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Given a graph $G$ and an integer $k$, the $H$-free Edge Deletion problem asks whether there exists a set of at most $k$ edges of $G$ whose deletion makes $G$ free of induced copies of $H$. Significant attention has been given to the kernelizability aspects of this problem -- i.e., for which graphs $H$ does the problem admit an "efficient preprocessing" procedure, known as a polynomial kernelization, where an instance $I$ of the problem with parameter $k$ is reduced to an equivalent instance $I'$ whose size and parameter value are bounded polynomially in $k$? Although such routines are known for many graphs $H$ where the class of $H$-free graphs has significant restricted structure, it is also clear that for most graphs $H$ the problem is incompressible, i.e., admits no polynomial kernelization parameterized by $k$ unless the polynomial hierarchy collapses. These results led Marx and Sandeep to the conjecture that $H$-free Edge Deletion is incompressible for any graph $H$ with at least five vertices, unless $H$ is complete or has at most one edge (JCSS 2022). This conjecture was reduced to the incompressibility of $H$-free Edge Deletion for a finite list of graphs $H$. We consider one of these graphs, which we dub the prison, and show that Prison-Free Edge Deletion has a polynomial kernel, refuting the conjecture. On the other hand, the same problem for the complement of the prison is incompressible.

We consider the MIN-r-LIN(R) problem: given a system S of length-r linear equations over a ring R, find a subset of equations Z of minimum cardinality such that S-Z is satisfiable. The problem is NP-hard and UGC-hard to approximate within any constant even when r=|R|=2, so we focus on parameterized approximability with solution size as the parameter. For a large class of infinite rings R called Euclidean domains, Dabrowski et al. [SODA-2023] obtained an FPT-algorithm for MIN-2-LIN(R) using an LP-based approach based on work by Wahlstr\"om [SODA-2017]. Here, we consider MIN-r-LIN(R) for finite commutative rings R, initiating a line of research with the ultimate goal of proving dichotomy theorems that separate problems that are FPT-approximable within a constant from those that are not. A major motivation is that our project is a promising step for more ambitious classification projects concerning finite-domain MinCSP and VCSP. Dabrowski et al.'s algorithm is limited to rings without zero divisors, which are only fields among finite commutative rings. Handling zero divisors seems to be an insurmountable obstacle for the LP-based approach. In response, we develop a constant-factor FPT-approximation algorithm for a large class of finite commutative rings, called Bergen rings, and thus prove approximability for chain rings, principal ideal rings, and Z_m for all m>1. We complement the algorithmic result with powerful lower bounds. For r>2, we show that the problem is not FPT-approximable within any constant (unless FPT=W[1]). We identify the class of non-Helly rings for which MIN-2-LIN(R) is not FPT-approximable. Under ETH, we also rule out (2-e)-approximation for every e>0 for non-lineal rings, which includes e.g. rings Z_{pq} where p and q are distinct primes. Towards closing the gaps between upper and lower bounds, we lay the foundation of a geometric approach for analysing rings.

We show new algorithms and constructions over linear delta-matroids. We observe an alternative representation for linear delta-matroids, as a contraction representation over a skew-symmetric matrix. This is equivalent to the more standard "twist representation" up to $O(n^\omega)$-time transformations, but is much more convenient for algorithmic tasks. For instance, the problem of finding a max-weight feasible set now reduces directly to the problem of finding a max-weight basis in a linear matroid. Supported by this representation, we provide new algorithms and constructions over linear delta-matroids. We show that the union and delta-sum of linear delta-matroids define linear delta-matroids, and a representation for the resulting delta-matroid can be constructed in randomized time $O(n^\omega)$. Previously, it was only known that these operations define delta-matroids. We also note that every projected linear delta-matroid can be represented as an elementary projection. This implies that several optimization problems over (projected) linear delta-matroids, including the coverage, delta-coverage, and parity problems, reduce (in their decision versions) to a single $O(n^{\omega})$-time matrix rank computation. Using the methods of Harvey, previously used by Cheung, Lao and Leung for linear matroid parity, we furthermore show how to solve the search versions in the same time. This improves on the $O(n^4)$-time augmenting path algorithm of Geelen, Iwata and Murota. Finally, we consider the maximum-cardinality delta-matroid intersection problem. Using Storjohann's algorithms for symbolic determinants, we show that such a solution can be found in $O(n^{\omega+1})$ time. This is the first polynomial-time algorithm for the problem, solving an open question of Kakimura and Takamatsu.

