ABSTRACT. In this work we study shortest path problems in multimode graphs, a generalization of the min-distance measure introduced by Abboud, Vassilevska W. and Wang in [SODA'16]. A multimode shortest path is the shortest path using one of multiple `modes' of transportation that cannot be combined. This represents real-world scenarios where different modes are not combinable, such as flights operated by different airline alliances. The problem arises naturally in machine learning in the context of learning with multiple embedding. More precisely, a $k$-multimode graph is a collection of $k$ graphs on the same vertex set and the $k$-mode distance between two vertices is defined as the minimum among the distances computed in each individual graph.

We focus on approximating fundamental graph parameters on these graphs, specifically diameter and radius. In undirected multimode graphs we first show an elegant linear time 3-approximation algorithm for 2-mode diameter. We then extend this idea into a general subroutine that can be used as a part of any $\alpha$-approximation, and use it to construct a 2 and 2.5 approximation algorithm for 2-mode diameter. For undirected radius, we introduce a general scheme that can compute a 3-approximation of the $k$-mode radius for any $k$ and runs in near linear time in the case of $k=O(1)$. In the directed case we establish an equivalence between approximating 2-mode diameter on DAGs and approximating the min-diameter, while for general graphs we develop novel techniques and provide a linear time algorithm to determine whether the diameter is finite.

We also develop many conditional fine-grained lower bounds for various multimode diameter and radius approximation problems. We are able to show that many of our algorithms are tight under popular fine-grained complexity hypotheses, including our linear time 3-approximation for $3$-mode undirected diameter and radius. As part of this effort we propose the first extension to the Hitting Set Hypothesis [SODA'16], which we call the $\ell$-Hitting Set Hypothesis. We use this hypothesis to prove the first parameterized lower bound tradeoff for radius approximation algorithms.

ABSTRACT. Given a graph G and two independent sets of same size, the Independent Set Reconfiguration Problem under token sliding asks whether one can, in a step by step manner, transform the first independent set into the second one. In each step we must preserve the condition of independence. Further, referring to solution vertices as tokens, we are only permitted to slide a token along an edge. Until the recent work of Ito et al. [Ito et al. MFCS 2022] this problem was only considered on undirected graphs. In this work, we study reconfiguration under token sliding focusing on DAGs.

We present a complete dichotomy of intractability in regard to the depth of the DAG, by proving that this problem is NP-complete for DAGs of depth 3 and W[1]-hard for depth 4 when parameterized by the number of tokens k, and that these bounds are tight. Further, we prove that it is fixed parameter tractable on DAGs parameterized by the combination of treewidth and k. We show that this result applies to undirected graphs, when the number of times a token can visit a vertex is restricted.

ABSTRACT. A periodic temporal graph, in its simplest form, is a graph in which every edge appears exactly once in the first $\Delta$ time steps, and then it reappears recurrently every $\Delta$ time steps, where $\Delta$ is a given period length. This model offers a natural abstraction of transportation networks where each transportation link connects two destinations periodically. From a network design perspective, a crucial task is to assign the time-labels on the edges in a way that optimizes some criterion. In this paper we introduce a very natural optimality criterion that captures how the temporal distances of all vertex pairs are ``stretched'', compared to their physical distances, i.e. their distances in the underlying static (non-temporal) graph. Given a static graph $G$, the task is to assign to each edge one time-label between 1 and $\Delta$ such that, in the resulting periodic temporal graph with period $\Delta$, the duration of the fastest temporal path from any vertex $u$ to any other vertex $v$ is at most $\alpha$ times the distance between $u$ and $v$ in $G$. Here, the value of $\alpha$ measures how much the shortest paths are allowed to be stretched once we assign the periodic time-labels.

Our results span three different directions: First, we provide a series of approximation and NP-hardness results. Second, we provide approximation and fixed-parameter algorithms. Among them, we provide a simple polynomial-time algorithm (the radius-algorithm) which always guarantees an approximation strictly smaller than $\Delta$, and which also computes the optimum stretch in some cases. Third, we consider a parameterized local search extension of the problem where we are given the temporal labeling of the graph, but we are allowed to change the time-labels of at most $k$ edges; for this problem we prove that it is W[2]-hard but admits an XP algorithm with respect to $k$.

