Inspired by Manjul Bhargava's theory of generalized factorials, Fedor Petrov and the author have defined the "Bhargava greedoid" -- a greedoid (a matroid-like set system on a finite set) assigned to any "ultra triple" (a somewhat extended variant of a finite ultrametric space). Here we show that the Bhargava greedoid of a finite ultra triple is always a "Gaussian elimination greedoid" over any sufficiently large (e.g., infinite) field; this is a greedoid analogue of a representable matroid. We find necessary and sufficient conditions on the size of the field to ensure this.

Binary relations derived from labeled rooted trees play an import role in mathematical biology as formal models of evolutionary relationships. The (symmetrized) Fitch relation formalizes xenology as the pairs of genes separated by at least one horizontal transfer event. As a natural generalization, we consider symmetrized Fitch maps, that is, symmetric maps $\varepsilon$ that assign a subset of colors to each pair of vertices in $X$ and that can be explained by a tree $T$ with edges that are labeled with subsets of colors in the sense that the color $m$ appears in $\varepsilon(x,y)$ if and only if $m$ appears in a label along the unique path between $x$ and $y$ in $T$. We first give an alternative characterization of the monochromatic case and then give a characterization of symmetrized Fitch maps in terms of compatibility of a certain set of quartets. We show that recognition of symmetrized Fitch maps is NP-complete but FPT in general. In the restricted case where $|\varepsilon(x,y)|\leq 1$ the problem becomes polynomial, since such maps coincide with class of monochromatic Fitch maps whose graph-representations form precisely the class of complete multi-partite graphs.

A hole in a graph is an induced cycle of length at least $4$, and an antihole is the complement of an induced cycle of length at least $4$. A hole or antihole is long if its length is at least $5$. For an integer $k$, the $k$-prism is the graph consisting of two cliques of size $k$ joined by a matching. The complexity of Maximum (Weight) Independent Set (MWIS) in long-hole-free graphs remains an important open problem. In this paper we give a polynomial time algorithm to solve MWIS in long-hole-free graphs with no $k$-prism (for any fixed integer $k$), and a subexponential algorithm for MWIS in long-hole-free graphs in general. As a special case this gives a polynomial time algorithm to find a maximum weight clique in perfect graphs with no long antihole, and no hole of length $6$. The algorithms use the framework of minimal chordal completions and potential maximal cliques.

We present an analytical upper bound on the number of required vehicles for vehicle routing problems with split deliveries and any number of capacitated depots. We show that a fleet size greater than the proposed bound is not achievable based on a set of common assumptions. This property of the upper bound is proved through a dynamic programming approach. We also discuss the validity of the bound for a wide variety of routing problems with or without split deliveries.

A rearrangement operation makes a small graph-theoretical change to a phylogenetic network to transform it into another one. For unrooted phylogenetic trees and networks, popular rearrangement operations are tree bisection and reconnection (TBR) and prune and regraft (PR) (called subtree prune and regraft (SPR) on trees). Each of these operations induces a metric on the sets of phylogenetic trees and networks. The TBR-distance between two unrooted phylogenetic trees $T$ and $T'$ can be characterised by a maximum agreement forest, that is, a forest with a minimum number of components that covers both $T$ and $T'$ in a certain way. This characterisation has facilitated the development of fixed-parameter tractable algorithms and approximation algorithms. Here, we introduce maximum agreement graphs as a generalisations of maximum agreement forests for phylogenetic networks. While the agreement distance -- the metric induced by maximum agreement graphs -- does not characterise the TBR-distance of two networks, we show that it still provides constant-factor bounds on the TBR-distance. We find similar results for PR in terms of maximum endpoint agreement graphs.

Let $H$ be a graph with $\Delta(H) \leq 2$, and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. We prove that if $H$ contains at most one odd cycle of length exceeding $3$, or if $H$ contains at most $3$ triangles, then $\chi(G) \leq 4$. This proves the Strong Coloring Conjecture for such graphs $H$. For graphs $H$ with $\Delta=2$ that are not covered by our theorem, we prove an approximation result towards the conjecture.

We show that every 1-planar graph with minimum degree at least 4 has girth at most $8$, and every 1-planar graph with minimum degree at least 3 has girth at most $198$.

Fitch graphs $G=(X,E)$ are digraphs that are explained by $\{\emptyset, 1\}$-edge-labeled rooted trees $T$ with leaf set $X$: there is an arc $(x,y) \in E$ if and only if the unique path in $T$ that connects the last common ancestor $\mathrm{lca}(x,y)$ of $x$ and $y$ with $y$ contains at least one edge with label "1". In practice, Fitch graphs represent xenology relations, i.e., pairs of genes $x$ and $y$ for which a horizontal gene transfer happened along the path from $\mathrm{lca}(x,y)$ to $y$. In this contribution, we generalize the concept of Fitch graphs and consider trees $T$ that are equipped with edge-labeling $\lambda: E\to \mathcal{P}(M)$ that assigns to each edge a subset $M'\subseteq M$ of colors. Given such a tree, we can derive a map $\varepsilon_{(T,\lambda)}$ (or equivalently a set of not necessarily disjoint binary relations), such that $i\in \varepsilon_{(T,\lambda)}(x,y)$ (or equivalently $(x,y)\in R_i$) with $x,y\in X$, if and only if there is at least one edge with color $i$ from $\mathrm{lca}(x,y)$ to $y$. The central question considered here: Is a given map $\varepsilon$ a Fitch map, i.e., is there there an edge-labeled tree $(T,\lambda)$ with $\varepsilon_{(T,\lambda)} = \varepsilon$, and thus explains $\varepsilon$? Here, we provide a characterization of Fitch maps in terms of certain neighborhoods and forbidden submaps. Further restrictions of Fitch maps are considered. Moreover, we show that the least-resolved tree explaining a Fitch map is unique (up to isomorphism). In addition, we provide a polynomial-time algorithm to decide whether $\varepsilon$ is a Fitch map and, in the affirmative case, to construct the (up to isomorphism) unique least-resolved tree $(T^*,\lambda^*)$ that explains $\varepsilon$.