The input in the Minimum-Cost Constraint Satisfaction Problem (MinCSP) over the Point Algebra contains a set of variables, a collection of constraints of the form $x < y$, $x = y$, $x \leq y$ and $x \neq y$, and a budget $k$. The goal is to check whether it is possible to assign rational values to the variables while breaking constraints of total cost at most $k$. This problem generalizes several prominent graph separation and transversal problems: MinCSP$(<)$ is equivalent to Directed Feedback Arc Set, MinCSP$(<,\leq)$ is equivalent to Directed Subset Feedback Arc Set, MinCSP$(=,\neq)$ is equivalent to Edge Multicut, and MinCSP$(\leq,\neq)$ is equivalent to Directed Symmetric Multicut. Apart from trivial cases, MinCSP$(\Gamma)$ for $\Gamma \subseteq \{<,=,\leq,\neq\}$ is NP-hard even to approximate within any constant factor under the Unique Games Conjecture. Hence, we study parameterized complexity of this problem under a natural parameterization by the solution cost $k$. We obtain a complete classification: if $\Gamma \subseteq \{<,=,\leq,\neq\}$ contains both $\leq$ and $\neq$, then MinCSP$(\Gamma)$ is W[1]-hard, otherwise it is fixed-parameter tractable. For the positive cases, we solve MinCSP$(<,=,\neq)$, generalizing the FPT results for Directed Feedback Arc Set and Edge Multicut as well as their weighted versions. Our algorithm works by reducing the problem into a Boolean MinCSP, which is in turn solved by flow augmentation. For the lower bounds, we prove that Directed Symmetric Multicut is W[1]-hard, solving an open problem.

We present representative sets-style statements for linear delta-matroids, which are set systems that generalize matroids, with important connections to matching theory and graph embeddings. Furthermore, our proof uses a new approach of sieving polynomial families, which generalizes the linear algebra approach of the representative sets lemma to a setting of bounded-degree polynomials. The representative sets statements for linear delta-matroids then follow by analyzing the Pfaffian of the skew-symmetric matrix representing the delta-matroid. Applying the same framework to the determinant instead of the Pfaffian recovers the representative sets lemma for linear matroids. Altogether, this significantly extends the toolbox available for kernelization. As an application, we show an exact sparsification result for Mader networks: Let $G=(V,E)$ be a graph and $\mathcal{T}$ a partition of a set of terminals $T \subseteq V(G)$, $|T|=k$. A $\mathcal{T}$-path in $G$ is a path with endpoints in distinct parts of $\mathcal{T}$ and internal vertices disjoint from $T$. In polynomial time, we can derive a graph $G'=(V',E')$ with $T \subseteq V(G')$, such that for every subset $S \subseteq T$ there is a packing of $\mathcal{T}$-paths with endpoints $S$ in $G$ if and only if there is one in $G'$, and $|V(G')|=O(k^3)$. This generalizes the (undirected version of the) cut-covering lemma, which corresponds to the case that $\mathcal{T}$ contains only two blocks. To prove the Mader network sparsification result, we furthermore define the class of Mader delta-matroids, and show that they have linear representations. This should be of independent interest.

We study the parameterized complexity of MinCSP for so-called equality languages, i.e., for finite languages over an infinite domain such as $\mathbb{N}$, where the relations are defined via first-order formulas whose only predicate is $=$. This is an important class of languages that forms the starting point of all study of infinite-domain CSPs under the commonly used approach pioneered by Bodirsky, i.e., languages defined as reducts of finitely bounded homogeneous structures. Moreover, MinCSP over equality languages forms a natural class of optimisation problems in its own right, covering such problems as Edge Multicut, Steiner Multicut and (under singleton expansion) Edge Multiway Cut. We classify MinCSP$(\Gamma)$ for every finite equality language $\Gamma$, under the natural parameter, as either FPT, W[1]-hard but admitting a constant-factor FPT-approximation, or not admitting a constant-factor FPT-approximation unless FPT=W[2]. In particular, we describe an FPT case that slightly generalises Multicut, and show a constant-factor FPT-approximation for Disjunctive Multicut, the generalisation of Multicut where the ``cut requests'' come as disjunctions over $d = O(1)$ individual cut requests $s_i \neq t_i$. We also consider singleton expansions of equality languages, i.e., enriching an equality language with the capability for assignment constraints $(x=i)$ for either finitely or infinitely many constants $i \in \mathbb{N}$, and fully characterize the complexity of the resulting MinCSP.

We introduce determinantal sieving, a new, remarkably powerful tool in the toolbox of algebraic FPT algorithms. Given a polynomial $P(X)$ on a set of variables $X=\{x_1,\ldots,x_n\}$ and a linear matroid $M=(X,\mathcal{I})$ of rank $k$, both over a field $\mathbb{F}$ of characteristic 2, in $2^k$ evaluations we can sieve for those terms in the monomial expansion of $P$ which are multilinear and whose support is a basis for $M$. Alternatively, using $2^k$ evaluations of $P$ we can sieve for those monomials whose odd support spans $M$. Applying this framework, we improve on a range of algebraic FPT algorithms, such as: 1. Solving $q$-Matroid Intersection in time $O^*(2^{(q-2)k})$ and $q$-Matroid Parity in time $O^*(2^{qk})$, improving on $O^*(4^{qk})$ over general fields (Brand and Pratt, ICALP 2021) 2. $T$-Cycle, Colourful $(s,t)$-Path, Colourful $(S,T)$-Linkage in undirected graphs, and the more general Rank $k$ $(S,T)$-Linkage problem, all in $O^*(2^k)$ time, improving on $O^*(2^{k+|S|})$ and $O^*(2^{|S|+O(k^2 \log(k+|\mathbb{F}|))})$ respectively (Fomin et al., SODA 2023) 3. Many instances of the Diverse X paradigm, finding a collection of $r$ solutions to a problem with a minimum mutual distance of $d$ in time $O^*(2^{r(r-1)d/2})$, improving solutions for $k$-Distinct Branchings from time $2^{O(k \log k)}$ to $O^*(2^k)$ (Bang-Jensen et al., ESA 2021), and for Diverse Perfect Matchings from $O^*(2^{2^{O(rd)}})$ to $O^*(2^{r^2d/2})$ (Fomin et al., STACS 2021) Here, all matroids are assumed to be represented over fields of characteristic 2. Over general fields, we achieve similar results at the cost of using exponential space by working over the exterior algebra. For a class of arithmetic circuits we call strongly monotone, this is even achieved without any loss of running time. However, the odd support sieving result appears to be specific to working over characteristic 2.

