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We perform a refined complexity-theoretic analysis of three classical problems in the context of Hierarchical Task Network Planning: the verification of a provided plan, whether an executable plan exists, and whether a given state can be reached. Our focus lies on identifying structural properties which yield tractability. We obtain new polynomial algorithms for all three problems on a natural class of primitive networks, along with corresponding lower bounds. We also obtain an algorithmic meta-theorem for lifting polynomial-time solvability from primitive to general task networks, and prove that its preconditions are tight. Finally, we analyze the parameterized complexity of the three problems., Comment: Updated version contains essentially the same results in a more concise and improved write-up

Atomic congestion games are a classic topic in network design, routing, and algorithmic game theory, and are capable of modeling congestion and flow optimization tasks in various application areas. While both the price of anarchy for such games as well as the computational complexity of computing their Nash equilibria are by now well-understood, the computational complexity of computing a system-optimal set of strategies -- that is, a centrally planned routing that minimizes the average cost of agents -- is severely understudied in the literature. We close this gap by identifying the exact boundaries of tractability for the problem through the lens of the parameterized complexity paradigm. After showing that the problem remains highly intractable even on extremely simple networks, we obtain a set of results which demonstrate that the structural parameters which control the computational (in)tractability of the problem are not vertex-separator based in nature (such as, e.g., treewidth), but rather based on edge separators. We conclude by extending our analysis towards the (even more challenging) min-max variant of the problem., Comment: Short version appeared at AAAI 2024. Long version accepted in the Journal of Artificial Intelligence

Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces: Let $\mathbf{v} \in \mathbb{Q}^d$ be a rational vector, $(T_{1}, T_{2} \ldots T_{m})$ a list of $d \times d$ rational matrices, $S \in \mathbb{Q}^{h \times d}$ a rational matrix not necessarily square and $k$ a parameter. The goal is to compute the number of ways one can choose $k$ matrices $T_{i_1}, T_{i_2}, \ldots, T_{i_k}$ from the list such that $ST_{i_k} \cdots T_{i_1}\mathbf{v} = \mathbf{0} \in \mathbb{Q}^h$. In this paper, we show that this problem is $\# W[2]$-hard for parameter $k$. As a consequence, computing the $k$-th homotopy group of a $d$-dimensional 1-connected topological space for $d > 3$ is $\# W[2]$-hard for parameter $k$. We also discuss a decision version of the problem and its several modifications for which we show $W[1]/W[2]$-hardness. This is in contrast to the parameterized $k$-sum problem, which is only $W[1]$-hard (Abboud-Lewi-Williams, ESA'14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized the matrix dimensions and the order of the field., Comment: Version 2: 18 pages, 5 figures; it contains result from arXiv:2209.09788; minor improvements, typos corrected

In a right-angle crossing (RAC) drawing of a graph, each edge is represented as a polyline and edge crossings must occur at an angle of exactly $90^\circ$, where the number of bends on such polylines is typically restricted in some way. While structural and topological properties of RAC drawings have been the focus of extensive research, little was known about the boundaries of tractability for computing such drawings. In this paper, we initiate the study of RAC drawings from the viewpoint of parameterized complexity. In particular, we establish that computing a RAC drawing of an input graph $G$ with at most $b$ bends (or determining that none exists) is fixed-parameter tractable parameterized by either the feedback edge number of $G$, or $b$ plus the vertex cover number of $G$., Comment: Accepted at GD 2023

Probably Approximately Correct (i.e., PAC) learning is a core concept of sample complexity theory, and efficient PAC learnability is often seen as a natural counterpart to the class P in classical computational complexity. But while the nascent theory of parameterized complexity has allowed us to push beyond the P-NP ``dichotomy'' in classical computational complexity and identify the exact boundaries of tractability for numerous problems, there is no analogue in the domain of sample complexity that could push beyond efficient PAC learnability. As our core contribution, we fill this gap by developing a theory of parameterized PAC learning which allows us to shed new light on several recent PAC learning results that incorporated elements of parameterized complexity. Within the theory, we identify not one but two notions of fixed-parameter learnability that both form distinct counterparts to the class FPT -- the core concept at the center of the parameterized complexity paradigm -- and develop the machinery required to exclude fixed-parameter learnability. We then showcase the applications of this theory to identify refined boundaries of tractability for CNF and DNF learning as well as for a range of learning problems on graphs., Comment: Accepted at AAAI 2023