We prove that for every integer $t\ge 1$ there exists a constant $c_t$ such that for every $K_t$-minor-free graph $G$, and every set $S$ of balls in $G$, the minimum size of a set of vertices of $G$ intersecting all the balls of $S$ is at most $c_t$ times the maximum number of vertex-disjoint balls in $S$. This was conjectured by Chepoi, Estellon, and Vax\`es in 2007 in the special case of planar graphs and of balls having the same radius.

This paper addresses a fundamental problem in random variate generation: given access to a random source that emits a stream of independent fair bits, what is the most accurate and entropy-efficient algorithm for sampling from a discrete probability distribution $(p_1, \dots, p_n)$, where the probabilities of the output distribution $(\hat{p}_1, \dots, \hat{p}_n)$ of the sampling algorithm must be specified using at most $k$ bits of precision? We present a theoretical framework for formulating this problem and provide new techniques for finding sampling algorithms that are optimal both statistically (in the sense of sampling accuracy) and information-theoretically (in the sense of entropy consumption). We leverage these results to build a system that, for a broad family of measures of statistical accuracy, delivers a sampling algorithm whose expected entropy usage is minimal among those that induce the same distribution (i.e., is "entropy-optimal") and whose output distribution $(\hat{p}_1, \dots, \hat{p}_n)$ is a closest approximation to the target distribution $(p_1, \dots, p_n)$ among all entropy-optimal sampling algorithms that operate within the specified $k$-bit precision. This optimal approximate sampler is also a closer approximation than any (possibly entropy-suboptimal) sampler that consumes a bounded amount of entropy with the specified precision, a class which includes floating-point implementations of inversion sampling and related methods found in many software libraries. We evaluate the accuracy, entropy consumption, precision requirements, and wall-clock runtime of our optimal approximate sampling algorithms on a broad set of distributions, demonstrating the ways that they are superior to existing approximate samplers and establishing that they often consume significantly fewer resources than are needed by exact samplers.

The interest in quantum walks has been steadily increasing during the last two decades. It is still worth to present new forms of quantum walks that might find practical applications and new physical behaviors. In this work, we define discrete-time quantum walks on arbitrary oriented graphs by partitioning a graph into tessellations, which is a collection of disjoint cliques that cover the vertex set. By using the adjacency matrices associated with the tessellations, we define local unitary operators, whose product is the evolution operator of our quantum walk model. We introduce a parameter, called $\alpha$, that quantifies the amount of orientation. We show that the parameter $\alpha$ can be tuned in order to increase the amount of quantum walk-based transport on oriented graphs.

The junction-tree representation provides an attractive structural property for organizing a decomposable graph. In this study, we present a novel stochastic algorithm, which we call the junction-tree expander, for sequential sampling of junction trees for decomposable graphs. We show that recursive application of the junction-tree expander, expanding incrementally the underlying graph with one vertex at a time, has full support on the space of junction trees with any given number of underlying vertices. A direct application of our suggested algorithm is demonstrated in a sequential Monte Carlo setting designed for sampling from distributions on spaces of decomposable graphs, where the junction-tree expander can be effectively employed as proposal kernel; see the companion paper Olsson et al. 2019 [16]. A numerical study illustrates the utility of our approach by two examples: in the first one, how the junction-tree expander can be incorporated successfully into a particle Gibbs sampler for Bayesian structure learning in decomposable graphical models; in the second one, we provide an unbiased estimator of the number of decomposable graphs for a given number of vertices. All the methods proposed in the paper are implemented in the Python library trilearn.

The study of complex networks has shown several applications to real-world networks. In this way, the demand for new graph abstractions in order to deal with multidimensional structures and their complexities increases. This article presents a theoretical investigation of incompressible multidimensional networks defined by generalized graph representations. In particular, we mathematically study the incompressibility (i.e., algorithmic randomness) of snapshot-dynamic networks and multiplex networks in comparison to the incompressibility of more general forms of multidimensional networks, from which snapshot-dynamic networks or multiplex networks are particular cases. In addition, from a worst-case compressibility analysis, we study some of the network topological properties of general multidimensional networks. To these ends, first we show that incompressible snapshot-dynamic (or multiplex) networks carry an amount of algorithmic information that is linearly dominated by the size of the set of time instants (or layers). This contrasts with the algorithmic information carried by incompressible general dynamic (or multilayer) networks that is of the quadratic order of the size of the set of time instants (or layers). Furthermore, we show that such incompressible general multidimensional networks have very short diameter, high k-connectivity, and degrees of the order of half of the network size within a strong-asymptotically dominated standard deviation. Then, we show that incompressible general multidimensional networks have transtemporal (crosslayer or, in general, non-sequential interdimensional) edges, i.e., edges linking vertices at non-sequential time instants (layers or, in general, elements of a node dimension).

