O ( 1 ) . We present a conceptually much simpler algorithm that runs in time $$2^{\mathcal {O}(\ell )} \cdot n^{\mathcal {O}(1)}$$ 2 O ( ℓ ) · n O ( 1 ) . We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is [1]-hard when parameterized by the leafage and complement this result by presenting a simple $$n^{\mathcal {O}(\ell )}$$ n O ( ℓ ) -time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in $$n^{{{\mathcal {O}}}(1)}$$ n O ( 1 ) -time.

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets σ, ρ of non-negative integers, a (σ, ρ)-set of a graph G is a set S of vertices such that | N (u) ∩ S| ∈ σ for every u ∈ S, and | N (v) ∩ S| ∈ ρ for every v ∉ S. The problem of finding a (σ, ρ)-set (of a certain size) unifies standard problems such as INDEPENDENT SET, DOMINATING SET, INDEPENDENT DOMINATING SET, and many others. For all pairs of finite or cofinite sets (σ, ρ), we determine (under standard complexity assumptions) the best possible value cσ,ρ such that there is an algorithm that counts (σ, ρ)-sets in time ctwσ,ρ · nO(1) (if a tree decomposition of width tw is given in the input). Let stop denote the largest element of σ if σ is finite, or the largest missing integer +1 if σ is cofinite; rtop is defined analogously for ρ. Surprisingly, cσ,ρ is often significantly smaller than the natural bound stop + rtop + 2 achieved by existing algorithms [van Rooij, 2020]. Toward defining cσ,ρ, we say that (σ,ρ) is m-structured if there is a pair (α,β) such that every integer in σ equals α mod m, and every integer in ρ equals β mod m. Then, setting • cσ,ρ = stop + rtop +2 if (σ, ρ) is not m-structured for any m ≥ 2 • cσ,ρ = max{stop,rtop} + 2 if (σ,ρ) is 2-structured, but not m-structured for any m ≥ 3, and stop = rtop is even, and • cσ,ρ = max{stop, rtop} + 1, otherwise we provide algorithms counting (σ, ρ)-sets in time ctwσ,ρ · nO(1). For example, for the EXACT INDEPENDENT DOMINATING SET problem (also known as PERFECT CODE) corresponding to σ = {0} and ρ = {1}, this improves the 3tw · nO(1) algorithm of van Rooij to 2tw· nO(1). Despite the unusually delicate definition of cσ,ρ, we show that our algorithms are most likely optimal, i.e., for any pair (σ, ρ) of finite or cofinite sets where the problem is non-trivial, and any ε > 0, a (cσ,ρ — ε)tw · nO(1)- algorithm counting the number of (σ, ρ)-sets would violate the COUNTING STRONG EXPONENTIAL-TIME HYPOTHESIS (#SETH). For finite sets σ and ρ, our lower bounds also extend to the decision version, showing that our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets. * The full version of this work can be accessed at https://arxiv.org/abs/2211.04278. Research supported by the European Research Council (ERC) consolidator grant No. 725978 SYSTEMATICGRAPH.

In this paper, we consider a general notion of convolution. Let $D$ be a finite domain and let $D^n$ be the set of $n$-length vectors (tuples) of $D$. Let $f : D \times D \to D$ be a function and let $\oplus_f$ be a coordinate-wise application of $f$. The $f$-Convolution of two functions $g,h : D^n \to \{-M,\ldots,M\}$ is $$(g \otimes_f h)(\textbf{v}) := \sum_{\substack{\textbf{v}_g,\textbf{v}_h \in D^n\\ \text{s.t. } \textbf{v}_g \oplus_f \textbf{v}_h}} g(\textbf{v}_g) \cdot h(\textbf{v}_h)$$ for every $\textbf{v} \in D^n$. This problem generalizes many fundamental convolutions such as Subset Convolution, XOR Product, Covering Product or Packing Product, etc. For arbitrary function $f$ and domain $D$ we can compute $f$-Convolution via brute-force enumeration in $\widetilde{O}(|D|^{2n}\mathrm{polylog}(M))$ time. Our main result is an improvement over this naive algorithm. We show that $f$-Convolution can be computed exactly in $\widetilde{O}((c \cdot |D|^2)^{n}\mathrm{polylog}(M))$ for constant $c := 3/4$ when $D$ has even cardinality. Our main observation is that a \emph{cyclic partition} of a function $f : D \times D \to D$ can be used to speed up the computation of $f$-Convolution, and we show that an appropriate cyclic partition exists for every $f$. Furthermore, we demonstrate that a single entry of the $f$-Convolution can be computed more efficiently. In this variant, we are given two functions $g,h : D^n \to \{-M,\ldots,M\}$ alongside with a vector $\textbf{v} \in D^n$ and the task of the $f$-Query problem is to compute integer $(g \otimes_f h)(\textbf{v})$. This is a generalization of the well-known Orthogonal Vectors problem. We show that $f$-Query can be computed in $\widetilde{O}(|D|^{\fracω{2} n}\mathrm{polylog}(M))$ time, where $ω\in [2,2.372)$ is the exponent of currently fastest matrix multiplication algorithm.

