We study the design of robust subexponential algorithms for classical connectivity problems on intersection graphs of similarly sized fat objects in $\mathbb{R}^d$. In this setting, each vertex corresponds to a geometric object, and two vertices are adjacent if and only if their objects intersect. We introduce a new tool for designing such algorithms, which we call a $λ$-linked partition. This is a partition of the vertex set into groups of highly connected vertices. Crucially, such a partition can be computed in polynomial time and does not require access to the geometric representation of the graph. We apply this framework to problems related to paths and cycles in graphs. First, we obtain the first robust ETH-tight algorithms for Hamiltonian Path and Hamiltonian Cycle, running in time $2^{O(n^{1-1/d})}$ on intersection graphs of similarly sized fat objects in $\mathbb{R}^d$. This resolves an open problem of de Berg et al. [STOC 2018] and completes the study of these problems on geometric intersection graphs from the viewpoint of ETH-tight exact algorithms. We further extend our approach to the parameterized setting and design the first robust subexponential parameterized algorithm for Long Path in any fixed dimension $d$. More precisely, we obtain a randomized robust algorithm running in time $2^{O(k^{1-1/d}\log^2 k)}\, n^{O(1)}$ on intersection graphs of similarly sized fat objects in $\mathbb{R}^d$, where $k$ is the natural parameter. Besides $λ$-linked partitions, our algorithm also relies on a low-treewidth pattern covering theorem that we establish for geometric intersection graphs, which may be viewed as a refinement of a result of Marx-Pilipczuk [ESA 2017]. This structural result may be of independent interest.

While most classical NP-hard graph problems cannot be solved in time 2^o(n) on general graphs under the Exponential Time Hypothesis (ETH), many exhibit the square-root phenomenon and admit optimal algorithms running in time 2^O(√n) on certain geometric intersection graphs, such as planar graphs or unit disk graphs. In 2018, de Berg et al. developed a general algorithmic framework for such problems on intersection graphs of similarly sized fat objects in ℝ^d, achieving running times of the form 2^O(n^{1-1/d}), along with matching lower bounds under ETH. In this paper, we identify problems that do not exhibit the square-root phenomenon, yet still admit subexponential algorithms on intersection graphs of similarly sized fat objects in ℝ^d, for every fixed dimension d ⩾ 2. We introduce the notion of a weak square-root phenomenon: problems that can be solved in time 2^Õ(n^{1-1/(d+1)}), and for which matching lower bounds hold under ETH. We develop both an algorithmic framework and a corresponding lower bound framework. As concrete examples, we show that the problems 2-Subcoloring and Two Sets Cut-Uncut exhibit this behavior. Our algorithms rely on a new win-win structural theorem, which can be informally stated as follows: every such graph admits a sublinear separator whose removal leaves connected components with sublinear independence number. To facilitate the design of these algorithms, we introduce a new graph parameter, the α-modulator number, which generalizes both the independence number and the vertex cover number.

Monitoring the quality of streaming video applications is important for Internet service providers (ISPs) to detect network issues and facilitate capacity planning. Machine Learning (ML) inference models have emerged as an effective solution to determine service quality using network traffic. However, while much focus has been on enhancing model performance, little attention has been given to deploying these models across entire networks. This paper introduces a new placement approach of quality inference models and their associated tasks to enhance the monitoring of video streaming applications over an entire mobile traffic network. Starting from the observation that inference tasks require the deployment of multiple components to, first, calculate input features from raw traffic, and then execute the inference models, we define the placement problem as an integer programming problem and, given its NP-hardness, we provide a heuristic solution, experimentally close to the optimum, based on the relaxation and the rounding of fractional solutions. We highlight that decoupling these components for the inference of network traffic can be beneficial in terms of total accuracy of the ML inference tasks. Finally, we experimentally show that our solution outperforms state-of-the-art placement techniques by ~30% of accuracy of the deployed inference models.