We develop a novel method to analyze the dynamics of stochastic rewriting systems evolving over finitary adhesive, extensive categories. Our formalism is based on the so-called rule algebra framework and exhibits an intimate relationship between the combinatorics of the rewriting rules (as encoded in the rule algebra) and the dynamics which these rules generate on observables (as encoded in the stochastic mechanics formalism). We introduce the concept of combinatorial conversion, whereby under certain technical conditions the evolution equation for (the exponential generating function of) the statistical moments of observables can be expressed as the action of certain differential operators on formal power series. This permits us to formulate the novel concept of moment-bisimulation, whereby two dynamical systems are compared in terms of their evolution of sets of observables that are in bijection. In particular, we exhibit non-trivial examples of graphical rewriting systems that are moment-bisimilar to certain discrete rewriting systems (such as branching processes or the larger class of stochastic chemical reaction systems). Our results point towards applications of a vast number of existing well-established exact and approximate analysis techniques developed for chemical reaction systems to the far richer class of general stochastic rewriting systems.

In this paper, we study several coloring problems on graphs from the viewpoint of parameterized complexity. We show that Precoloring Extension and Equitable Coloring problems are fixed-parameter tractable (FPT) parameterized by the distance to threshold graphs. We also study the List k-Coloring and show that the problem is NP-complete on split graphs and it is FPT parameterized by solution size on split graphs.

The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order $\sigma$, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering $\sigma$, i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force $f(k)n^{2^{k-1}}$-time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where $k$ is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on $k$ in the exponent of $n$ can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS '17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on $K_{t,t}$-free graphs for b-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest.

Searching for a line on the plane with $n$ unit speed robots is a classic online problem that dates back to the 50's, and for which competitive ratio upper bounds are known for every $n\geq 1$. In this work we improve the best lower bound known for $n=2$ robots from 1.5993 to 3. Moreover we prove that the competitive ratio is at least $\sqrt{3}$ for $n=3$ robots, and at least $1/\cos(\pi/n)$ for $n\geq 4$ robots. Our lower bounds match the best upper bounds known for $n\geq 4$, hence resolving these cases. To the best of our knowledge, these are the first lower bounds proven for the cases $n\geq 3$ of this several decades old problem.

In this paper, domination in Signed Petri net(SPN) has been introduced.We identify some of the Petri net structures where a dominating set can exist.Applications of producer consumer problem, searching of food by bees and finding similarity in research papers are given to understand the areas where the proposed theory can be used.

In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP($\mathcal{C}$,-), in which the goal is, given a relational structure $\mathbf{A}$ from a class $\mathcal{C}$ of structures and an arbitrary structure $\mathbf{B}$, to find the number of homomorphisms from $\mathbf{A}$ to $\mathbf{B}$. Flum and Grohe showed that #CSP($\mathcal{C}$,-) is solvable in polynomial time if $\mathcal{C}$ has bounded treewidth [FOCS'02]. Building on the work of Grohe [JACM'07] on decision CSPs, Dalmau and Jonsson then showed that, if $\mathcal{C}$ is a recursively enumerable class of relational structures of bounded arity, then assuming FPT $\neq$ #W[1], there are no other cases of #CSP($\mathcal{C}$,-) solvable exactly in polynomial time (or even fixed-parameter time) [TCS'04]. We show that, assuming FPT $\neq$ W[1] (under randomised parametrised reductions) and for $\mathcal{C}$ satisfying certain general conditions, #CSP($\mathcal{C}$,-) is not solvable even approximately for $\mathcal{C}$ of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP($\mathcal{C}$,-). In particular, our condition generalises the case when $\mathcal{C}$ is closed under taking minors.

Graph connectivity and network design problems are among the most fundamental problems in combinatorial optimization. The minimum spanning tree problem, the two edge-connected spanning subgraph problem (2-ECSS) and the tree augmentation problem (TAP) are all examples of fundamental well-studied network design tasks that postulate different initial states of the network and different assumptions on the reliability of network components. In this paper we motivate and study \emph{Flexible Graph Connectivity} (FGC), a problem that mixes together both the modeling power and the complexities of all aforementioned problems and more. In a nutshell, FGC asks to design a connected network, while allowing to specify different reliability levels for individual edges. While this non-uniform nature of the problem makes it appealing from the modeling perspective, it also renders most existing algorithmic tools for dealing with network design problems unfit for approximating FGC. In this paper we develop a general algorithmic approach for approximating FGC that yields approximation algorithms with ratios that are very close to the best known bounds for many special cases, such as 2-ECSS and TAP. Our algorithm and analysis combine various techniques including a weight-scaling algorithm, a charging argument that uses a variant of exchange bijections between spanning trees and a factor revealing min-max-min optimization problem.

