arXiv subjects

Magnus Wahlström

Publications and source records attributed to Magnus Wahlström.

At least 19 recordsLinked to original sources

arXiv2026

The Richness of CSP Non-redundancy

In the field of constraint satisfaction problems (CSP), a clause is called redundant if its satisfaction is implied by satisfying all other clauses. An instance of CSP$(P)$ is called non-redundant if it does not contain any redundant clause. The non-redundancy (NRD) of a predicate $P$ is the maximum number of clauses in a non-redundant instance of CSP$(P)$, as a function of the number of variables $n$. Recent progress has shown that non-redundancy is crucially linked to many other important questions in computer science and mathematics including sparsification, kernelization, query complexity, universal algebra, and extremal combinatorics. Given that non-redundancy is a nexus for many of these important problems, the central goal of this paper is to more deeply understand non-redundancy. Our first main result shows that for every rational number $r \ge 1$, there exists a finite CSP predicate $P$ such that the non-redundancy of $P$ is $Θ(n^r)$. Our second main result explores the concept of conditional non-redundancy first coined by Brakensiek and Guruswami [STOC 2025]. We completely classify the conditional non-redundancy of all binary predicates (i.e., constraints on two variables) by connecting these non-redundancy problems to the structure of high-girth graphs in extremal combinatorics. Inspired by these concrete results, we build off the work of Carbonnel [CP 2022] to develop an algebraic theory of conditional non-redundancy. As an application of this algebraic theory, we revisit the notion of Mal'tsev embeddings, which is the most general technique known to date for establishing that a predicate has linear non-redundancy. For example, we provide the first example of predicate with a Mal'tsev embedding that cannot be attributed to the structure of an Abelian group, but rather to the structure of the quantum Pauli group.

cs.DM↗

arXiv2026

Odd Properly Colored Cycles in Edge-Colored Graphs

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.

math.CO↗

arXiv2026

Super-linear Lower Bounds for CSP Non-Redundancy via Shrinking Instances

We say that an instance of a constraint satisfaction problem (CSP) is non-redundant if the satisfaction of each clause cannot be implied by the satisfaction of the other clauses in the instance. The non-redundancy (NRD) of a CSP is the maximal number of clauses a non-redundant instance can have for a given number of variables. NRD is closely tied to the behavior of CSPs in various computational models including their sparsification, kernelization, and streaming complexity. A primary open question in the study of non-redundancy is the identification of which CSP predicates have near-linear NRD. Recent works by Carbonnel [CP 2022], Khanna, Putterman and Sudan [STOC 2025], Brakensiek and Guruswami [STOC 2025] and Brakensiek, Guruswami, Jansen, Lagerkvist, and Wahlström [2025] have introduced various forms of gadget reductions between CSPs to relate their non-redundancy. The primary contribution of this work is to recontextualize many of these gadget reductions in a framework which we call hypergraph projections. By studying a quantity we call the shrinking factor of these hypergraph projections, we can more precisely predict when a gadget reduction between predicates can yield a super-linear NRD lower bound, greatly improving on the analysis of previous works. To illustrate the power of our framework, we identify some concrete CSP predicates whose non-redundancy is at the cusp of our understanding and show how our methods give lower bounds that could not have been achieved with previous methods. We also demonstrate how these gadget reductions can be automatically deduced using SAT solvers, thereby opening up novel computational avenues for discovering further relationships between the non-redundancy of various CSPs.