One of the first application of the recently introduced technique of \emph{flow-augmentation} [Kim et al., STOC 2022] is a fixed-parameter algorithm for the weighted version of \textsc{Directed Feedback Vertex Set}, a landmark problem in parameterized complexity. In this note we explore applicability of flow-augmentation to other weighted graph separation problems parameterized by the size of the cutset. We show the following. -- In weighted undirected graphs \textsc{Multicut} is FPT, both in the edge- and vertex-deletion version. -- The weighted version of \textsc{Group Feedback Vertex Set} is FPT, even with an oracle access to group operations. -- The weighted version of \textsc{Directed Subset Feedback Vertex Set} is FPT. Our study reveals \textsc{Directed Symmetric Multicut} as the next important graph separation problem whose parameterized complexity remains unknown, even in the unweighted setting., Comment: 17 pages, 1 figure

We study the problem Symmetric Directed Multicut from a parameterized complexity perspective. In this problem, the input is a digraph $D$, a set of cut requests $C=\{(s_1,t_1),\ldots,(s_\ell,t_\ell)\}$ and an integer $k$, and the task is to find a set $X \subseteq V(D)$ of size at most $k$ such that for every $1 \leq i \leq \ell$, $X$ intersects either all $(s_i,t_i)$-paths or all $(t_i,s_i)$-paths. Equivalently, every strongly connected component of $D-X$ contains at most one vertex out of $s_i$ and $t_i$ for every $i$. This problem is previously known from research in approximation algorithms, where it is known to have an $O(\log k \log \log k)$-approximation. We note that the problem, parameterized by $k$, directly generalizes multiple interesting FPT problems such as (Undirected) Vertex Multicut and Directed Subset Feedback Vertex Set. We are not able to settle the existence of an FPT algorithm parameterized purely by $k$, but we give three partial results: An FPT algorithm parameterized by $k+\ell$; an FPT-time 2-approximation parameterized by $k$; and an FPT algorithm parameterized by $k$ for the special case that the cut requests form a clique, Symmetric Directed Multiway Cut. The existence of an FPT algorithm parameterized purely by $k$ remains an intriguing open possibility., Comment: A shortened version of this article has been accepted for presentation and publication at The 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

Checking whether a system of linear equations is consistent is a basic computational problem with ubiquitous applications. When dealing with inconsistent systems, one may seek an assignment that minimizes the number of unsatisfied equations. This problem is NP-hard and UGC-hard to approximate within any constant even for two-variable equations over the two-element field. We study this problem from the point of view of parameterized complexity, with the parameter being the number of unsatisfied equations. We consider equations defined over Euclidean domains - a family of commutative rings that generalize finite and infinite fields including the rationals, the ring of integers, and many other structures. We show that if every equation contains at most two variables, the problem is fixed-parameter tractable. This generalizes many eminent graph separation problems such as Bipartization, Multiway Cut and Multicut parameterized by the size of the cutset. To complement this, we show that the problem is W[1]-hard when three or more variables are allowed in an equation, as well as for many commutative rings that are not Euclidean domains. On the technical side, we introduce the notion of important balanced subgraphs, generalizing important separators of Marx [Theor. Comput. Sci. 2006] to the setting of biased graphs. Furthermore, we use recent results on parameterized MinCSP [Kim et al., SODA 2021] to efficiently solve a generalization of Multicut with disjunctive cut requests.

We study the parameterized problem of satisfying ``almost all'' constraints of a given formula $F$ over a fixed, finite Boolean constraint language $\Gamma$, with or without weights. More precisely, for each finite Boolean constraint language $\Gamma$, we consider the following two problems. In Min SAT$(\Gamma)$, the input is a formula $F$ over $\Gamma$ and an integer $k$, and the task is to find an assignment $\alpha \colon V(F) \to \{0,1\}$ that satisfies all but at most $k$ constraints of $F$, or determine that no such assignment exists. In Weighted Min SAT$(\Gamma$), the input additionally contains a weight function $w \colon F \to \mathbb{Z}_+$ and an integer $W$, and the task is to find an assignment $\alpha$ such that (1) $\alpha$ satisfies all but at most $k$ constraints of $F$, and (2) the total weight of the violated constraints is at most $W$. We give a complete dichotomy for the fixed-parameter tractability of these problems: We show that for every Boolean constraint language $\Gamma$, either Weighted Min SAT$(\Gamma)$ is FPT; or Weighted Min SAT$(\Gamma)$ is W[1]-hard but Min SAT$(\Gamma)$ is FPT; or Min SAT$(\Gamma)$ is W[1]-hard. This generalizes recent work of Kim et al. (SODA 2021) which did not consider weighted problems, and only considered languages $\Gamma$ that cannot express implications $(u \to v)$ (as is used to, e.g., model digraph cut problems). Our result generalizes and subsumes multiple previous results, including the FPT algorithms for Weighted Almost 2-SAT, weighted and unweighted $\ell$-Chain SAT, and Coupled Min-Cut, as well as weighted and directed versions of the latter. The main tool used in our algorithms is the recently developed method of directed flow-augmentation (Kim et al., STOC 2022)., Comment: v2. Major update to all three flow-augmentation papers

