Fixed-parameter Tractability Research Articles

Fixed-Parameter Tractability of Maximum Colored Path and Beyond

We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is as follows. We give a randomized algorithm, that given a colored \(n\) -vertex undirected graph, vertices \(s\) and \(t\) , and an integer \(k\) , finds an \((s,t)\) -path containing at least \(k\) different colors in time \(2^{k}n^{\mathcal{O}(1)}\) . This is the first FPT algorithm for this problem, and it generalizes the algorithm of Björklund, Husfeldt, and Taslaman on finding a path through \(k\) specified vertices. It also implies the first \(2^{k}n^{\mathcal{O}(1)}\) time algorithm for finding an \((s,t)\) -path of length at least \(k\) . Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an \(n\) -vertex undirected graph \(G\) , a matroid \(M\) whose elements correspond to the vertices of \(G\) and which is represented over a finite field of order \(q\) , a positive integer weight function on the vertices of \(G\) , two sets of vertices \(S,T\subseteq V(G)\) , and integers \(p,k,w\) , and the task is to find \(p\) vertex-disjoint paths from \(S\) to \(T\) so that the union of the vertices of these paths contains an independent set of \(M\) of cardinality \(k\) and weight \(w\) , while minimizing the sum of the lengths of the paths. We give a \(2^{p+\mathcal{O}(k^{2}\log(q+k))}n^{\mathcal{O}(1)}w\) time randomized algorithm for this problem.

The Maximum Zero-Sum Partition problem

We study the Maximum Zero-Sum Partition problem (or MZSP), defined as follows: given a multiset S={a1,a2,…,an} of integers ai∈Z⁎ (where Z⁎ denotes the set of non-zero integers) such that ∑i=1nai=0, find a maximum cardinality partition {S1,S2,…,Sk} of S such that, for every 1≤i≤k, ∑aj∈Siaj=0. Solving MZSP is useful in genomics for computing evolutionary distances between pairs of species. Our contributions are a series of algorithmic results concerning MZSP, in terms of complexity, (in)approximability, with a particular focus on the fixed-parameter tractability of MZSP with respect to either (i) the size k of the solution, (ii) the number of negative (resp. positive) values in S and (iii) the largest integer in S.

Parameterized complexity of coverage in multi-interface IoT networks: Pathwidth

The Internet of Things (IoT) has emerged as one of the growing fields in digital technology over the past decade. A primary goal of IoT is to connect physical objects to the Internet to provide various services. Due to the vast number and diversity of these objects, referred to as devices, IoT must tackle both traditional and novel theoretical and practical network problems. Among these, multi-interface problems are well-known and have been extensively studied.This research focuses on one of the newest multi-interface models that fits well within the IoT context. It is known as the Coverage in the budget-constrained multi-interface problem, where the budget represents the total amount of energy in the network, and coverage refers to the model’s goal of activating all required communications among IoT devices. Since most IoT devices are battery-powered, energy consumption must be considered to extend the network’s lifespan. This means selecting the most energy-efficient interface configuration that allows all desired connections to function. To achieve this, both global energy consumption and the local number of active interfaces are limited. Moreover, this model also incentivize devices to turn on the available interfaces to create a more performant network. Finally, this model also takes into account the performance of the networks assigning a profit to devices that activate interfaces and realize connections.This problem can be represented using an undirected graph where each vertex represents a device, and each edge represents a desired connection. Every device is equipped with a set of available interfaces that can be used to facilitate transmission among the devices. The final goal is to activate a subset of the available interfaces that maximize the total profit, while not violating the constraints.This problem has been recognized as NP-hard, which is why we decided to investigate the decision version from the perspective of fixed-parameter tractability (FPT) theory. FPT is an advanced area of complexity theory that aims to identify the core complexity of a combinatorial problem by incorporating parameters into the time complexity domain.We provide two fixed-parameter tractability results, each describing an FPT algorithm. One algorithm is based on the well-known pathwidth parameter, the number of available interfaces, and the maximum available energy. The other algorithm considers pathwidth, the number of available interfaces, and an upper bound on the optimal profit. Finally, we show that these two algorithms can be applied to the maximization version of the problem.