ABSTRACT. Regular model checking is a well-established technique for the verification of regular transition systems (RTS): transition systems whose initial configurations and transition relation can be effectively encoded as regular languages. In 2008, To and Libkin studied RTSs in which the reachability relation (the reflexive and transitive closure of the transition relation) is also effectively regular, and showed that the recurrent reachability problem (whether a regular set L of configurations is reached infinitely often) is polynomial in the size of RTS and the transducer for the reachability relation. We extend the work of To and Libkin by studying the decidability and complexity of verifying almost-sure reachability and recurrent reachability---that is, whether L is reachable or recurrently reachable w.p. 1. We then apply our results to the more common case in which only a regular overapproximation of the reachability relation is available. In particular, we extend recent complexity results on verifying safety using regular abstraction frameworks---a technique recently introduced by Czerner, the authors, and Welzel-Mohr---to liveness and almost-sure properties.

ABSTRACT. Several reachability problems in finite automata, such as completeness of NFAs and synchronisation of total DFAs, correspond to fundamental properties of sets of nonnegative matrices. In particular, the two mentioned properties correspond to matrix mortality and ergodicity, which ask whether there exists a product of the input matrices that is equal to, respectively, the zero matrix and a matrix with a column of strictly positive entries only. The case where the input automaton is strongly connected (that is, the corresponding set of nonnegative matrices is irreducible) frequently appears in applications and often admits better properties than the general case. In this paper, we address the existence of such properties from the computational complexity point of view, and develop a versatile technique to show that several NL-complete problems remain NL-complete in the strongly connected case. Namely, we show that deciding if a binary total DFA is synchronising is NL-complete even if it is promised to be strongly connected, and that deciding completeness of a binary unambiguous NFA with very limited nondeterminism is NL-complete under the same promise.

ABSTRACT. In a separability problem, we are given two sets $K$ and $L$ from a class $\mathcal{C}$, and we want to decide whether there exists a set $S$ from a class $\mathcal{S}$ such that $K\subseteq S$ and $S\cap L=\emptyset$. In this case, we speak of separability of sets in $\mathcal{C}$ by sets in $\mathcal{S}$.

We study two types of separability problems. First, we consider separability of semilinear sets (i.e.\ subsets of $\mathbb{N}^d$ for some $d$) by sets definable by quantifier-free monadic Presburger formulas (or equivalently, the recognizable subsets of $\mathbb{N}^d$). Here, a formula is monadic if each atom uses at most one variable. Second, we consider separability of languages of Parikh automata by regular languages. A Parikh automaton is a machine with access to counters that can only be incremented, and have to meet a semilinear constraint at the end of the run. Both of these separability problems are known to be decidable with elementary complexity.

Our main results are that both problems are coNP-complete. In the case of semilinear sets, coNP-completeness holds regardless of whether the input sets are specified by existential Presburger formulas, quantifier-free formulas, or semilinear representations. Our results imply that recognizable separability of rational subsets of $\Sigma^*\times\mathbb{N}^d$ (shown decidable by Choffrut and Grigorieff) is coNP-complete as well. Another application is that regularity of deterministic Parikh automata (where the target set is specified using a quantifier-free Presburger formula) is coNP-complete as well.

ABSTRACT. For a fixed integer t>=1, a (t-)long claw, denoted S_{t,t,t}, is the unique tree with three leaves, each at distance exactly t from the vertex of degree three. Majewski et al. [ICALP 2022, ACM ToCT 2024] proved an analog of the Gyárfás’ path argument for S_{t,t,t}-free graphs: given an n-vertex S_{t,t,t}-free graph, one can delete neighborhoods of O(log n) vertices so that the remainder admits an extended strip decomposition (an appropriate generalization of partition into connected components) into particles of multiplicatively smaller size. In this work we refine the argument of Majewski et al. to its arguably final form: we show that a constant number of neighborhoods suffice.

The statement of Majewski et al. is one of the two pillars of a recent quasi-polynomial time algorithm for Maximum Weight Independent Set in S_{t,t,t}-free graphs [Gartland et al., STOC20 2024]; our work immediately improves the quasi-polynomial function in the running time bound. Furthermore, our result significantly simplifies known polynomial-time algorithms for Maximum Weight Independent Set in S_{t,t,t}-free graphs with an additional sparsity assumption such as bounded degree or excluding a fixed biclique as a subgraph.

ABSTRACT. We initiate the study of Hamiltonian cycles up to symmetries of the underlying graph. Our focus lies on the extremal case of Hamiltonian-transitive graphs, i.e., Hamiltonian graphs where, for every pair of Hamiltonian cycles, there is a graph automorphism mapping one cycle to the other. This generalizes the extensively studied uniquely Hamiltonian graphs. In this paper, we show that Cayley graphs of abelian groups are not Hamiltonian-transitive (under some mild conditions and some non-surprising exceptions), i.e., they contain at least two structurally different Hamiltonian cycles. To show this, we reduce Hamiltonian-transitivity to properties of the prime factors of a Cartesian product decomposition, which we believe is interesting in its own right. We complement our results by constructing infinite families of regular Hamiltonian-transitive graphs and take a look at the opposite extremal case by constructing a family with many different Hamiltonian cycles up to symmetry.