Decompositional parameters such as treewidth are commonly used to obtain fixed-parameter algorithms for NP-hard graph problems. For problems that are W[1]-hard parameterized by treewidth, a natural alternative would be to use a suitable analogue of treewidth that is based on edge cuts instead of vertex separators. While tree-cut width has been coined as such an analogue of treewidth for edge cuts, its algorithmic applications have often led to disappointing results: out of twelve problems where one would hope for fixed-parameter tractability parameterized by an edge-cut based analogue to treewidth, eight were shown to be W[1]-hard parameterized by tree-cut width. As our main contribution, we develop an edge-cut based analogue to treewidth called edge-cut width. Edge-cut width is, intuitively, based on measuring the density of cycles passing through a spanning tree of the graph. Its benefits include not only a comparatively simple definition, but mainly that it has interesting algorithmic properties: it can be computed by a fixed-parameter algorithm, and it yields fixed-parameter algorithms for all the aforementioned problems where tree-cut width failed to do so., Comment: 27 pages, 4 figures

An influential 1990 paper of Hochbaum and Shanthikumar made it common wisdom that "convex separable optimization is not much harder than linear optimization" [JACM 1990]. We exhibit two fundamental classes of mixed integer (linear) programs that run counter this intuition. Namely those whose constraint matrices have small coefficients and small primal or dual treedepth: While linear optimization is easy [Brand, Kouteck\'y, Ordyniak, AAAI 2021], we prove that separable convex optimization IS much harder. Moreover, in the pure integer and mixed integer linear cases, these two classes have the same parameterized complexity. We show that they yet behave quite differently in the separable convex mixed integer case. Our approach employs the mixed Graver basis introduced by Hemmecke [Math. Prog. 2003]. We give the first non-trivial lower and upper bounds on the norm of mixed Graver basis elements. In previous works involving the integer Graver basis, such upper bounds have consistently resulted in efficient algorithms for integer programming. Curiously, this does not happen in our case. In fact, we even rule out such an algorithm.

Linear programs (LPs) can be solved by polynomially many moves along the circuit direction improving the objective the most, so-called deepest-descent steps (dd-steps). Computing these steps is NP-hard (De Loera et al., arXiv, 2019), a consequence of the hardness of deciding the existence of an optimal circuit-neighbor (OCNP) on LPs with non-unique optima. We prove OCNP is easy under the promise of unique optima, but already $O(n^{1-\varepsilon})$-approximating dd-steps remains hard even for totally unimodular $n$-dimensional 0/1-LPs with a unique optimum. We provide a matching $n$-approximation.

We study the following problem and its applications: given a homogeneous degree-$d$ polynomial $g$ as an arithmetic circuit, and a $d \times d$ matrix $X$ whose entries are homogeneous linear polynomials, compute $g(\partial/\partial x_1, \ldots, \partial/\partial x_n) \det X$. By considering special cases of this problem we obtain faster parameterized algorithms for several problems, including the matroid $k$-parity and $k$-matroid intersection problems, faster \emph{deterministic} algorithms for testing if a linear space of matrices contains an invertible matrix (Edmonds's problem) and detecting $k$-internal outbranchings, and more. We also match the runtime of the fastest known deterministic algorithm for detecting subgraphs of bounded pathwidth, while using a new approach. Our approach raises questions in algebraic complexity related to Waring rank and the exponent of matrix multiplication $\omega$. In particular, we study a new complexity measure on the space of homogeneous polynomials, namely the bilinear complexity of a polynomial's apolar algebra. Our algorithmic improvements are reflective of the fact that for the degree-$n$ determinant polynomial this quantity is at most $O(n 2^{\omega n})$, whereas all known upper bounds on the Waring rank of this polynomial exceed $n!$.