Fixed parameter tractable algorithms for watchman route related problems on line segment arrangements

In this paper, we study the parameterized complexity of some variants of the watchman route problem on line segment arrangements. We show that finding a minimum length Watchman Tree/Route on line Segment arrangements (WTS/WRS) is Fixed Parameter Tractable (FPT) with respect to the number of vertices in the tree/route as the parameter. We then prove that another variant of WTS referred to as Minimum Corridor Connection (MCC), is FPT with respect to the number of regions as the parameter. We also consider WRS with the link distance metric (LWRS) and propose an FPT algorithm for a restricted version in which each line segment in the input arrangement has a bound on the number of intersections it makes with other segments in the arrangement.

Backdoor DNFs

We introduce backdoor DNFs, as a tool to measure the theoretical hardness of CNF formulas. Like backdoor sets and backdoor trees, backdoor DNFs are defined relative to a tractable class of CNF formulas. Each conjunctive term of a backdoor DNF defines a partial assignment that moves the input CNF formula into the base class. Backdoor DNFs are more expressive and potentially smaller than their predecessors backdoor sets and backdoor trees. We establish the fixed-parameter tractability of the backdoor DNF detection problem. Our results hold for the fundamental base classes Horn and 2CNF, and their combination. We complement our theoretical findings by an empirical study. Our experiments show that backdoor DNFs provide a significant improvement over their predecessors.

Temporal reachability minimization: Delaying vs. deleting

We study spreading processes in temporal graphs, that is, graphs whose connections change over time. More precisely, we investigate how such a spreading process, emerging from a given set of sources, can be contained to a small part of the graph. We consider two ways of modifying the graph, which are (1) deleting and (2) delaying connections. We show a close relationship between the two associated problems. It is known that both problems are W[1]-hard when parameterized by the number of modifications. We consider the number of vertices to which the spread is contained as a parameter. Surprisingly, we prove W[1]-hardness for the deletion variant but fixed-parameter tractability for the delaying variant. Furthermore, we give a polynomial time algorithm for both problem variants when the graph has a tree structure and show how to generalize this result to an FPT-algorithm for the so-called timed feedback vertex number as a parameter.

Counting Answers to Unions of Conjunctive Queries: Natural Tractability Criteria and Meta-Complexity

We study the problem of counting answers to unions of conjunctive queries (UCQs) under structural restrictions on the input query. Concretely, given a class C of UCQs, the problem #UCQ (C) provides as input a UCQ Ψ ∈ C and a database D and the problem is to compute the number of answers of Ψ in D. Chen and Mengel [PODS'16] have shown that for any recursively enumerable class C, the problem #UCQ (C) is either fixed-parameter tractable or hard for one of the parameterised complexity classes W[1] or #W[1]. However, their tractability criterion is unwieldy in the sense that, given any concrete class C of UCQs, it is not easy to determine how hard it is to count answers to queries in C. Moreover, given a single specific UCQ Ψ, it is not easy to determine how hard it is to count answers to Ψ. In this work, we address the question of finding a natural tractability criterion: The combined conjunctive query of a UCQ Ψ=φ 1 ∨ ... ∨ φ l is the conjunctive query ^ Ψ = φ_1 ∧ ... ∧ φ l . We show that under natural closure properties of C, the problem #UCQ (C) is fixed-parameter tractable if and only if the combined conjunctive queries of UCQs in C, and their contracts, have bounded treewidth. A contract of a conjunctive query is an augmented structure, taking into account how the quantified variables are connected to the free variables --- if all variables are free, then a conjunctive query is equal to its contract; in this special case the criterion for fixed-parameter tractability of #UCQ (C) thus simplifies to the combined queries having bounded treewidth. Finally, we give evidence that a closure property on C is necessary for obtaining a natural tractability criterion: We show that even for a single UCQ Ψ, the meta problem of deciding whether #UCQ (Ψ) can be solved in time O(|D| d ) is NP-hard for any fixed d ≥ 1. Moreover, we prove that a known exponential-time algorithm for solving the meta problem is optimal under assumptions from fine-grained complexity theory. As a corollary of our reduction, we also establish that approximating the Weisfeiler-Leman-Dimension of a UCQ is NP-hard.

Is this network proper forest-based?