An instance $I$ of the Stable Matching Problem (SMP) is given by a bipartite graph with a preference list of neighbors for every vertex. A swap in $I$ is the exchange of two consecutive vertices in a preference list. A swap can be viewed as a smallest perturbation of $I$. Boehmer et al. (2021) designed a polynomial-time algorithm to find the minimum number of swaps required to turn a given maximal matching into a stable matching. We generalize this result to the many-to-many version of SMP. We do so first by introducing a new representation of SMP as an extended bipartite graph and subsequently by reducing the problem to submodular minimization. It is a natural problem to establish the computational complexity of deciding whether at most $k$ swaps are enough to turn $I$ into an instance where one of the maximum matchings is stable. Using a hardness result of Gupta et al. (2020), we prove that this problem is NP-hard and, moreover, this problem parameterised by $k$ is W[1]-hard. We also obtain a lower bound on the running time for solving the problem using the Exponential Time Hypothesis.

We show a flow-augmentation algorithm in directed graphs: There exists a randomized polynomial-time algorithm that, given a directed graph $G$, two vertices $s,t \in V(G)$, and an integer $k$, adds (randomly) to $G$ a number of arcs such that for every minimal $st$-cut $Z$ in $G$ of size at most $k$, with probability $2^{-\mathrm{poly}(k)}$ the set $Z$ becomes a minimum $st$-cut in the resulting graph. We also provide a deterministic counterpart of this procedure. The directed flow-augmentation tool allows us to prove fixed-parameter tractability of a number of problems parameterized by the cardinality of the deletion set, whose parameterized complexity status was repeatedly posed as open problems: (1) Chain SAT, defined by Chitnis, Egri, and Marx [ESA'13, Algorithmica'17], (2) a number of weighted variants of classic directed cut problems, such as Weighted $st$-Cut} or Weighted Directed Feedback Vertex Set. By proving that Chain SAT is FPT, we confirm a conjecture of Chitnis, Egri, and Marx that, for any graph $H$, if the List $H$-Coloring problem is polynomial-time solvable, then the corresponding vertex-deletion problem is fixed-parameter tractable., Comment: v2. Major update of three flow-augmentation papers. Includes a deterministic version. Weighted Almost 2-SAT algorithm has been removed, as it is superseded by a more general algorithm of Flow-Augmentation III (arXiv:2207.07422)

Let $G$ be a graph and $S, T \subseteq V(G)$ be (possibly overlapping) sets of terminals, $|S|=|T|=k$. We are interested in computing a vertex sparsifier for terminal cuts in $G$, i.e., a graph $H$ on a smallest possible number of vertices, where $S \cup T \subseteq V(H)$ and such that for every $A \subseteq S$ and $B \subseteq T$ the size of a minimum $(A,B)$-vertex cut is the same in $G$ as in $H$. We assume that our graphs are unweighted and that terminals may be part of the min-cut. In previous work, Kratsch and Wahlstr\"om (FOCS 2012/JACM 2020) used connections to matroid theory to show that a vertex sparsifier $H$ with $O(k^3)$ vertices can be computed in randomized polynomial time, even for arbitrary digraphs $G$. However, since then, no improvements on the size $O(k^3)$ have been shown. In this paper, we draw inspiration from the renowned Bollob\'as's Two-Families Theorem in extremal combinatorics and introduce the use of total orderings into Kratsch and Wahlstr\"om's methods. This new perspective allows us to construct a sparsifier $H$ of $\Theta(k^2)$ vertices for the case that $G$ is a DAG. We also show how to compute $H$ in time near-linear in the size of $G$, improving on the previous $O(n^{\omega+1})$. Furthermore, $H$ recovers the closest min-cut in $G$ for every partition $(A,B)$, which was not previously known. Finally, we show that a sparsifier of size $\Omega(k^2)$ is required, both for DAGs and for undirected edge cuts.