A subcoloring of a graph is a partition of its vertex set into subsets (called colors), each inducing a disjoint union of cliques. It is a natural generalization of the classical proper coloring, in which each color must instead induce an independent set. Similarly to proper coloring, we define the subchromatic number of a graph as the minimum integer k such that it admits a subcoloring with k colors, and the corresponding problem k-Subcoloring which asks whether a graph has subchromatic number at most k. In this paper, we initiate the study of the subcoloring of (unit) disk graphs. One motivation stems from the fact that disk graphs can be seen as a dense generalization of planar graphs where, intuitively, each vertex can be blown into a large clique-much like subcoloring generalizes proper coloring. Interestingly, it can be observed that every unit disk graph admits a subcoloring with at most 7 colors. We first prove that the subchromatic number can be 3-approximated in polynomial-time in unit disk graphs. We then present several hardness results for special cases of unit disk graphs which somehow prevents the use of classical approaches for improving this result. We show in particular that 2-Subcoloring remains NP-hard in triangle-free unit disk graphs, as well as in unit disk graphs representable within a strip of bounded height. We also solve an open question of Broersma, Fomin, Nešetřil, and Woeginger (2002) by proving that 3-Subcoloring remains NP-hard in co-comparability graphs (which contain unit disk graphs representable within a strip of height √3/2). Finally, we prove that every n-vertex disk graph admits a subcoloring with at most O(log³(n)) colors and present a O(log²(n))-approximation algorithm for computing the subchromatic number of such graphs. This is achieved by defining a decomposition and a special type of co-comparability disk graph, called Δ-disk graphs, which might be of independent interest.

Abstract. We recently introduced the notion of twin-width, a novel graph invariant, and showed that first-order model checking can be solved in time [Formula: see text] for [Formula: see text]-vertex graphs given with a witness that the twin-width is at most [Formula: see text], called [Formula: see text]-contraction sequence or [Formula: see text]-sequence, and formulas of size [Formula: see text] [Bonnet et al., JACM ’22]. The inevitable price to pay for such a general result is that [Formula: see text] is a tower of exponentials of height roughly [Formula: see text]. In this paper, we show that algorithms based on twin-width need not be impractical. We present [Formula: see text]-time algorithms for [Formula: see text]-independent set, [Formula: see text]-scattered set, [Formula: see text]-clique, and [Formula: see text]-dominating set when an [Formula: see text]-sequence of the graph is given in input. We further show how to solve the weighted version of [Formula: see text]-independent set, subgraph isomorphism, and induced subgraph isomorphism in the slightly worse running time [Formula: see text]. Up to logarithmic factors in the exponent, all these running times are optimal unless the exponential time hypothesis fails. Like our first-order model checking algorithm, these new algorithms are based on a dynamic programming scheme following the sequence of contractions forward. We then show a second algorithmic use of the contraction sequence by starting at its end and rewinding it. As an example of such a reverse scheme, we present a polynomial-time algorithm that properly colors the vertices of a graph with relatively few colors, thereby establishing that bounded twin-width classes are [Formula: see text]-bounded. This significantly extends the [Formula: see text]-boundedness of bounded rank-width classes and does so with a very concise proof. It readily yields a constant approximation for max independent set on [Formula: see text]-free graphs of bounded twin-width and a [Formula: see text]-approximation for min coloring on bounded twin-width graphs. We further observe that a constant approximation for max independent set on bounded twin-width graphs (but arbitrarily large clique number) would actually imply a polynomial-time approximation scheme. The third algorithmic use of twin-width builds on the second one. Playing the contraction sequence backward, we show that bounded twin-width graphs can be edge-partitioned into a linear number of bicliques such that both sides of the bicliques are on consecutive vertices in a fixed vertex ordering. This property is trivially shared with graphs of bounded average degree. Given that biclique edge-partition, we show how to solve the unweighted single-source shortest paths, and hence all-pairs shortest paths, in time [Formula: see text] and time [Formula: see text], respectively. In sharp contrast, even diameter does not admit a truly subquadratic algorithm on bounded twin-width graphs unless the strong exponential time hypothesis fails. The fourth algorithmic use of twin-width builds on the so-called versatile tree of contractions [Bonnet et al., Comb. Theory ’22], a branching and more robust witness of low twin-width. We present constant-approximation algorithms for min dominating set and related problems on bounded twin-width graphs by showing that the integrality gap is constant. This is done by going down the versatile tree and stopping according to a problem-dependent criterion. At the reached node, a greedy approach yields the desired approximation.