The burning number of a graph was recently introduced by Bonato et al. Although they mention that the burning number generalises naturally to directed graphs, no further research on this has been done. Here, we introduce graph burning for directed graphs, and we study bounds for the corresponding burning number and the hardness of finding this number. We derive sharp bounds from simple algorithms and examples. The hardness question yields more surprising results: finding the burning number of a directed tree is NP-hard, but FPT; however, it is W[2]-complete for DAGs. Finally, we give a fixed-parameter algorithm to find the burning number of a digraph, with a parameter inspired by research in phylogenetic networks.

An orientation of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. An orientation of a graph $G$ is called a strong orientation, if from each vertex there is a directed path to every other vertex. The oriented diameter of a graph $G$ is the smallest diameter among all the strong orientations of $G$. The maximum oriented diameter of a family of graphs $\mathscr{F}$ is the maximum oriented diameter among all the graphs in $\mathscr{F}$. Chv\'atal and Thomassen gave a lower bound of $\frac{1}{2}d^2+d$ and an upper bound of $2d^2+2d$ for the maximum oriented diameter of the family of $2$-edge connected graphs of diameter $d$. We improve this upper bound to $ 1.373 d^2 + 6.971d-1 $, which outperforms the former upper bound for all values of $d$ greater than or equal to $8$. For the family of $2$-edge connected graphs of diameter $3$, Kwok, Liu and West obtained improved lower and upper bounds of $9$ and $11$ respectively. For the family of $2$-edge connected graphs of diameter $4$, the bounds provided by Chv\'atal and Thomassen are $12$ and $40$ and no better bounds were known. By extending the method we used for diameter $d$ graphs, along with a generalized version of a technique used by Chv\'atal and Thomassen, we have improved this upper bound to $21$.

We consider the problem of uniformly generating a spanning tree, of a connected undirected graph. This process is useful to compute statistics, namely for phylogenetic trees. We describe a Markov chain for producing these trees. For cycle graphs we prove that this approach significantly outperforms existing algorithms. For general graphs we obtain no analytical bounds, but experimental results show that the chain still converges quickly. This yields an efficient algorithm, also due to the use of proper fast data structures. To bound the mixing time of the chain we describe a coupling, which we analyse for cycle graphs and simulate for other graphs.

The minimum $s$-$t$ cut problem in graphs is one of the most fundamental problems in combinatorial optimization, and graph cuts underlie algorithms throughout discrete mathematics, theoretical computer science, operations research, and data science. While graphs are a standard model for pairwise relationships, hypergraphs provide the flexibility to model multi-way relationships, and are now a standard model for complex data and systems. However, when generalizing from graphs to hypergraphs, the notion of a "cut hyperedge" is less clear, as a hyperedge's nodes can be split in several ways. Here, we develop a framework for hypergraph cuts by considering the problem of separating two terminal nodes in a hypergraph in a way that minimizes a sum of penalties at split hyperedges. In our setup, different ways of splitting the same hyperedge have different penalties, and the penalty is encoded by what we call a splitting function. Our framework opens a rich space on the foundations of hypergraph cuts. We first identify a natural class of cardinality-based hyperedge splitting functions that depend only on the number of nodes on each side of the split. In this case, we show that the general hypergraph $s$-$t$ cut problem can be reduced to a tractable graph $s$-$t$ cut problem if and only if the splitting functions are submodular. We also identify a wide regime of non-submodular splitting functions for which the problem is NP-hard. We also analyze extensions to multiway cuts with at least three terminal nodes and identify a natural class of splitting functions for which the problem can be reduced in an approximation-preserving way to the node-weighted multiway cut problem in graphs, again subject to a submodularity property. Finally, we outline several open questions on general hypergraph cut problems.

For most practical optimisation problems local search outperforms random sampling - despite the "No Free Lunch Theorem". This paper introduces a property of search landscapes termed Neighbours' Similar Fitness (NSF) that underlies the good performance of neighbourhood search in terms of local improvement. Though necessary, NSF is not sufficient to ensure that searching for improvement among the neighbours of a good solution is better than random search. The paper introduces an additional (natural) property which supports a general proof that, for NSF landscapes, neighbourhood search beats random search.

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The independent domination number $i(G)$ of $G$ is the minimum cardinality of an independent dominating set in $G$. In 2013 Goddard and Henning [Discrete Math 313 (2013), 839--854] conjectured that if $G$ is a connected cubic graph of order $n$, then $i(G) \le \frac{3}{8}n$, except if $G$ is the complete bipartite graph $K_{3,3}$ or the $5$-prism $C_5 \, \Box \, K_2$. Further they construct two infinite families of connected cubic graphs with independent domination three-eighths their order. They remark that perhaps it is even true that for $n > 10$ these two families are only families for which equality holds. In this paper, we provide a new family of connected cubic graphs $G$ of order $n$ such that $i(G) = \frac{3}{8}n$. We also show that if $G$ is a subcubic graph of order $n$ with no isolated vertex, then $i(G) \le \frac{1}{2}n$, and we characterize the graphs achieving equality in this bound.

We prove that the cop number of any $2K_2$-free graph is at most 2, which was previously conjectured by Sivaraman and Testa.