We present an undirected version of the recently introduced flow-augmentation technique: Given an undirected multigraph $G$ with distinguished vertices $s,t \in V(G)$ and an integer $k$, one can in randomized $k^{O(1)} \cdot (|V(G)| + |E(G)|)$ time sample a set $A \subseteq \binom{V(G)}{2}$ such that the following holds: for every inclusion-wise minimal $st$-cut $Z$ in $G$ of cardinality at most $k$, $Z$ becomes a minimum-cardinality cut between $s$ and $t$ in $G+A$ (i.e., in the multigraph $G$ with all edges of $A$ added) with probability $2^{-O(k \log k)}$. Compared to the version for directed graphs [STOC 2022], the version presented here has improved success probability ($2^{-O(k \log k)}$ instead of $2^{-O(k^4 \log k)}$), linear dependency on the graph size in the running time bound, and an arguably simpler proof. An immediate corollary is that the Bi-objective $st$-Cut problem can be solved in randomized FPT time $2^{O(k \log k)} (|V(G)|+|E(G)|)$ on undirected graphs., Comment: v2. Major update of all three flow-augmentation papers. The partial dichotomy part has been removed from this work, as it is superseded by the full dichotomy of Flow-Augmentation III (arXiv:2207.07422)

We study the Many Visits TSP problem, where given a number $k(v)$ for each of $n$ cities and pairwise (possibly asymmetric) integer distances, one has to find an optimal tour that visits each city $v$ exactly $k(v)$ times. The currently fastest algorithm is due to Berger, Kozma, Mnich and Vincze [SODA 2019, TALG 2020] and runs in time and space $\mathcal{O}^*(5^n)$. They also show a polynomial space algorithm running in time $\mathcal{O}^*(16^{n+o(n)})$. In this work, we show three main results: (i) A randomized polynomial space algorithm in time $\mathcal{O}^*(2^nD)$, where $D$ is the maximum distance between two cities. By using standard methods, this results in $(1+\epsilon)$-approximation in time $\mathcal{O}^*(2^n\epsilon^{-1})$. Improving the constant $2$ in these results would be a major breakthrough, as it would result in improving the $\mathcal{O}^*(2^n)$-time algorithm for Directed Hamiltonian Cycle, which is a 50 years old open problem. (ii) A tight analysis of Berger et al.'s exponential space algorithm, resulting in $\mathcal{O}^*(4^n)$ running time bound. (iii) A new polynomial space algorithm, running in time $\mathcal{O}(7.88^n)$.

We show the existence of an exact mimicking network of $k^{O(\log k)}$ edges for minimum multicuts over a set of terminals in an undirected graph, where $k$ is the total capacity of the terminals, as well as a method for computing a mimicking network of quasipolynomial size in polynomial time. As a consequence of the latter, several problems are shown to have quasipolynomial kernels, including Edge Multiway Cut, Group Feedback Edge Set for an arbitrary group, and Edge Multicut parameterized by the solution and the number of cut requests. The result combines the matroid-based irrelevant edge approach used in the kernel for $s$-Multiway Cut with a recursive decomposition and sparsification of the graph along sparse cuts. This is the first progress on the kernelization of Multiway Cut problems since the kernel for $s$-Multiway Cut for constant value of $s$ (Kratsch and Wahlstr\"om, FOCS 2012)., Comment: Updated version with simplified proof and new constructive result

Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by $k$: (1) Given a graph $G$, a clique modulator $D$ (a clique modulator is a set of vertices, whose removal results in a clique) of size $k$ for $G$, and a list $L(v)$ of colors for every $v\in V(G)$, decide whether $G$ has a proper list coloring; (2) Given a graph $G$, a clique modulator $D$ of size $k$ for $G$, and a pre-coloring $\lambda_P: X \rightarrow Q$ for $X \subseteq V(G),$ decide whether $\lambda_P$ can be extended to a proper coloring of $G$ using only colors from $Q.$ For Problem 1 we design an $O^*(2^k)$-time randomized algorithm and for Problem 2 we obtain a kernel with at most $3k$ vertices. Banik et al. (IWOCA 2019) proved the the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph $G$, an integer $k$, and a list $L(v)$ of exactly $n-k$ colors for every $v \in V(G),$ decide whether there is a proper list coloring for $G.$ We obtain a kernel with $O(k^2)$ vertices and colors and a compression to a variation of the problem with $O(k)$ vertices and $O(k^2)$ colors.

We show an algorithm that, given an $n$-vertex graph $G$ and a parameter $k$, in time $2^{O(k \log k)} n^{O(1)}$ finds a tree decomposition of $G$ with the following properties: * every adhesion of the tree decomposition is of size at most $k$, and * every bag of the tree decomposition is $(i,i)$-unbreakable in $G$ for every $1 \leq i \leq k$. Here, a set $X \subseteq V(G)$ is $(a,b)$-unbreakable in $G$ if for every separation $(A,B)$ of order at most $b$ in $G$, we have $|A \cap X| \leq a$ or $|B \cap X| \leq a$. The resulting tree decomposition has arguably best possible adhesion size boundsand unbreakability guarantees. Furthermore, the parametric factor in the running time bound is significantly smaller than in previous similar constructions. These improvements allow us to present parameterized algorithms for Minimum Bisection, Steiner Cut, and Steiner Multicut with improved parameteric factor in the running time bound. The main technical insight is to adapt the notion of lean decompositions of Thomas and the subsequent construction algorithm of Bellenbaum and Diestel to the parameterized setting., Comment: v2: New co-author (Magnus) and improved results on vertex unbreakability of bags, v3: final changes, including new abstract