In evolutionary biology, networks are becoming increasingly used to represent evolutionary histories for species that have undergone non-treelike or reticulate evolution. Such networks are essentially directed acyclic graphs with a leaf set that corresponds to a collection of species, and in which non-leaf vertices with indegree 1 correspond to speciation events and vertices with indegree greater than 1 correspond to reticulate events such as gene transfer. Recently forest-based networks have been introduced, which are essentially (multi-rooted) networks that can be formed by adding some arcs to a collection of phylogenetic trees (or phylogenetic forest), where each arc is added in such a way that its ends always lie in two different trees in the forest. In this paper, we consider the complexity of deciding whether a given network is proper forest-based, that is, whether it can be formed by adding arcs to some underlying phylogenetic forest which contains the same number of trees as there are roots in the network. More specifically, we show that it is NP-complete to decide whether a tree-child network with m roots is proper forest-based, for each m≥2. Moreover, for binary networks the problem remains NP-complete when m≥3 but becomes polynomial-time solvable for m=2. We also give a fixed parameter tractable (FPT) algorithm, with parameters the maximum outdegree of a vertex, the number of roots, and the number of indegree 2 vertices, for deciding if a semi-binary network is proper forest-based. A key element in proving our results is a new characterization for when a network with m roots is proper forest-based in terms of certain m-colorings.

The parameterized complexity of finding minimum bounded chains

Finding the smallest d-chain with a specific (d−1)-boundary in a simplicial complex is known as the Minimum Bounded Chain problem (MBCd). MBCd is NP-hard for all d≥2. In this paper, we prove that it is also W[1]-hard for all d≥2, if we parameterize the problem by solution size. We also give an algorithm solving MBC1 in polynomial time and introduce and implement two fixed parameter tractable (FPT) algorithms solving MBCd for all d. The first algorithm uses a shortest path approach and is parameterized by solution size and coface degree. The second algorithm is a dynamic programming approach based on treewidth, which has the same runtime as a lower bound we prove under the exponential time hypothesis.

New algorithms for fair k-center problem with outliers and capacity constraints

The fair k-center problem has been paid lots of attention recently. In the fair k-center problem, we are given a set X of points in a metric space and a parameter k∈Z+, where the points in X are divided into several groups, and each point is assigned a color to denote which group it is in. The goal is to partition X into k clusters such that the number of cluster centers with each color is equal to a given value, and the k-center problem objective is minimized. In this paper, we consider the fair k-center problem with outliers and capacity constraints, denoted as the fair k-center with outliers (FkCO) problem and the capacitated fair k-center (CFkC) problem, respectively. The outliers constraints allow up to z outliers to be discarded when computing the objective function, while the capacity constraints require that each cluster has size no more than L. In this paper, we design an Fixed-Parameter Tractability (FPT) approximation algorithm and a polynomial approximation algorithm for the above two problems. In particular, our algorithms give (1+ϵ)-approximations with FPT time for the FkCO and CFkC problems in doubling metric space. Moreover, we also propose a 3-approximation algorithm in polynomial time for the FkCO problem with some reasonable assumptions.

Exact Algorithms and Lowerbounds for Multiagent Path Finding: Power of Treelike Topology

In the Multiagent Path Finding (MAPF for short) problem, we focus on efficiently finding non-colliding paths for a set of k agents on a given graph G, where each agent seeks a path from its source vertex to a target. An important measure of the quality of the solution is the length of the proposed schedule l, that is, the length of a longest path (including the waiting time). In this work, we propose a systematic study under the parameterized complexity framework. The hardness results we provide align with many heuristics used for this problem, whose running time could potentially be improved based on our Fixed-Parameter Tractability (FPT) results. We show that MAPF is W[1]-hard with respect to k (even if k is combined with the maximum degree of the input graph). The problem remains NP-hard in planar graphs even if the maximum degree and the makespan l are fixed constants. On the positive side, we show an FPT algorithm for k+l. As we continue, the structure of G comes into play. We give an FPT algorithm for parameter k plus the diameter of the graph G. The MAPF problem is W[1]-hard for cliquewidth of G plus l while it is FPT for treewidth of G plus l.

A General Theoretical Framework for Learning Smallest Interpretable Models

We develop a general algorithmic framework that allows us to obtain fixed-parameter tractability for computing smallest symbolic models that represent given data. Our framework applies to all ML model types that admit a certain extension property. By showing this extension property for decision trees, decision sets, decision lists, and binary decision diagrams, we obtain that minimizing these fundamental model types is fixed-parameter tractable. Our framework even applies to ensembles, which combine individual models by majority decision.