Solving (mixed) integer linear programs, (M)ILPs for short, is a fundamental optimization task. While hard in general, recent years have brought about vast progress for solving structurally restricted, (non-mixed) ILPs: $n$-fold, tree-fold, 2-stage stochastic and multi-stage stochastic programs admit efficient algorithms, and all of these special cases are subsumed by the class of ILPs of small treedepth. In this paper, we extend this line of work to the mixed case, by showing an algorithm solving MILP in time $f(a,d) \textrm{poly}(n)$, where $a$ is the largest coefficient of the constraint matrix, $d$ is its treedepth, and $n$ is the number of variables. This is enabled by proving bounds on the denominators of the vertices of bounded-treedepth (non-integer) linear programs. We do so by carefully analyzing the inverses of invertible submatrices of the constraint matrix. This allows us to afford scaling up the mixed program to the integer grid, and applying the known methods for integer programs. We trace the limiting boundary of our approach, showing that naturally related classes of linear programs have vertices of unbounded fractionality. Finally, we show that restricting the structure of only the integral variables in the constraint matrix does not yield tractable special cases.

We devise an algorithm that approximately computes the number of paths of length $k$ in a given directed graph with $n$ vertices up to a multiplicative error of $1 \pm \varepsilon$. Our algorithm runs in time $\varepsilon^{-2} 4^k(n+m) \operatorname{poly}(k)$. The algorithm is based on associating with each vertex an element in the exterior (or, Grassmann) algebra, called an extensor, and then performing computations in this algebra. This connection to exterior algebra generalizes a number of previous approaches for the longest path problem and is of independent conceptual interest. Using this approach, we also obtain a deterministic $2^{k}\cdot\operatorname{poly}(n)$ time algorithm to find a $k$-path in a given directed graph that is promised to have few of them. Our results and techniques generalize to the subgraph isomorphism problem when the subgraphs we are looking for have bounded pathwidth. Finally, we also obtain a randomized algorithm to detect $k$-multilinear terms in a multivariate polynomial given as a general algebraic circuit. To the best of our knowledge, this was previously only known for algebraic circuits not involving negative constants., Comment: To appear at STOC 2018: Symposium on Theory of Computing, June 23-27, 2018, Los Angeles, CA, USA

We investigate the complexity of counting trees, forests and bases of matroids from a parameterized point of view. It turns out that the problems of computing the number of trees and forests with $k$ edges are $\# W[1]$-hard when parameterized by $k$. Together with the recent algorithm for deterministic matrix truncation by Lokshtanov et al. (ICALP 2015), the hardness result for $k$-forests implies $\# W[1]$-hardness of the problem of counting bases of a matroid when parameterized by rank or nullity, even if the matroid is restricted to be representable over a field of characteristic $2$. We complement this result by pointing out that the problem becomes fixed parameter tractable for matroids represented over a fixed finite field., Comment: 14 pages

Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahl\'en [9] and Husfeldt and Taslaman [12], in combination with Curticapean [7], extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line $y=1$, which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given $n$-vertex graph cannot be determined in time $exp(o(n))$ unless #ETH fails. Another dichotomy theorem we strengthen is the one of Creignou and Hermann [6] for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that all #P-hard cases are also hard under #ETH. The main ingredient is to prove that the number of independent sets in bipartite graphs with $n$ vertices cannot be computed in time $exp(o(n))$ unless #ETH fails. In order to prove our results, we use the block interpolation idea by Curticapean [7] and transfer it to systems of linear equations that might not directly correspond to interpolation., Comment: 16 pages, 1 figure

Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l with coefficients of small bit size. For computing l, we need to project the solutions into one dimension along O(n) distinct directions but no further algebraic manipulations. The solutions are then directly reconstructed from the considered projections. The first step is deterministic, whereas the second step uses randomization, thus being Las-Vegas. The theoretical analysis of our approach shows that the overall cost for the two problems considered above is dominated by the cost of carrying out the projections. We also give bounds on the bit complexity of our algorithms that are exclusively stated in terms of the number of variables, the total degree and the bitsize of the input polynomials.