We study multi-budgeted variants of the classic minimum cut problem and graph separation problems that turned out to be important in parameterized complexity: Skew Multicut and Directed Feedback Arc Set. In our generalization, we assign colors $1,2,...,\ell$ to some edges and give separate budgets $k_{1},k_{2},...,k_{\ell}$. Let $E_{i}$ be the set of edges of color $i$. The solution $C$ for the multi-budgeted variant of a graph separation problem not only needs to satisfy the usual separation requirements, but also needs to satisfy that $|C\cap E_{i}|\leq k_{i}$ for every $i\in \{1,...,\ell\}$. Contrary to the classic minimum cut problem, the multi-budgeted variant turns out to be NP-hard even for $\ell = 2$. We propose FPT algorithms parameterized by $k=k_{1}+...+k_{\ell}$ for all three problems. To this end, we develop a branching procedure for the multi-budgeted minimum cut problem that measures the progress of the algorithm not by reducing $k$ as usual, by but elevating the capacity of some edges and thus increasing the size of maximum source-to-sink flow. Using the fact that a similar strategy is used to enumerate all important separators of a given size, we merge this process with the flow-guided branching and show an FPT bound on the number of (appropriately defined) important multi-budgeted separators. This allows us to extend our algorithm to the Skew Multicut and Directed Feedback Arc Set problems. Furthermore, we show connections of the multi-budgeted variants with weighted variants of the directed cut problems and the Chain $\ell$-SAT problem, whose parameterized complexity remains an open problem. We show that these problems admit a bounded-in-parameter number of "maximally pushed" solutions (in a similar spirit as important separators are maximally pushed), giving somewhat weak evidence towards their tractability., Comment: Full version of the paper presented at IPEC 2018

Abasi et al. (2014) and Gabizon et al. (2015) studied the following problems. In the $r$-Simple $k$-Path problem, given a digraph $G$ on $n$ vertices and integers $r,k$, decide whether $G$ has an $r$-simple $k$-path, which is a walk where every vertex occurs at most $r$ times and the total number of vertex occurrences is $k$. In the $(r,k)$-Monomial Detection problem, given an arithmetic circuit that encodes some polynomial $P$ on $n$ variables and integers $k,r$, decide whether $P$ has a monomial of degree $k$ where the degree of each variable is at most~$r$. In the $p$-Set $(r,q)$-Packing problem, given a universe $V$, positive integers $p,q,r$, and a collection $\cal H$ of sets of size $p$ whose elements belong to $V$, decide whether there exists a subcollection ${\cal H}'$ of $\cal H$ of size $q$ where each element occurs in at most $r$ sets of ${\cal H}'$. Abasi et al. and Gabizon et al. proved that the three problems are single-exponentially fixed-parameter tractable (FPT) when parameterized by $(k/r)\log r$, where $k=pq$ for $p$-Set $(r,q)$-Packing and asked whether the $\log r$ factor in the exponent can be avoided. We consider their question from a wider perspective: are the above problems FPT when parameterized by $k/r$ only? We resolve the wider question by (a) obtaining a $2^{O((k/r)^2\log(k/r))} (n+\log k)^{O(1)}$-time algorithm for $r$-Simple $k$-Path on digraphs and a $2^{O(k/r)} (n+\log k)^{O(1)}$-time algorithm for $r$-Simple $k$-Path on undirected graphs (i.e., for undirected graphs we answer the original question in affirmative), (b) showing that $p$-Set $(r,q)$-Packing is FPT, and (c) proving that $(r,k)$-Monomial Detection is para-NP-hard. For $p$-Set $(r,q)$-Packing, we obtain a polynomial kernel for any fixed $p$, which resolves a question posed by Gabizon et al. regarding the existence of polynomial kernels for problems with relaxed disjointness constraints., Comment: To appear in ACM Trans. on Algorithms

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.

We consider the problems ZERO EXTENSION and METRIC LABELLING under the paradigm of parameterized complexity. These are natural, well-studied problems with important applications, but have previously not received much attention from parameterized complexity. Depending on the chosen cost function $\mu$, we find that different algorithmic approaches can be applied to design FPT-algorithms: for arbitrary $\mu$ we parameterized by the number of edges that cross the cut (not the cost) and show how to solve ZERO EXTENSION in time $O(|D|^{O(k^2)} n^4 \log n)$ using randomized contractions. We improve this running time with respect to both parameter and input size to $O(|D|^{O(k)} m)$ in the case where $\mu$ is a metric. We further show that the problem admits a polynomial sparsifier, that is, a kernel of size $O(k^{|D|+1})$ that is independent of the metric $\mu$. With the stronger condition that $\mu$ is described by the distances of leaves in a tree, we parameterize by a gap parameter $(q - p)$ between the cost of a true solution $q$ and a `discrete relaxation' $p$ and achieve a running time of $O(|D|^{q-p} |T|m + |T|\phi(n,m))$ where $T$ is the size of the tree over which $\mu$ is defined and $\phi(n,m)$ is the running time of a max-flow computation. We achieve a similar running for the more general METRIC LABELLING, while also allowing $\mu$ to be the distance metric between an arbitrary subset of nodes in a tree using tools from the theory of VCSPs. We expect the methods used in the latter result to have further applications.