We recently introduced the graph invariant twin-width, and showed that first-order model checking can be solved in time $f(d,k)n$ for $n$-vertex graphs given with a witness that the twin-width is at most $d$, called $d$-contraction sequence or $d$-sequence, and formulas of size $k$ [Bonnet et al., FOCS '20]. The inevitable price to pay for such a general result is that $f$ is a tower of exponentials of height roughly $k$. In this paper, we show that algorithms based on twin-width need not be impractical. We present $2^{O(k)}n$-time algorithms for $k$-Independent Set, $r$-Scattered Set, $k$-Clique, and $k$-Dominating Set when an $O(1)$-sequence is provided. We further show how to solve weighted $k$-Independent Set, Subgraph Isomorphism, and Induced Subgraph Isomorphism, in time $2^{O(k \log k)}n$. These algorithms are based on a dynamic programming scheme following the sequence of contractions forward. We then show a second algorithmic use of the contraction sequence, by starting at its end and rewinding it. As an example of this reverse scheme, we present a polynomial-time algorithm that properly colors the vertices of a graph with relatively few colors, establishing that bounded twin-width classes are $\chi$-bounded. This significantly extends the $\chi$-boundedness of bounded rank-width classes, and does so with a very concise proof. The third algorithmic use of twin-width builds on the second one. Playing the contraction sequence backward, we show that bounded twin-width graphs can be edge-partitioned into a linear number of bicliques, such that both sides of the bicliques are on consecutive vertices, in a fixed vertex ordering. Given that biclique edge-partition, we show how to solve the unweighted Single-Source Shortest Paths and hence All-Pairs Shortest Paths in sublinear time $O(n \log n)$ and time $O(n^2 \log n)$, respectively.

We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for $k$-Dominating Set on graphs of twin-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for Connected $k$-Dominating Set and Total $k$-Dominating Set (albeit with a worse upper bound on the twin-width). The $k$-Independent Set problem admits the same lower bound by a much simpler argument, previously observed [ICALP '21], which extends to $k$-Independent Dominating Set, $k$-Path, $k$-Induced Path, $k$-Induced Matching, etc. On the positive side, we obtain a simple quadratic vertex kernel for Connected $k$-Vertex Cover and Capacitated $k$-Vertex Cover on graphs of bounded twin-width. Interestingly the kernel applies to graphs of Vapnik-Chervonenkis density 1, and does not require a witness sequence. We also present a more intricate $O(k^{1.5})$ vertex kernel for Connected $k$-Vertex Cover. Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most optimization/decision graph problems can be solved in polynomial time on graphs of twin-width at most 1.

The recently introduced twin-width of a graph G is the minimum integer d such that G has a d-contraction sequence, that is, a sequence of |V(G)| – 1 iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most d, where a red edge appears between two sets of identified vertices if they are not homogeneous in G (not fully adjacent nor fully non-adjacent). We show that if a graph admits a d-contraction sequence, then it also has a linear-arity tree of f(d)-contractions, for some function f. Informally if we accept to worsen the twin-width bound, we can choose the next contraction from a set of Θ(|V(G)|) pairwise disjoint pairs of vertices. This has two main consequences. First it permits to show that every bounded twin-width class is small, i.e., has at most n!cn graphs labeled by [n], for some constant c. This unifies and extends the same result for bounded treewidth graphs [Beineke and Pippert, JCT '69], proper subclasses of permutations graphs [Marcus and Tardos, JCTA '04], and proper minor-free classes [Norine et al., JCTB '06]. It implies in turn that bounded-degree graphs, interval graphs, and unit disk graphs have unbounded twin-width. The second consequence is an O(log n)-adjacency labeling scheme for bounded twin-width graphs, confirming several cases of the implicit graph conjecture. We then explore the small conjecture that, conversely, every small hereditary class has bounded twin-width. The conjecture passes many tests. Inspired by sorting networks of logarithmic depth, we show that logΘ(log log d) n-subdivisions of Kn (a small class when d is constant) have twin-width at most d. We obtain a rather sharp converse with a surprisingly direct proof: the logd+1 n-subdivision of Kn has twin-width at least d. Secondly graphs with bounded stack or queue number (also small classes) have bounded twin-width. These sparse classes are surprisingly rich since they contain certain (small) classes of expanders. Thirdly we show that cubic expanders obtained by iterated random 2-lifts from K4 [Bilu and Linial, Combinatorica '06] also have bounded twin-width. These graphs are related to so-called separable permutations and also form a small class. We suggest a promising connection between the small conjecture and group theory. Finally we define a robust notion of sparse twin-width. We show that for a hereditary class of bounded twin-width the five following conditions are equivalent: every graph in (1) is Kt,t-free for some fixed t, (2) has an adjacency matrix without a d-by-d division with a 1 entry in each d2 cells for some fixed d, (3) has at most linearly many edges, (4) the subgraph closure of has bounded twin-width, and (5) has bounded expansion. We discuss how sparse classes with similar behavior with respect to clique subdivisions compare to bounded sparse twin-width.