Let $L$ be any finite distributive lattice and $B$ be any boolean predicate defined on $L$ such that the set of elements satisfying $B$ is a sublattice of $L$. Consider any subset $M$ of $L$ of size $k$ of elements of $L$ that satisfy $B$. Then, we show that $k$ generalized median elements generated from $M$ also satisfy $B$. We call this result generalized median theorem on finite distributive lattices. When this result is applied to the stable matching, we get Teo and Sethuraman's median stable matching theorem. Our proof is much simpler than that of Teo and Sethuraman. When the generalized median theorem is applied to the assignment problem, we get an analogous result for market clearing price vectors.

Network Function Virtualization (NFV) is a promising virtualization technology that has the potential to significantly reduce the expenses and improve the service agility. NFV makes it possible for Internet Service Providers (ISPs) to employ various Virtual Network Functions (VNFs) without installing new equipments. One of the most attractive approaches in NFV technology is a so-called Joint Placement and Allocation of Virtual Network Functions (JPA-VNF) which considers the balance between VNF investment with Quality of Services (QoS). We introduce a novel capability function to measure the potential of locating VNF instances for each server in the proposed OJPA-HS model. This model allows the servers in the network to be heterogeneous, at the same time combines and generalizes many classical JPA-VNF models. Despite its NP-hardness, we present a provable best-possible deterministic online algorithm based on dynamic programming (DP). To conquer the high complexity of DP, we propose two additional randomized heuristics, the Las Vegas (LV) and Monte Carlo (MC) randomized algorithms, which performs even as good as DP with much smaller complexity. Besides, MC is a promising heuristic in practice as it has the advantage to deal with big data environment. Extensive numerical experiments are constructed for the proposed algorithms in the paper.

The \emph{induced odd cycle packing number} $\text{iocp}(G)$ of a graph $G$ is the maximum integer $k$ such that $G$ contains an induced subgraph consisting of $k$ pairwise vertex-disjoint odd cycles. Motivated by applications to geometric graphs, Bonamy et al. proved that graphs of bounded induced odd cycle packing number, bounded VC dimension, and linear independence number admit a randomized EPTAS for the independence number. We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm that for any integers $k\ge 0$ and $t\ge 1$ and any $n$-vertex graph $G$ of induced odd cycle packing number returns in time $O_{k,t}(n^{k+4})$ an independent set of $G$ whose size is at least $\alpha(G)-n/t$ with high probability. In addition, we present $\chi$-boundedness results for graphs with bounded odd cycle packing number, and use them to design a QPTAS for the independence number only assuming bounded induced odd cycle packing number.

Polling systems have been widely studied, however most of these studies focus on polling systems with renewal processes for arrivals and random variables for service times. There is a need driven by practical applications to study polling systems with arbitrary arrivals (not restricted to time-varying or in batches) and revealed service time upon a job's arrival. To address that need, our work considers a polling system with generic setting and for the first time provides the worst case analysis for online scheduling policies in this system. We provide conditions for the existence of constant competitive ratio for this system, and also the competitive ratios for several well-studied policies such as cyclic exhaustive, gated and \emph{l}-limited policies, Stochastic Largest Queue policy, One Machine policy and Gittins Index policy for polling systems. We show that any policy with (1) purely static, (2) queue-length based or (3) job-processing-time based routing discipline does not have a competitive ratio smaller than $k$, where $k$ is the number of queues. Finally, a mixed strategy is provided for a practical scenario where setup time is large but bounded.

We present the single-particle sector of a quantum cellular automaton, namely a quantum walk, on a simple dynamical triangulated $2-$manifold. The triangulation is changed through Pachner moves, induced by the walker density itself, allowing the surface to transform into any topologically equivalent one. This model extends the quantum walk over triangular grid, introduced in a previous work, by one of the authors, whose space-time limit recovers the Dirac equation in (2+1)-dimensions. Numerical simulations show that the number of triangles and the local curvature grow as $t^\alpha e^{-\beta t^2}$, where $\alpha$ and $\beta$ parametrize the way geometry changes upon the local density of the walker, and that, in the long run, flatness emerges. Finally, we also prove that the global behavior of the walker, remains the same under spacetime random fluctuations.

We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such as equality and greater-than. We apply our generalized theorem to solve two open problems: * We present the first result that demonstrates a separation in proof power for cutting planes with unbounded versus polynomially bounded coefficients. Specifically, we exhibit CNF formulas that can be refuted in quadratic length and constant line space in cutting planes with unbounded coefficients, but for which there are no refutations in subexponential length and subpolynomial line space if coefficients are restricted to be of polynomial magnitude. * We give the first explicit separation between monotone Boolean formulas and monotone real formulas. Specifically, we give an explicit family of functions that can be computed with monotone real formulas of nearly linear size but require monotone Boolean formulas of exponential size. Previously only a non-explicit separation was known. An important technical ingredient, which may be of independent interest, is that we show that the Nullstellensatz degree of refuting the pebbling formula over a DAG G over any field coincides exactly with the reversible pebbling price of G. In particular, this implies that the standard decision tree complexity and the parity decision tree complexity of the corresponding falsified clause search problem are equal.