Grouped domination parameterized by vertex cover, twin cover, and beyond

A dominating set S of graph G is called an r-grouped dominating set if S can be partitioned into S1,S2,…,Sk such that the size of each unit Si is r and the subgraph of G induced by Si is connected. The concept of r-grouped dominating sets generalizes several well-studied variants of dominating sets with requirements for connected component sizes, such as the ordinary dominating sets (r=1), paired dominating sets (r=2), and connected dominating sets (r is arbitrary and k=1). In this paper, we investigate the computational complexity of r-Grouped Dominating Set, which is the problem of deciding whether a given graph has an r-grouped dominating set with at most k units. For general r, r-Grouped Dominating Set is hard to solve in various senses because the hardness of the connected dominating set is inherited. We thus focus on the case in which r is a constant or a parameter, but we see that r-Grouped Dominating Set for every fixed r>0 is still hard to solve. From the observations about the hardness, we consider the parameterized complexity concerning well-studied graph structural parameters. We first see that r-Grouped Dominating Set is fixed-parameter tractable for r and treewidth, which is derived from the fact that the condition of r-grouped domination for a constant r can be represented as monadic second-order logic (MSO2). This fixed-parameter tractability is good news, but the running time is not practical. We then design an O⁎(min⁡{(2τ(r+1))τ,(2τ)2τ})-time algorithm for general r≥2, where τ is the twin cover number, which is a parameter between vertex cover number and clique-width. For paired dominating set and trio dominating set, i.e., r∈{2,3}, we can speed up the algorithm, whose running time becomes O⁎((r+1)τ). We further argue the relationship between FPT results and graph parameters, which draws the parameterized complexity landscape of r-Grouped Dominating Set.

Long directed detours: Reduction to 2-Disjoint Paths

In the Longest(s,t)-Detour problem, we look for an (s,t)-path that is at least k vertices longer than a shortest one. We study the parameterized complexity of Longest(s,t)-Detour when parameterized by k: this falls into the research paradigm of ‘parameterization above guarantee’. Whereas the problem is known to be fixed-parameter tractable (FPT) on undirected graphs, the status of Longest(s,t)-Detour on directed graphs remains highly unclear: it is not even known to be solvable in polynomial time for k=1. Recently, Fomin et al. made progress in this direction by showing that the problem is FPT on every class of directed graphs where the 3-Disjoint Paths problem is solvable in polynomial time. We improve upon their result by weakening this assumption: we show that only a polynomial-time algorithm for 2-Disjoint Paths is required.

An impossible combinatorial counting method in distance geometry

The Distance Geometry Problem asks for a geometric representation of a given weighted graph in RK so that vertices are points and edges are segments with lengths equal to the corresponding weights. Two problem variants are defined by a vertex order given as part of the input, which allows the use of a branching algorithm based on K-lateration: find two possible positions for the next vertex j using the positions of K predecessors and their distances to j, then explore each position recursively, pruning positions at need. Whereas the first variant only requires the K predecessors to exist, the second variant also requires them to be contiguous and immediately preceding j. For this variant, fixed-parameter tractability of the algorithm can be established by means of a solution counting method that only depends on the graph edges (rather than their weights). Only in the first variant, however, it is possible to efficiently construct a suitable vertex order directly from the graph. Since both fixed-parameter tractability and efficient vertex order construction are desirable properties, one would need an analogous solution counting method for the first variant. In this paper we prove that such a counting method cannot be devised for the first variant.