We study the complexity of SAT($\Gamma$) problems for potentially infinite languages $\Gamma$ closed under variable negation (sign-symmetric languages). Via an algebraic connection, this reduces to the study of restricted partial polymorphisms of $\Gamma$ we refer to as \emph{pSDI-operations} (for partial, self-dual and idempotent). First, we study the language classes themselves. We classify the structure of the least restrictive pSDI-operations, corresponding to the most powerful languages $\Gamma$, and find that these operations can be divided into \emph{levels}, corresponding to a rough notion of difficulty; and that within each level there is a strongest operation (the partial $k$-NU operation, preserving $(k-1)$-SAT) and a weakest operation (the $k$-universal operation $u_k$, preserving problems definable via bounded-degree polynomials). We show that every sign-symmetric $\Gamma$ not preserved by $u_k$ implements all $k$-clauses; thus if $\Gamma$ is not preserved by $u_k$ for any $k$, then SAT($\Gamma$) is trivially SETH-hard and cannot be solved faster than $O^*(2^n)$ unless SETH fails. Second, we study upper and lower bounds for SAT($\Gamma$) for such languages. We show that several classes in the hierarchy correspond to problems which can be solved faster than $2^n$ using previously known algorithmic strategies such as Subset Sum-style meet-in-the-middle and fast matrix multiplication. Furthermore, if the sunflower conjecture holds for sunflowers with k sets, then the partial k-NU language has an improved algorithm via local search. Complementing this, we show that for every class there is a concrete lower bound $c$ such that SAT($\Gamma$) cannot be solved faster than $O^*(c^n)$ for all problems in the class unless SETH fails. This gives the first known case of a SAT-problem which simultaneously has non-trivial upper and lower bounds under SETH.

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.

In recent years, several powerful techniques have been developed to design {\em randomized} polynomial-space parameterized algorithms. In this paper, we introduce an enhancement of color coding to design deterministic polynomial-space parameterized algorithms. Our approach aims at reducing the number of random choices by exploiting the special structure of a solution. Using our approach, we derive the following deterministic algorithms (see Introduction for problem definitions). 1. Polynomial-space $O^*(3.86^k)$-time (exponential-space $O^*(3.41^k)$-time) algorithm for {\sc $k$-Internal Out-Branching}, improving upon the previously fastest {\em exponential-space} $O^*(5.14^k)$-time algorithm for this problem. 2. Polynomial-space $O^*((2e)^{k+o(k)})$-time (exponential-space $O^*(4.32^k)$-time) algorithm for {\sc $k$-Colorful Out-Branching} on arc-colored digraphs and {\sc $k$-Colorful Perfect Matching} on planar edge-colored graphs. To obtain our polynomial space algorithms, we show that $(n,k,\alpha k)$-splitters ($\alpha\ge 1$) and in particular $(n,k)$-perfect hash families can be enumerated one by one with polynomial delay.

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.

An out-branching and an in-branching of a digraph $D$ are called $k$-distinct if each of them has $k$ arcs absent in the other. Bang-Jensen, Saurabh and Simonsen (2016) proved that the problem of deciding whether a strongly connected digraph $D$ has $k$-distinct out-branching and in-branching is fixed-parameter tractable (FPT) when parameterized by $k$. They asked whether the problem remains FPT when extended to arbitrary digraphs. Bang-Jensen and Yeo (2008) asked whether the same problem is FPT when the out-branching and in-branching have the same root. By linking the two problems with the problem of whether a digraph has an out-branching with at least $k$ leaves (a leaf is a vertex of out-degree zero), we first solve the problem of Bang-Jensen and Yeo (2008). We then develop a new digraph decomposition called the rooted cut decomposition and using it we prove that the problem of Bang-Jensen et al. (2016) is FPT for all digraphs. We believe that the \emph{rooted cut decomposition} will be useful for obtaining other results on digraphs.

We give a combinatorial condition for the existence of efficient, LP-based FPT algorithms for a broad class of graph-theoretical optimisation problems. Our condition is based on the notion of biased graphs known from matroid theory. Specifically, we show that given a biased graph $\Psi=(G,\mathcal{B})$, where $\mathcal{B}$ is a class of balanced cycles in $G$, the problem of finding a set $X$ of at most $k$ vertices in $G$ which intersects every unbalanced cycle in $G$ admits an FPT algorithm using an LP-branching approach, similar to those previously seen for VCSP problems (Wahlstr\"om, SODA 2014). This framework captures many of the problems previously solved via the VCSP approach to LP-branching, as well as new generalisations, such as Group Feedback Vertex Set for infinite groups (e.g., for graphs whose edges are labelled by matrices). A major advantage compared to previous work is that it is immediate to check the applicability of the result for a given problem, whereas testing applicability of the VCSP approach for a specific VCSP requires determining the existence of an embedding language with certain algebraically defined properties, which is not known to be decidable in general. Additionally, we study the approximation question, and show that every problem of this category admits an $O(\log \text{OPT})$-approximation., Comment: New version with new coauthor (Euiwoong Lee) and approximation results

It is well-known that an undirected graph has no odd cycle if and only if it is bipartite. A less obvious, but similar result holds for directed graphs: a strongly connected digraph has no odd cycle if and only if it is bipartite. Can this result be further generalized to more general graphs such as edge-colored graphs? In this paper, we study this problem and show how to decide if there exists an odd properly colored cycle in a given edge-colored graph. As a by-product, we show how to detect if there is a perfect matching in a graph with even (or odd) number of edges in a given edge set.