cs.DM↗

arXiv2026

Optimal FPT-Approximability for Modular Linear Equations

We show optimal FPT-approximability results for solving almost satisfiable systems of modular linear equations, completing the picture of the parameterized complexity and FPT-approximability landscape for the Min-$r$-Lin$(\mathbb{Z}_m)$ problem for every $r$ and $m$. In Min-$r$-Lin$(\mathbb{Z}_m)$, we are given a system $S$ of linear equations modulo $m$, each on at most $r$ variables, and the goal is to find a subset $Z \subseteq S$ of minimum cardinality such that $S - Z$ is satisfiable. The problem is UGC-hard to approximate within any constant factor for every $r \geq 2$ and $m \geq 2$, which motivates studying it through the lens of parameterized complexity with solution size as the parameter. From previous work (Dabrowski et al. SODA'23/TALG and ESA'25) we know that Min-$r$-Lin$(\mathbb{Z}_m)$ is W[1]-hard to FPT-approximate within any constant factor when $r \geq 3$, and that Min-$2$-Lin$(\mathbb{Z}_m)$ is in FPT when $m$ is prime and W[1]-hard when $m$ has at least two distinct prime factors. The case when $m = p^d$ for some prime $p$ and $d \geq 2$ has remained an open problem. We resolve this problem in this paper and prove the following: (1) We prove that Min-$2$-Lin$(\mathbb{Z}_{p^d})$ is in FPT for every prime $p$ and $d \geq 1$. This implies that Min-$2$-Lin$(\mathbb{Z}_{m})$ can be FPT-approximated within a factor of $ω(m)$, where $ω$ is the number of distinct prime factors of $m$. (2) We show that, under the ETH, Min-$2$-Lin$(\mathbb{Z}_m)$ cannot be FPT-approximated within $ω(m) - ε$ for any $ε> 0$. Our main algorithmic contribution is a new technique coined balanced subgraph covering, which generalizes important balanced subgraphs of Dabrowski et al. (SODA'23/TALG) and shadow removal of Marx and Razgon (STOC'11/SICOMP). For the lower bounds, we develop a framework for proving optimality of FPT-approximation factors under the ETH.

cs.DS↗

arXiv2025

Representative set statements for delta-matroids and the Mader delta-matroid

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.

cs.DS↗

arXiv2025

Determinantal Sieving

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. Long $(s,t)$-Path in $O^*(1.66^k)$ time, improving on $O^*(2^k)$, and Rank $k$ $(S,T)$-Linkage in so-called frameworks in $O^*(2^k)$ time, improving on $O^*(2^{|S|+O(k^2 \log(k+|\mathbb{F}|))})$ over general fields (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.

cs.DS↗

arXiv2025

Parameterized Approximability for Modular Linear Equations

We consider the Min-$r$-Lin$(Z_m)$ problem: given a system $S$ of length-$r$ linear equations modulo $m$, find $Z \subseteq S$ of minimum cardinality such that $S-Z$ is satisfiable. The problem is NP-hard and UGC-hard to approximate in polynomial time within any constant factor even when $r = m = 2$. We focus on parameterized approximation with solution size as the parameter. Dabrowski et al. showed that Min-$2$-Lin$(Z_m)$ is in FPT if $m$ is prime (i.e. $Z_m$ is a field), and it is W[1]-hard if $m$ is not a prime power. We show that Min-$2$-Lin$(Z_{p^n})$ is FPT-approximable within a factor of $2$ for every prime $p$ and integer $n \geq 2$. This implies that Min-$2$-Lin$(Z_m)$, $m \in Z^+$, is FPT-approximable within a factor of $2ω(m)$ where $ω(m)$ counts the number of distinct prime divisors of $m$. The idea behind the algorithm is to solve ever tighter relaxations of the problem, decreasing the set of possible values for the variables at each step. Working over $Z_{p^n}$ and viewing the values in base-$p$, one can roughly think of a relaxation as fixing the number of trailing zeros and the least significant nonzero digits of the values assigned to the variables. To solve the relaxed problem, we construct a certain graph where solutions can be identified with a particular collection of cuts. The relaxation may hide obstructions that will only become visible in the next iteration of the algorithm, which makes it difficult to find optimal solutions. To deal with this, we use a strategy based on shadow removal to compute solutions that (1) cost at most twice as much as the optimum and (2) allow us to reduce the set of values for all variables simultaneously. We complement the algorithmic result with two lower bounds, ruling out constant-factor FPT-approximation for Min-$3$-Lin$(R)$ over any nontrivial ring $R$ and for Min-$2$-Lin$(R)$ over some finite commutative rings $R$.