In this paper, we prove a conjecture of Aharoni and Howard on the existence of rainbow (transversal) matchings in sufficiently large families $\mathcal F_1,\ldots, \mathcal F_s$ of tuples in $\{1,\ldots, n\}^k$, provided $s\ge 470.$

In 1985, Restivo and Salemi presented a list of five problems concerning power free languages. Problem $4$ states: Given $\alpha$-power-free words $u$ and $v$, decide whether there is a transition from $u$ to $v$. Problem $5$ states: Given $\alpha$-power-free words $u$ and $v$, find a transition word $w$, if it exists. Let $\Sigma_k$ denote an alphabet with $k$ letters. Let $L_{k,\alpha}$ denote the $\alpha$-power free language over the alphabet $\Sigma_k$, where $\alpha$ is a rational number or a rational "number with $+$". If $\alpha$ is a "number with $+$" then suppose $k\geq 3$ and $\alpha\geq 2$. If $\alpha$ is "only" a number then suppose $k=3$ and $\alpha>2$ or $k>3$ and $\alpha\geq 2$. We show that: If $u\in L_{k,\alpha}$ is a right extendable word in $L_{k,\alpha}$ and $v\in L_{k,\alpha}$ is a left extendable word in $L_{k,\alpha}$ then there is a (transition) word $w$ such that $uwv\in L_{k,\alpha}$. We also show a construction of the word $w$.

In the Topological Minor Deletion (TM-Deletion) problem input consists of an undirected graph $G$, a family of undirected graphs ${\cal F}$ and an integer $k$. The task is to determine whether $G$ contains a set of vertices $S$ of size at most $k$, such that the graph $G\setminus S$ obtained from $G$ by removing the vertices of $S$, contains no graph from ${\cal F}$ as a topological minor. We give an algorithm for TM-Deletionwith running time $f(h^\star,k)\cdot |V(G)|^{4}$. Here $h^\star$ is the maximum size of a graph in ${\cal F}$ and $f$ is a computable function of $h^\star$ and $k$. This is the first fixed parameter tractable algorithm (FPT) for the problem. In fact, even for the restricted case of planar inputs the first FPT algorithm was found only recently by Golovach et al. [SODA 2020]. For this case we improve upon the algorithm of Golovach et al. [SODA 2020] by designing an FPT algorithm with explicit dependence on $k$ and $h^\star$.

Let $\mathbb{K}$ be a finite commutative ring, and let $\mathbb{L}$ be a commutative $\mathbb{K}$-algebra. Let $A$ and $B$ be two $n \times n$-matrices over $\mathbb{L}$ that have the same characteristic polynomial. The main result of this paper (Thm.~\ref{thm.finpowmat.main}) states that the set $\left\{ A^0,A^1,A^2,\ldots\right\}$ is finite if and only if the set $\left\{ B^0,B^1,B^2,\ldots\right\}$ is finite. We apply this result to Cellular Automata (CA). Indeed, it gives a complete and easy-to-check characterization of sensitivity to initial conditions and equicontinuity for linear CA over the alphabet $\mathbb{K}^n$ for $\mathbb{K} = \mathbb{Z}/m\Z$ (Thm.~\ref{froblca}), \ie, CA in which the local rule is defined by $n\times n$-matrices with elements in $\mathbb{Z}/m\Z$. To prove our main result, we derive an integrality criterion for matrices (Thm\ref{thm.finpowmat.char-int} and Prop.\ref{prop.finpowmat.char-int-conv}) that is likely of independent interest. Namely, let $\mathbb{K}$ be any commutative ring (not necessarily finite), and let $\mathbb{L}$ be a commutative $\mathbb{K}$-algebra. Consider any $n \times n$-matrix $A$ over $\mathbb{L}$. Then, $A \in \mathbb{L}^{n \times n}$ is integral over $\mathbb{K}$ (that is, there exists a monic polynomial $f \in \mathbb{K}\left[t\right]$ satisfying $f\left(A\right) = 0$) if and only if all coefficients of the characteristic polynomial of $A$ are integral over $\mathbb{K}$. The proof of this fact relies on a strategic use of exterior powers (a trick pioneered by Gert Almkvist). Furthermore, we extend the decidability result concerning sensitivity and equicontinuity to the wider class of additive CA over a finite abelian group. For such CA, we also prove the decidability of injectivity, surjectivity, topological transitivity and all the properties (as, for instance, ergodicity) that are equivalent to the latter.

The basic goal of threshold group testing is to identify up to $d$ defective items among a population of $n$ items, where $d$ is usually much smaller than $n$. The outcome of a test on a subset of items is positive if the subset has at least $u$ defective items, negative if it has up to $\ell$ defective items, where $0 \leq \ell < u$, and arbitrary otherwise. This is called threshold group testing with a gap. There are a few reported studies on test designs and decoding algorithms for identifying defective items. Most of the previous studies have not been feasible because there are numerous constraints on their problem settings or the decoding complexities of their proposed schemes are relatively large. Therefore, it is compulsory to reduce the number of tests as well as the decoding complexity, i.e., the time for identifying the defective items, for achieving practical schemes. The work presented here makes five contributions. The first is a corrected theorem for a non-adaptive algorithm for threshold group testing proposed by Chen and Fu. The second is an improvement in the construction of disjunct matrices, which are the main tools for tackling (threshold) group testing. Specifically, we present a better upper bound on the number of tests for disjunct matrices compared to related work. The third and fourth contributions are a reduction in the number of tests and a reduction in the decoding time for identifying defective items in a noisy setting on test outcomes. The fifth contribution is a demonstration of the resulting improvements by simulation for previous work and the proposed schemes.