In the classic Maximum Weight Independent Set problem we are given a graph $G$ with a nonnegative weight function on vertices, and the goal is to find an independent set in $G$ of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any $P_6$-free graph, that is, a graph that has no path on $6$ vertices as an induced subgraph. This improves the polynomial-time algorithm on $P_5$-free graphs of Lokshtanov et al. (SODA 2014), and the quasipolynomial-time algorithm on $P_6$-free graphs of Lokshtanov et al (SODA 2016). The main technical contribution leading to our main result is enumeration of a polynomial-size family $\mathcal{F}$ of vertex subsets with the following property: for every maximal independent set $I$ in the graph, $\mathcal{F}$ contains all maximal cliques of some minimal chordal completion of $G$ that does not add any edge incident to a vertex of $I$.

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA '14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, Kt-free unit d-dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of d-contractions, witness that the twin-width is at most d. We show that FO model checking, that is deciding if a given first-order formula φ evaluates to true for a given binary structure G on a domain D, is FPT in |φ| on classes of bounded twin-width, provided the witness is given. More precisely, being given a d-contraction sequence for G, our algorithm runs in time f(d,|φ|)·|D| where f is a computable but non-elementary function. We also prove that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS '15].

Constraints such as separation-of-duty are widely used to specify requirements that supplement basic authorization policies. However, the existence of constraints (and authorization policies) may mean that a user is unable to fulfill her/his organizational duties because access to resources has been denied. In short, there is a tension between the need to protect resources (using policies and constraints) and the availability of resources. Recent work on workflow satisfiability and resiliency in access control asks whether this tension compromises the ability of an organization to achieve its objectives. In this paper, we develop a new method of specifying constraints which subsumes much related work and allows a wider range of constraints to be specified. The use of such constraints leads naturally to a range of questions related to “policy existence”, where a positive answer means that an organization's objectives can be realized. We analyze the complexity of these policy existence questions and, for particular sub-classes of constraints defined by our language, develop fixed-parameter tractable algorithms to solve them.11.An extended abstract of this paper appeared in the Proceedings of the Seventh ACM Conference on Data and Application Security and Privacy [1]. Research was partially supported by Leverhulme Trust grant RPG-2018-161 and Royal Society Wolfson Research Merit Award.

Maximum Independent Set (MIS for short) is in general graphs the paradigmatic $W[1]$-hard problem. In stark contrast, polynomial-time algorithms are known when the inputs are restricted to structured graph classes such as, for instance, perfect graphs (which includes bipartite graphs, chordal graphs, co-graphs, etc.) or claw-free graphs. In this paper, we introduce some variants of co-graphs with parameterized noise, that is, graphs that can be made into disjoint unions or complete sums by the removal of a certain number of vertices and the addition/deletion of a certain number of edges per incident vertex, both controlled by the parameter. We give a series of FPT Turing-reductions on these classes and use them to make some progress on the parameterized complexity of MIS in $H$-free graphs. We show that for every fixed $t \geqslant 1$, MIS is FPT in $P(1,t,t,t)$-free graphs, where $P(1,t,t,t)$ is the graph obtained by substituting all the vertices of a four-vertex path but one end of the path by cliques of size $t$. We also provide randomized FPT algorithms in dart-free graphs and in cricket-free graphs. This settles the FPT/W[1]-hard dichotomy for five-vertex graphs $H$.

In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains NP-hard for most graphs H, we study its fixed-parameter tractability and make progress towards a dichotomy between FPT and W[1]-hard cases. We first show that MIS remains W[1]-hard in graphs forbidding simultaneously K_{1, 4}, any finite set of cycles of length at least 4, and any finite set of trees with at least two branching vertices. In particular, this answers an open question of Dabrowski et al. concerning C_4-free graphs. Then we extend the polynomial algorithm of Alekseev when H is a disjoint union of edges to an FPT algorithm when H is a disjoint union of cliques. We also provide a framework for solving several other cases, which is a generalization of the concept of iterative expansion accompanied by the extraction of a particular structure using Ramsey's theorem. Iterative expansion is a maximization version of the so-called iterative compression. We believe that our framework can be of independent interest for solving other similar graph problems. Finally, we present positive and negative results on the existence of polynomial (Turing) kernels for several graphs H.