cs.DS↗

arXiv2025

Going Beyond Twin-width? CSPs with Unbounded Domain and Few Variables

We study a model of constraint satisfaction problems geared towards instances with few variables but with domain of unbounded size (udCSP). Our model is inspired by recent work on FPT algorithms for MinCSP where frequently both upper and lower bounds on the parameterized complexity of a problem correspond to $k$-variable udCSPs; e.g., the FPT algorithms for Boolean MinCSP (Kim et al., SODA 2023) and Directed Multicut with three cut requests (Hatzel et al., SODA 2023) both reduce to k-variable udCSPs, and the canonical W[1]-hardness construction in the area, Paired Min Cut by Marx and Razgon (IPL 2009), is effectively a k-variable udCSP. The udCSP framework represents constraints with unbounded domains via a collection $\mathcal{M}$ of unary maps into a finite-domain base language $Γ$. We develop an algebraic theory for studying the complexity of udCSP$(Γ,\mathcal{M})$ with a Galois connection based on partial multifunctions. We study three types of maps: unrestricted, one-hot, and monotone. For unrestricted maps, the problem is W[1]-hard for all but trivial cases, and for one-hot maps, the characterization coincides with Marx' FPT dichotomy for Boolean Weighted CSPs (Computational Complexity 2005). For the case of monotone maps Mo, we show that the complexity depends on restricted identifies we call ordered polymorphisms; we identify the "connector" polymorphism as the likely FPT boundary. We show that its absence implies that udCSP($Γ$,Mo) defines all permutations, and the problem is W[1]-hard; while its presence for a binary language implies bounded twin-width, and the problem is FPT (Twin-Width IV; Bonnet et al., JACM 2024). For non-binary languages, where twin-width does not apply, the polymorphism coincides with a notion of bounded projected grid-rank; however, we leave the FPT question for this case open.

cs.DS↗

arXiv2025

Kernelization for list $H$-coloring for graphs with small vertex cover

For a fixed graph $H$, in the List $H$-Coloring problem, we are given a graph $G$ along with list $L(v) \subseteq V(H)$ for every $v \in V(G)$, and we have to determine if there exists a list homomorphism $φ$ from $(G,L)$ to $H$, i.e., an edge preserving mapping $φ: V(G)\to V(H)$ that satisfies $φ(v)\in L(v)$ for every $v\in V(G)$. Note that if $H$ is the complete graph on $q$ vertices, the problem is equivalent to List $q$-Coloring. We investigate the kernelization properties of List $H$-Coloring parameterized by the vertex cover number of $G$: given an instance $(G,L)$ and a vertex cover of $G$ of size $k$, can we reduce $(G,L)$ to an equivalent instance $(G',L')$ of List $H$-Coloring where the size of $G'$ is bounded by a low-degree polynomial $p(k)$ in $k$? This question has been investigated previously by Jansen and Pieterse [Algorithmica 2019], who provided an upper bound, which turns out to be optimal if $H$ is a complete graph, i.e., for List $q$-Coloring. This result was one of the first applications of the method of kernelization via bounded-degree polynomials. We define two new integral graph invariants, $c^*(H)$ and $d^*(H)$, with $d^*(H) \leq c^*(H) \leq d^*(H)+1$, and show that for every graph $H$, List $H$-Coloring -- has a kernel with $\mathcal{O}(k^{c^*(H)})$ vertices, -- admits no kernel of size $\mathcal{O}(k^{d^*(H)-\varepsilon})$ for any $\varepsilon > 0$, unless the polynomial hierarchy collapses. -- Furthermore, if $c^*(H) > d^*(H)$, then there is a kernel with $\mathcal{O}(k^{c^*(H)-\varepsilon})$ vertices where $\varepsilon \geq 2^{1-c^*(H)}$. Additionally, we show that for some classes of graphs, including powers of cycles and graphs $H$ where $Δ(H) \leq c^*(H)$ (which in particular includes cliques), the bound $d^*(H)$ is tight, using the polynomial method. We conjecture that this holds in general.

math.CO↗

arXiv2025

Flow-augmentation III: Complexity dichotomy for Boolean CSPs parameterized by the number of unsatisfied constraints