Given $n$ positive integers $a_1,a_2,\dots,a_n$, and a positive integer right hand side $\beta$, we consider the feasibility version of the subset sum problem which is the problem of determining whether a subset of $a_1,a_2,\dots,a_n$ adds up to $\beta$. We show that if the right hand side $\beta$ is chosen as $\lfloor r\sum_{j=1}^n a_j \rfloor$ for a constant $0 < r < 1$ and if the $a_j$'s are independentand identically distributed from a discrete uniform distribution taking values ${1,2,\dots,\lfloor 10^{n/2} \rfloor }$, then the probability that the instance of the subset sum problem generated requires the creation of an exponential number of branch-and-bound nodes when one branches on the individual variables in any order goes to $1$ as $n$ goes to infinity.

The classical NP-hard \emph{feedback arc set problem} (FASP) and \emph{feedback vertex set problem} (FVSP) ask for a minimum set of arcs $\varepsilon \subseteq E$ or vertices $\nu \subseteq V$ whose removal $G\setminus \varepsilon$, $G\setminus \nu$ makes a given multi-digraph $G=(V,E)$ acyclic, respectively. The corresponding decision problems are part of the $21$ NP-complete problems of R. M. Karp's famous list. Though both problems are known to be APX-hard, approximation algorithms or proofs of inapproximability are unknown. We propose a new $\mathcal{O}(|V||E|^4)$-heuristic for the directed FASP. While $r \approx 1.3606$ is known to be a lower bound for the APX-hardness, at least by validation, we show that $r \leq 2$. Applying the approximation preserving $L$-reduction from the directed FVSP to the FASP thereby allows computing feedback vertex sets with the same accuracy. Benchmarking the algorithm with state of the art alternatives yields that, for the first time, the most relevant instance class of large sparse graphs can be solved efficiently within tight error bounds. Our derived insights might provide a path to prove the APX-completeness of both problems.

Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a $(\Delta(G)+1)$-edge coloring where $\Delta(G)$ denotes the maximum degreee of $G$. Recently, Cabello raised the following question: given two graphs $G_1,G_2$ of maximum degree $\Delta$ on the same set of vertices $V$, is it possible to edge-color their (edge) union with $\Delta+2$ colors in such a way the restriction of $G$ to respectively the edges of $G_1$ and the edges of $G_2$ are edge-colorings? More generally, given $\ell$ graphs, how many colors do we need to color their union in such a way the restriction of the coloring to each graph is proper? In this short note, we prove that we can always color the union of the graphs $G_1,\ldots,G_\ell$ of maximum degree $\Delta$ with $\Omega(\sqrt{\ell} \cdot \Delta)$ colors and that there exist graphs for which this bound is tight up to a constant multiplicative factor. Moreover, for two graphs, we prove that at most $\frac 32 \Delta +4$ colors are enough which is, as far as we know, the best known upper bound.

This paper is expository and is accessible to students. We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible to a non-specialist or a student. The simplest invariants naturally appear in an attempt to unknot a knot or unlink a link. Then we present certain `skein' recursive relations for the simplest invariants, which allow to introduce stronger invariants. We state the Vassiliev-Kontsevich theorem in a way convenient for calculating the invariants themselves, not only the dimension of the space of the invariants. No prerequisites are required; we give rigorous definitions of the main notions in a way not obstructing intuitive understanding.

We present an $n\Delta^{O(k^2)}$ time algorithm to obtain an optimal solution for $1$-dimensional cutting stock problem: the bin packing problem of packing $n$ items onto unit capacity bins under the restriction that the number of item sizes $k$ is fixed, where $\Delta$ is the reciprocal of the size of the smallest item. We employ elementary ideas in both the design and analysis our algorithm.

A {\em theta} is a graph made of three internally vertex-disjoint chordless paths $P_1 = a \dots b$, $P_2 = a \dots b$, $P_3 = a \dots b$ of length at least~2 and such that no edges exist between the paths except the three edges incident to $a$ and the three edges incident to $b$. A {\em pyramid} is a graph made of three chordless paths $P_1 = a \dots b_1$, $P_2 = a \dots b_2$, $P_3 = a \dots b_3$ of length at least~1, two of which have length at least 2, vertex-disjoint except at $a$, and such that $b_1b_2b_3$ is a triangle and no edges exist between the paths except those of the triangle and the three edges incident to~$a$. An \emph{even hole} is a chordless cycle of even length. For three non-negative integers $i\leq j\leq k$, let $S_{i,j,k}$ be the tree with a vertex $v$, from which start three paths with $i$, $j$, and $k$ edges respectively. We denote by $K_t$ the complete graph on $t$ vertices.

We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges form a planar graph. By planar duality this is equivalent to packing cuts in a planar graph such that each cut contains exactly one demand edge. We also show that the natural linear programming relaxations have constant integrality gap, yielding an approximate max-multiflow min-multicut theorem.