Constraints such as separation-of-duty are widely used to specify requirements that supplement basic authorization policies. However, the existence of constraints (and authorization policies) may mean that a user is unable to fulfill her/his organizational duties because access to resources is denied. In short, there is a tension between the need to protect resources (using policies and constraints) and the availability of resources. Recent work on workflow satisfiability and resiliency in access control asks whether this tension compromises the ability of an organization to achieve its objectives. In this paper, we develop a new method of specifying constraints which subsumes much related work and allows a wider range of constraints to be specified. The use of such constraints leads naturally to a range of questions related to "policy existence", where a positive answer means that an organization's objectives can be realized. We provide an overview of our results establishing that some policy existence questions, notably for those instances that are restricted to user-independent constraints, are fixed-parameter tractable.

There has been a considerable amount of interest in recent years in the problem of workflow satisfiability, which asks whether the existence of constraints in a workflow specification means that it is impossible to allocate authorized users to each step in the workflow. Recent developments have seen the workflow satisfiability problem (WSP) studied in the context of workflow specifications in which the set of steps may vary from one instance of the workflow to another. This, in turn, means that some constraints may only apply to certain workflow instances. Inevitably, WSP becomes more complex for such workflow specifications. In this paper, we present the first fixed parameter algorithms to solve WSP for workflow specifications of this type. Moreover, we significantly extend the range of constraints that can be used in workflow specifications of this type.

A computerized workflow management system may enforce a security policy, specified in terms of authorized actions and constraints, thereby restricting which users can perform particular steps in a workflow. The existence of a security policy may mean that a workflow is unsatisfiable, in the sense that it is impossible to find a valid plan (an assignment of steps to authorized users such that all constraints are satisfied). Work in the literature focuses on the workflow satisfiability problem, a decision problem that outputs a valid plan if the instance is satisfiable (and a negative result otherwise). In this paper, we introduce the Bi-Objective Workflow Satisfiability Problem (BO-WSP), which enables us to solve optimization problems related to workflows and security policies. In particular, we are able to compute a “least bad” plan when some components of the security policy may be violated. In general, BO-WSP is intractable from both the classical and parameterized complexity point of view (where the parameter is the number of steps). We prove that computing a Pareto front for BO-WSP is fixed-parameter tractable (FPT) if we restrict our attention to user-independent constraints. This result has important practical consequences, since most constraints of practical interest in the literature are user-independent. Our proof is constructive and defines an algorithm, the implementation of which we describe and evaluate. We also present a second algorithm to compute a Pareto front which solves multiples instances of a related problem using mixed integer programming (MIP). We compare the performance of both our algorithms on synthetic instances, and show that the FPT algorithm outperforms the MIP-based one by several orders of magnitude on most instances. Finally, we study the important question of workflow resiliency and prove new results establishing that known decision problems are fixed-parameter tractable when restricted to user-independent constraints. We then propose a new way of modeling the availability of users and demonstrate that many questions related to resiliency in the context of this new model may be reduced to instances of BO-WSP.

Resiliency is a relatively new topic in the context of access control. Informally, it refers to the extent to which a multi-user computer system, subject to an authorization policy, is able to continue functioning if a number of authorized users are unavailable. Several interesting problems connected to resiliency were introduced by Li, Wang and Tripunitara [13], many of which were found to be intractable. In this paper, we show that these resiliency problems have unexpected connections with the workflow satisfiability problem (WSP). In particular, we show that an instance of the resiliency checking problem (RCP) may be reduced to an instance of WSP. We then demonstrate that recent advances in our understanding of WSP enable us to develop fixed-parameter tractable algorithms for RCP. Moreover, these algorithms are likely to be useful in practice, given recent experimental work demonstrating the advantages of bespoke algorithms to solve WSP. We also generalize RCP in several different ways, showing in each case how to adapt the reduction to WSP. Li et al also showed that the coexistence of resiliency policies and static separation-of-duty policies gives rise to further interesting questions. We show how our reduction of RCP to WSP may be extended to solve these problems as well and establish that they are also fixed-parameter tractable.