ABSTRACT. We construct an algorithm that inputs an MSO-interpretation from finite words to graphs, and decides if there exists a k ∈ N such that the class of graphs induced by the interpretation is not well-quasi-ordered by the induced subgraph relation when vertices are freely labelled using {1,…, k}. In case no such k exists, we also prove that the class of graphs is not well-quasi-ordered by the induced subgraph relation when vertices are freely labelled using any well-quasi-ordered set of labels. As a byproduct of our analysis, we prove that for classes of bounded linear clique-width, a weak version of a conjecture by Pouzet holds.

ABSTRACT. In a seminal work, Buhrman et al. (STOC 2014) defined the class CSPACE(s,c) of problems solvable in space s with an additional catalytic tape of size c, which is a tape whose initial content must be restored at the end of the computation. They showed that uniform TC^1 circuits are computable in catalytic logspace, i.e., CL=CSPACE(O(log(n)), 2^{O(log(n))})$, thus giving strong evidence that catalytic space gives L strict additional power. Their study focuses on an arithmetic model called register programs, which has been a focal point in development since then.

Understanding CL remains a major open problem, as TC^1 remains the most powerful containment to date. In this work, we study the power of catalytic space and register programs to compute circuits of larger depth. Using register programs, we show that for every epsilon > 0, SAC^2 \subseteq CSPACE(O(\frac{\log^2{n}}{\log\log{n}}), 2^{O(\log^{1+\epsilon} n)})

This is an O(log log n) factor improvement on the free space needed to compute \SAC^2, which can be accomplished with near-polynomial catalytic space.

We also exhibit non-trivial register programs for matrix powering, which is a further step towards showing NC^2 \subseteq CL.

ABSTRACT. For every fixed graph H, it is known that homomorphism counts from H and colorful H-subgraph counts can be determined in O(n^{t+1}) time on n-vertex input graphs G, where t is the treewidth of H. On the other hand, a running time of n^{o(t / log t)} would refute the exponential-time hypothesis. Komarath et al. studied algebraic variants of these counting problems, i.e., homomorphism and subgraph polynomials for fixed graphs H. These polynomials are weighted sums over the objects counted above, where each object is weighted by the product of variables corresponding to edges contained in the object. As shown by Komarath et al., the monotone circuit complexity of the homomorphism polynomial for H is Θ(n^{tw(H)+1}).

In this paper, we characterize the power of monotone bounded-depth circuits for homomorphism and colorful subgraph polynomials. This leads us to a natural hierarchy of graph parameters tw_Δ(H), for fixed Δ ∈ ℕ, which capture the width of tree-decompositions for H when the underlying tree is required to have depth at most Δ. We prove that monotone circuits of product-depth Δ computing the homomorphism polynomial for H require size Θ(n^{tw_Δ(H†)+1}), where H† is the graph obtained from H by removing all degree-1 vertices. This allows us to derive an optimal depth hierarchy theorem for monotone bounded-depth circuits through graph-theoretic arguments.

ABSTRACT. In this paper, we provide a uniform framework for investigating small circuit classes and bounds through the lens of ordinary differential equations (ODEs). Following an approach recently introduced to capture the class of polynomial-time computable functions via ODE-based recursion schemas and later applied to the context of functions computed by unbounded fan-in circuits of constant depth (FAC^0), we study multiple relevant small circuit classes. In particular, we show that natural restrictions on linearity and derivation along functions with specific growth rate correspond to kinds of functions that can be proved to be in various classes, ranging from FAC^0 to FAC^1. This reveals an intriguing link between constraints over linear-length ODEs and circuit computation, providing new tools to tackle the complex challenge of establishing bounds for classes in the circuit hierarchies and possibly enhancing our understanding of the role of counters in this setting. Additionally, we establish several completeness results, in particular obtaining the first ODE-based characterizations for the classes of functions computable in constant depth with unbounded fan-in and Mod 2 gates (FACC[2]) and in logarithmic depth with bounded fan-in Boolean gates (FNC^1)

ABSTRACT. We deal with the problem of computing the exact crossing number of almost planar graphs and the closely related problem of computing the exact anchored crossing number of a pair of planar graphs. It was shown by [Cabello and Mohar, 2013] that both problems are NP-hard; although they required an unbounded number of high-degree vertices (in the first problem) or an unbounded number of anchors (in the second problem) to prove their result. Somehow surprisingly, only three vertices of degree greater than 3, or only three anchors, are sufficient to maintain hardness of these problems, as we prove here. The new result also improves the previous result on hardness of joint crossing number on surfaces by [Hliněný and Salazar, 2015]. Our result is best possible in the anchored case since the anchored crossing number of a pair of planar graphs with two anchors each is trivial, and close to being best possible in the almost planar case since the crossing number is polytime computable for almost planar graphs of maximum degree 3 [Riskin 1996, Cabello and Mohar 2011]. The complexity of crossing number of almost planar graphs with one or two vertices of degree greater than 3 is, interestingly, still wide open.