On the Parameterized Intractability of Determinant Maximization

In the Determinant Maximization problem, given an n×n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \ imes n$$\\end{document} positive semi-definite matrix A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf {A}} $$\\end{document} in Qn×n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}^{n \ imes n}$$\\end{document} and an integer k, we are required to find a k×k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \ imes k$$\\end{document} principal submatrix of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf {A}} $$\\end{document} having the maximum determinant. This problem is known to be NP-hard and further proven to be W[1]-hard with respect to k by Koutis (Inf Process Lett 100:8–13, 2006); i.e., a f(k)nO(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(k)n^{{{\\,\\mathrm{\\mathcal {O}}\\,}}(1)}$$\\end{document}-time algorithm is unlikely to exist for any computable function f. However, there is still room to explore its parameterized complexity in the restricted case, in the hope of overcoming the general-case parameterized intractability. In this study, we rule out the fixed-parameter tractability of Determinant Maximization even if an input matrix is extremely sparse or low rank, or an approximate solution is acceptable. We first prove that Determinant Maximization is NP-hard and W[1]-hard even if an input matrix is an arrowhead matrix; i.e., the underlying graph formed by nonzero entries is a star, implying that the structural sparsity is not helpful. By contrast, Determinant Maximization is known to be solvable in polynomial time on tridiagonal matrices (Al-Thani and Lee, in: LAGOS, 2021). Thereafter, we demonstrate the W[1]-hardness with respect to the rankr of an input matrix. Our result is stronger than Koutis’ result in the sense that any k×k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \ imes k$$\\end{document} principal submatrix is singular whenever k>r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k > r$$\\end{document}. We finally give evidence that it is W[1]-hard to approximate Determinant Maximization parameterized by k within a factor of 2-ck\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{-c\\sqrt{k}}$$\\end{document} for some universal constant c>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c > 0$$\\end{document}. Our hardness result is conditional on the Parameterized Inapproximability Hypothesis posed by Lokshtanov et al. (in: SODA, 2020), which asserts that a gap version of Binary Constraint Satisfaction Problem is W[1]-hard. To complement this result, we develop an ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}-additive approximation algorithm that runs in ε-r2·rO(r3)·nO(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon ^{-r^2} \\cdot r^{{{\\,\\mathrm{\\mathcal {O}}\\,}}(r^3)} \\cdot n^{{{\\,\\mathrm{\\mathcal {O}}\\,}}(1)}$$\\end{document} time for the rank r of an input matrix, provided that the diagonal entries are bounded.

Characterization of matrices with bounded Graver bases and depth parameters and applications to integer programming

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to a row-equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse row-equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-norm of the Graver basis is bounded by a function of the maximum ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix row-equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such a row-equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-norm of the Graver basis of the constraint matrix, when parameterized by the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix row-equivalent to the constraint matrix.

Orthologic with Axioms

We study the proof theory and algorithms for orthologic, a logical system based on ortholattices, which have shown practical relevance in simplification and normalization of verification conditions. Ortholattices weaken Boolean algebras while having polynomial-time equivalence checking that is sound with respect to Boolean algebra semantics. We generalize ortholattice reasoning and obtain an algorithm for proving a larger class of classically valid formulas. As the key result, we analyze a proof system for orthologic augmented with axioms. An important feature of the system is that it limits the number of formulas in a sequent to at most two, which makes the extension with axioms non-trivial. We show a generalized form of cut elimination for this system, which implies a sub-formula property. From there we derive a cubic-time algorithm for provability from axioms, or equivalently, for validity in finitely presented ortholattices. We further show that propositional resolution of width 5 proves all formulas provable in orthologic with axioms. We show that orthologic system subsumes resolution of width 2 and arbitrarily wide unit resolution and is complete for reasoning about generalizations of propositional Horn clauses. Moving beyond ground axioms, we introduce effectively propositional orthologic (by analogy with EPR for classical logic), presenting its semantics as well as a sound and complete proof system. Our proof system implies the decidability of effectively propositional orthologic, as well as its fixed-parameter tractability for a bounded maximal number of variables in each axiom. As a special case, we obtain a generalization of Datalog with negation and disjunction.

The maximum 2-edge-colorable subgraph problem and its fixed-parameter tractability

A $k$-edge-coloring of a graph is an assignment of colors $\{1,...,k\}$ to edges of the graph such that adjacent edges receive different colors. In the maximum $k$-edge-colorable subgraph problem we are given a graph and an integer $k$, the goal is to find a $k$-edge-colorable subgraph with maximum number of edges together with its $k$-edge-coloring. In this paper, we consider the maximum 2-edge-colorable subgraph problem and present some results that deal with the fixed-parameter tractability of this problem.

Disentangling the Computational Complexity of Network Untangling

We study the network untangling problem introduced by Rozenshtein et al. (Data Min. Knowl. Disc. 35(1), 213–247, 2021), which is a variant of Vertex Cover on temporal graphs–graphs whose edge set changes over discrete time steps. They introduce two problem variants. The goal is to select at most k time intervals for each vertex such that all time-edges are covered and (depending on the problem variant) either the maximum interval length or the total sum of interval lengths is minimized. This problem has data mining applications in finding activity timelines that explain the interactions of entities in complex networks. Both variants of the problem are NP-hard. In this paper, we initiate a multivariate complexity analysis involving the following parameters: number of vertices, lifetime of the temporal graph, number of intervals per vertex, and the interval length bound. For both problem versions, we (almost) completely settle the parameterized complexity for all combinations of those four parameters, thereby delineating the border of fixed-parameter tractability.