We study the parameterized problem of satisfying ``almost all'' constraints of a given formula $F$ over a fixed, finite Boolean constraint language $Γ$, with or without weights. More precisely, for each finite Boolean constraint language $Γ$, we consider the following two problems. In Min SAT$(Γ)$, the input is a formula $F$ over $Γ$ and an integer $k$, and the task is to find an assignment $α\colon V(F) \to \{0,1\}$ that satisfies all but at most $k$ constraints of $F$, or determine that no such assignment exists. In Weighted Min SAT$(Γ$), 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 $α$ such that (1) $α$ 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 $Γ$, either Weighted Min SAT$(Γ)$ is FPT; or Weighted Min SAT$(Γ)$ is W[1]-hard but Min SAT$(Γ)$ is FPT; or Min SAT$(Γ)$ is W[1]-hard. This generalizes recent work of Kim et al. (SODA 2021) which did not consider weighted problems, and only considered languages $Γ$ 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).

cs.CC↗

arXiv2025

FPT algorithms over linear delta-matroids with applications

Matroids, particularly linear ones, have been a powerful tool in parameterized complexity for algorithms and kernelization. They have sped up or replaced dynamic programming. Delta-matroids generalize matroids by encapsulating structures such as non-maximum matchings in general graphs and various path-packing and topological configurations. Linear delta-matroids (represented by skew-symmetric matrices) offer significant expressive power and enable powerful algorithms. We investigate parameterized complexity aspects of problems defined over linear delta-matroids or with delta-matroid constraints. Our analysis of basic intersection and packing problems reveals a different complexity landscape compared to the familiar matroid case. In particular, there is a stark contrast between the cardinality parameter $k$ and the rank parameter $r$. For example, finding an intersection of size $k$ of three linear delta-matroids is W[1]-hard when parameterized by $k$, while more general problems (e.g., finding a set packing of size $k$ feasible in a linear delta-matroid) are FPT when parameterized by $r$. We extend the recent determinantal sieving procedure of Eiben, Koana and Wahlström (SODA 2024) to sieve a polynomial for a monomial whose support is feasible in a linear delta-matroid by $r$. Second, we investigate a class of problems that remains FPT when parameterized by $k$, even on delta-matroids of unbounded rank. We begin with Delta-matroid Triangle Cover - finding a feasible set of size $k$ that can be covered by a vertex-disjoint packing of triangles (sets of size 3) from a given collection. This approach allows us to find a packing of $K_3$'s and $K_2$'s in a graph with a maximum number of edges, parameterized above the matching number. As applications, we settle questions on the FPT status of Cluster Subgraph and Strong Triadic Closure parameterized above the matching number.

cs.DS↗

arXiv2025

Polynomial Kernel and Incompressibility for Prison-Free Edge Deletion and Completion

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.

cs.DS↗

arXiv2024

Towards a Parameterized Approximation Dichotomy of MinCSP for Linear Equations over Finite Commutative Rings

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 Wahlström [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.

cs.DS↗

arXiv2024

Parameterized Complexity of MinCSP over the Point Algebra

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$(Γ)$ for $Γ\subseteq \{<,=,\leq,\neq\}$ 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 $Γ\subseteq \{<,=,\leq,\neq\}$ contains both $\leq$ and $\neq$, then MinCSP$(Γ)$ 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.

cs.DS↗

arXiv2024

Faster algorithms on linear delta-matroids

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 $O(n^ω)$-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 $O(n^ω)$. 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 $O(n^ω)$-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 $O(n^{ω+1})$ time. This is the first polynomial-time algorithm for the problem, solving an open question of Kakimura and Takamatsu.

cs.DS↗

arXiv2023

Parameterized Complexity of Equality MinCSP

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$(Γ)$ for every finite equality language $Γ$, 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.

cs.DS↗

arXiv2023

Flow-augmentation II: Undirected graphs

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.

cs.DS↗

arXiv2023

Flow-augmentation I: Directed graphs

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.

cs.DS↗