ABSTRACT. Cops and Robbers is a game played on a graph where a set of cops attempt to capture a single robber. The game proceeds in rounds, where each round first consists of the cops' turn, followed by the robbers turn. In the cops' turn, every cop can choose to either stay on the same vertex or move to an adjacent vertex, and likewise the robber in his turn. The robber is considered to be captured if, at any point in time, there is some cop on the same vertex as the robber. The cops win if they can capture the robber within a finite number of rounds; else the robber wins.

A natural question in this game concerns the cop-number of a graph---the minimum number of cops needed to capture the robber. It has long been known that graphs embeddable (without crossings) on surfaces of bounded genus have bounded cop-number. In contrast, it was shown recently that the class of 1-planar graphs---graphs that can be drawn on the plane with at most one crossing per edge---does not have bounded cop-number. In this paper, we initiate a study of the effect that distance between crossing pairs of edges have on the cop-number of a graph. In particular, we look at Distance $d$ Cops and Robbers, a variant of the classical game, where the robber is considered to be captured if there is a cop within distance $d$ of the robber.

Let $c_d(G)$ denote the minimum number of cops required in the graph $G$ to capture a robber within distance $d$. We looks at various classes of graphs, such as 1-plane graphs, $k$-plane graphs (graphs where each edge is crossed at most $k$ times), and even general graph drawings, and show that if every crossing pair of edges can be connected by a path of small length, then $c_d(G)$ is bounded, for small values of $d$. For instance, we show that if a graph $G$ can be drawn such that no pair of crossing edges form a matching, then $c_4(G) \leq 21$. And if the drawing permits a stronger assumption that the endpoints of every crossing induce the complete graph $K_4$, then $c_3(G) \leq 9$. The tools and techniques that we develop in this paper are sufficiently general, enabling us to examine graphs drawn not only on the sphere but also on orientable and non-orientable surfaces.

ABSTRACT. Color Refinement, also known as Naive Vertex Classification, is a classical method to distinguish graphs by iteratively computing a coloring of their vertices. While it is traditionally used as an imperfect way to test for isomorphism, the algorithm has permeated many other, seemingly unrelated, areas of computer science. The method is algorithmically simple, and it has a well-understood distinguishing power: it is logically characterized by Cai, Fürer and Immerman (1992), who showed that it distinguishes precisely those graphs that can be distinguished by a sentence of first-order logic with counting quantifiers and only two variables. A combinatorial characterization was given by Dvořák (2010), who showed that it distinguishes precisely those graphs that differ in the number of homomorphisms from some tree.

In this paper, we introduce Relational Color Refinement (RCR, for short), a generalization of the Color Refinement method from graphs to arbitrary relational structures, whose distinguishing power admits the equivalent combinatorial and logical characterizations as Color Refinement has on graphs: we show that RCR distinguishes precisely those structures that differ in the number of homomorphisms from an acyclic connected relational structure. Further, we show that RCR distinguishes precisely those structures that be distinguished by a sentence of the guarded fragment of first-order logic with counting quantifiers. Additionally, we show that for every fixed finite relational signature, RCR can be implemented to run on structures of that signature in time O(N log N), where N denotes the number of tuples present in the structure.

ABSTRACT. Lazy search trees (Sandlund & Wild FOCS 2020, Sandlund & Zhang SODA 2022) are sorted dictionaries whose update and query performance smoothly interpolates between that of efficient priority queues and binary search trees – automatically, depending on actual use; no adjustments are necessary to the data structure to realize the cost savings. In this paper, we design lazy B-trees, a variant of lazy search trees suitable for external memory that generalizes the speedup of B-trees over binary search trees wrt. input/output operations to the same smooth interpolation regime.

A key technical difficulty to overcome is the lack of a (fully satisfactory) external variant of biased search trees, on which lazy search trees crucially rely. We give a construction for the subset of performance guarantees sufficient to realize external-memory lazy search trees, which we deem of independent interest.

As one special case, lazy B-trees can be used as an external-memory priority queue, in which case they are competitive with some tailed-made heaps; indeed, they offer faster decrease-key and insert operations than known data structures.