In the general AntiFactor problem, a graph G and, for every vertex v of G, a set X_v ⊆ ℕ of forbidden degrees is given. The task is to find a set S of edges such that the degree of v in S is not in the set X_v. Standard techniques (dynamic programming plus fast convolution) can be used to show that if M is the largest forbidden degree, then the problem can be solved in time (M+2)^{tw}⋅n^{O(1)} if a tree decomposition of width tw is given. However, significantly faster algorithms are possible if the sets X_v are sparse: our main algorithmic result shows that if every vertex has at most x forbidden degrees (we call this special case AntiFactor_x), then the problem can be solved in time (x+1)^{O(tw)}⋅n^{O(1)}. That is, AntiFactor_x is fixed-parameter tractable parameterized by treewidth tw and the maximum number x of excluded degrees. Our algorithm uses the technique of representative sets, which can be generalized to the optimization version, but (as expected) not to the counting version of the problem. In fact, we show that #AntiFactor₁ is already #W[1]-hard parameterized by the width of the given decomposition. Moreover, we show that, unlike for the decision version, the standard dynamic programming algorithm is essentially optimal for the counting version. Formally, for a fixed nonempty set X, we denote by X-AntiFactor the special case where every vertex v has the same set X_v = X of forbidden degrees. We show the following lower bound for every fixed set X: if there is an ε > 0 such that #X-AntiFactor can be solved in time (max X+2-ε)^{tw}⋅n^{O(1)} given a tree decomposition of width tw, then the Counting Strong Exponential-Time Hypothesis (#SETH) fails.

Many problems in interprocedural program analysis can be modeled as the context-free language (CFL) reachability problem on graphs and can be solved in cubic time. Despite years of efforts, there are no known truly sub-cubic algorithms for this problem. We study the related certification task: given an instance of CFL reachability, are there small and efficiently checkable certificates for the existence and for the non-existence of a path? We show that, in both scenarios, there exist succinct certificates ( O ( n 2 ) in the size of the problem) and these certificates can be checked in subcubic (matrix multiplication) time. The certificates are based on grammar-based compression of paths (for reachability) and on invariants represented as matrix inequalities (for non-reachability). Thus, CFL reachability lies in nondeterministic and co-nondeterministic subcubic time. A natural question is whether faster algorithms for CFL reachability will lead to faster algorithms for combinatorial problems such as Boolean satisfiability (SAT). As a consequence of our certification results, we show that there cannot be a fine-grained reduction from SAT to CFL reachability for a conditional lower bound stronger than n ω , unless the nondeterministic strong exponential time hypothesis (NSETH) fails. In a nutshell, reductions from SAT are unlikely to explain the cubic bottleneck for CFL reachability. Our results extend to related subcubic equivalent problems: pushdown reachability and 2NPDA recognition; as well as to all-pairs CFL reachability. For example, we describe succinct certificates for pushdown non-reachability (inductive invariants) and observe that they can be checked in matrix multiplication time. We also extract a new hardest 2NPDA language, capturing the “hard core” of all these problems.

For the General Factor problem we are given an undirected graph $G$ and for each vertex $v\in V(G)$ a finite set $B_v$ of non-negative integers. The task is to decide if there is a subset $S\subseteq E(G)$ such that $deg_S(v)\in B_v$ for all vertices $v$ of $G$. The maxgap of a finite integer set $B$ is the largest $d\ge 0$ such that there is an $a\ge 0$ with $[a,a+d+1]\cap B=\{a,a+d+1\}$. Cornuéjols (1988) showed that if the maxgap of all sets $B_v$ is at most 1, then the decision version of General Factor is poly-time solvable. Dudycz and Paluch (2018) extended this result for the minimization and maximization versions. Using convolution techniques from van Rooij (2020), we improve upon the previous algorithm by Arulselvan et al. (2018) and present an algorithm counting the number of solutions of a certain size in time $O^*((M+1)^k)$, given a tree decomposition of width $k$, where $M=\max_v \max B_v$. We prove that this algorithm is essentially optimal for all cases that are not polynomial time solvable for the decision, minimization or maximization versions. We prove that such improvements are not possible even for $B$-Factor, which is General Factor on graphs where all sets $B_v$ agree with the fixed set $B$. We show that for every fixed $B$ where the problem is NP-hard, our new algorithm cannot be significantly improved: assuming the Strong Exponential Time Hypothesis (SETH), no algorithm can solve $B$-Factor in time $O^*((\max B+1-ε)^k)$ for any $ε>0$. We extend this bound to the counting version of $B$-Factor for arbitrary, non-trivial sets $B$, assuming #SETH. We also investigate the parameterization of the problem by cutwidth. Unlike for treewidth, a larger set $B$ does not make the problem harder: Given a linear layout of width $k$ we give a $O^*(2^k)$ algorithm for any $B$ and provide a matching lower bound that this is optimal for the NP-hard cases.

The currently fastest algorithm for regular expression pattern matching and membership improves the classical O(nm) time algorithm by a factor of about log^{3/2}n. Instead of focussing on general patterns we analyse homogeneous patterns of bounded depth in this work. For them a classification splitting the types in easy (strongly sub-quadratic) and hard (essentially quadratic time under SETH) is known. We take a very fine-grained look at the hard pattern types from this classification and show a dichotomy: few types allow super-poly-logarithmic improvements while the algorithms for the other pattern types can only be improved by a constant number of log-factors, assuming the Formula-SAT Hypothesis.