It is well-known that the Chinese postman problem on undirected and directed graphs is polynomial-time solvable. We extend this result to edge-colored multigraphs. Our result is in sharp contrast to the Chinese postman problem on mixed graphs, i.e., graphs with directed and undirected edges, for which the problem is NP-hard.

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 bounds

The Workflow Satisfiability Problem (WSP) is a problem of practical interest that arises whenever tasks need to be performed by authorized users, subject to constraints defined by business rules. We are required to decide whether there exists a plan -- an assignment of tasks to authorized users -- such that all constraints are satisfied. The WSP is, in fact, the conservative Constraint Satisfaction Problem (i.e., for each variable, here called task, we have a unary authorization constraint) and is, thus, NP-complete. It was observed by Wang and Li (2010) that the number k of tasks is often quite small and so can be used as a parameter, and several subsequent works have studied the parameterized complexity of WSP regarding parameter k. We take a more detailed look at the kernelization complexity of WSP(\Gamma) when \Gamma\ denotes a finite or infinite set of allowed constraints. Our main result is a dichotomy for the case that all constraints in \Gamma\ are regular: (1) We are able to reduce the number n of users to n' <= k. This entails a kernelization to size poly(k) for finite \Gamma, and, under mild technical conditions, to size poly(k+m) for infinite \Gamma, where m denotes the number of constraints. (2) Already WSP(R) for some R \in \Gamma\ allows no polynomial kernelization in k+m unless the polynomial hierarchy collapses., Comment: An extended abstract appears in the proceedings of IPEC 2014

In this book chapter we survey known approaches and algorithms to compute discrepancy measures of point sets. After providing an introduction which puts the calculation of discrepancy measures in a more general context, we focus on the geometric discrepancy measures for which computation algorithms have been designed. In particular, we explain methods to determine $L_2$-discrepancies and approaches to tackle the inherently difficult problem to calculate the star discrepancy of given sample sets. We also discuss in more detail three applications of algorithms to approximate discrepancies.

A recent trend in parameterized algorithms is the application of polytope tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). However, although interesting results have been achieved, the methods require the underlying polytope to have very restrictive properties (half-integrality and persistence), which are known only for few problems (essentially Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994)). Taking a slightly different approach, we view half-integrality as a \emph{discrete} relaxation of a problem, e.g., a relaxation of the search space from $\{0,1\}^V$ to $\{0,1/2,1\}^V$ such that the new problem admits a polynomial-time exact solution. Using tools from CSP (in particular Thapper and \v{Z}ivn\'y, 2012) to study the existence of such relaxations, we provide a much broader class of half-integral polytopes with the required properties, unifying and extending previously known cases. In addition to the insight into problems with half-integral relaxations, our results yield a range of new and improved FPT algorithms, including an $O^*(|\Sigma|^{2k})$-time algorithm for node-deletion Unique Label Cover with label set $\Sigma$ and an $O^*(4^k)$-time algorithm for Group Feedback Vertex Set, including the setting where the group is only given by oracle access. All these significantly improve on previous results. The latter result also implies the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, answering an open question of Cygan et al. (2012). Additionally, we propose a network flow-based approach to solve some cases of the relaxation problem. This gives the first linear-time FPT algorithm to edge-deletion Unique Label Cover., Comment: Added results on linear-time FPT algorithms (not present in SODA paper)

We give an algebraic, determinant-based algorithm for the K-Cycle problem, i.e., the problem of finding a cycle through a set of specified elements. Our approach gives a simple FPT algorithm for the problem, matching the $O^*(2^{|K|})$ running time of the algorithm of Bj\"orklund et al. (SODA, 2012). Furthermore, our approach is open for treatment by classical algebraic tools (e.g., Gaussian elimination), and we show that it leads to a polynomial compression of the problem, i.e., a polynomial-time reduction of the $K$-Cycle problem into an algebraic problem with coding size $O(|K|^3)$. This is surprising, as several related problems (e.g., k-Cycle and the Disjoint Paths problem) are known not to admit such a reduction unless the polynomial hierarchy collapses. Furthermore, despite the result, we are not aware of any witness for the K-Cycle problem of size polynomial in $|K|+\log n$, which seems (for now) to separate the notions of polynomial compression and polynomial kernelization (as a polynomial kernelization for a problem in NP necessarily implies a small witness)., Comment: 12 pages

The MULTICUT problem, given a graph G, a set of terminal pairs T={(s_i,t_i) | 1 <= i <= r} and an integer p, asks whether one can find a cutset consisting of at most p non-terminal vertices that separates all the terminal pairs, i.e., after removing the cutset, t_i is not reachable from s_i for each 1 <= i <= r. The fixed-parameter tractability of MULTICUT in undirected graphs, parameterized by the size of the cutset only, has been recently proven by Marx and Razgon (STOC'11) and, independently, by Bousquet et al. (STOC'11), after resisting attacks as a long-standing open problem. In this paper we prove that MULTICUT is fixed-parameter tractable on directed acyclic graphs, when parameterized both by the size of the cutset and the number of terminal pairs. We complement this result by showing that this is implausible for parameterization by the size of the cutset only, as this version of the problem remains W[1]-hard.