Motivated by emerging resource allocation and data placement problems such as web caches and peer-to-peer systems, we consider and study a class of resource allocation problems over a network of agents (nodes). In this model, nodes can store only a limited number of resources while accessing the remaining ones through their closest neighbors. We consider this problem under both optimization and game-theoretic frameworks. In the case of optimal resource allocation we will first show that when there are only k=2 resources, the optimal allocation can be found efficiently in O(n^2\log n) steps, where n denotes the total number of nodes. However, for k>2 this problem becomes NP-hard with no polynomial time approximation algorithm with a performance guarantee better than 1+1/102k^2, even under metric access costs. We then provide a 3-approximation algorithm for the optimal resource allocation which runs only in linear time O(n). Subsequently, we look at this problem under a selfish setting formulated as a noncooperative game and provide a 3-approximation algorithm for obtaining its pure Nash equilibria under metric access costs. We then establish an equivalence between the set of pure Nash equilibria and flip-optimal solutions of the Max-k-Cut problem over a specific weighted complete graph. Using this reduction, we show that finding the lexicographically smallest Nash equilibrium for k> 2 is NP-hard, and provide an algorithm to find it in O(n^3 2^n) steps. While the reduction to weighted Max-k-Cut suggests that finding a pure Nash equilibrium using best response dynamics might be PLS-hard, it allows us to use tools from quadratic programming to devise more systematic algorithms towards obtaining Nash equilibrium points.

Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E'$. The \emph{Dominating Induced Matching} (\emph{DIM}) problem asks for the existence of a d.i.m.\ in $G$; this problem is also known as the \emph{Efficient Edge Domination} problem; it is the Efficient Domination problem for line graphs. The DIM problem is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 and is solvable in linear time for $P_7$-free graphs, and in polynomial time for $S_{1,2,4}$-free graphs as well as for $S_{2,2,2}$-free graphs. In this paper, combining two distinct approaches, we solve it in polynomial time for $S_{2,2,3}$-free graphs.

We construct a theory of holant clones to capture the notion of expressibility in the holant framework. Their role is analogous to the role played by functional clones in the study of weighted counting Constraint Satisfaction Problems. We explore the landscape of conservative holant clones and determine the situations in which a set $\mathcal{F}$ of functions is "universal in the conservative case", which means that all functions are contained in the holant clone generated by $\mathcal{F}$ together with all unary functions. When $\mathcal{F}$ is not universal in the conservative case, we give concise generating sets for the clone. We demonstrate the usefulness of the holant clone theory by using it to give a complete complexity-theory classification for the problem of approximating the solution to conservative holant problems. We show that approximation is intractable exactly when $\mathcal{F}$ is universal in the conservative case.

This note presents several results in graph theory inspired by the author's work in the proof theory of linear logic; these results are purely combinatorial and do not involve logic. We show that trails avoiding forbidden transitions, properly arc-colored directed trails and rainbow paths for complete multipartite color classes can be found in linear time, whereas finding rainbow paths is NP-complete for any other restriction on color classes. For the tractable cases, we also state new structural properties equivalent to Kotzig's theorem on the existence of bridges in unique perfect matchings. Another result on graphs equipped with unique perfect matchings that we prove here is the combinatorial counterpart of a theorem due to Bellin in linear logic: a connection between blossoms and bridge deletion orders.

In this paper, we consider the problem of learning an unknown graph via queries on groups of nodes, with the result indicating whether or not at least one edge is present among those nodes. While learning arbitrary graphs with $n$ nodes and $k$ edges is known to be hard in the sense of requiring $\Omega( \min\{ k^2 \log n, n^2\})$ tests (even when a small probability of error is allowed), we show that learning an Erd\H{o}s-R\'enyi random graph with an average of $\bar{k}$ edges is much easier; namely, one can attain asymptotically vanishing error probability with only $O(\bar{k}\log n)$ tests. We establish such bounds for a variety of algorithms inspired by the group testing problem, with explicit constant factors indicating a near-optimal number of tests, and in some cases asymptotic optimality including constant factors. In addition, we present an alternative design that permits a near-optimal sublinear decoding time of $O(\bar{k} \log^2 \bar{k} + \bar{k} \log n)$.

Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}{b_i} \in [0,1)$ is an irreducible fraction. Let $\mathcal{F} = \{A_1, \ldots , A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is a \emph{r-wise fractional $L$-intersecting family} if for every distinct $i_1,i_2, \ldots,i_r \in [m]$, there exists an $\frac{a}{b} \in L$ such that $|A_{i_1} \cap A_{i_2} \cap \ldots \cap A_{i_r}| \in \{ \frac{a}{b}|A_{i_1}|, \frac{a}{b} |A_{i_2}|,\ldots, \frac{a}{b} |A_{i_r}| \}$. In this paper, we introduce and study the notion of r-wise fractional $L$-intersecting families. This is a generalization of notion of fractional $L$-intersecting families studied in \cite{niranj2019}.

In 2013, Fici and Zamboni proved a number of theorems about finite and infinite words having only a small number of factors that are palindromes. In this paper we rederive some of their results, and obtain some new ones, by a different method based on finite automata.

Assume that $X$ is a connected regular undirected graph of finite order $n$. Let $N_k$ denote the number of geodesic cycles on $X$ of length $k$. The numbers $\{N_k\}_{k=1}^\infty$ first appeared in the Ihara zeta function of $X$. The Hasse--Weil bounds on $\{N_k\}_{k=1}^\infty$ provide a necessary and sufficient condition for $X$ as a Ramanujan graph. For a given $k$, we propose a fast algorithm that computes the number $N_k$ in $O(n^\omega \lg k )$ time, where $\omega<2.3729$ is the exponent of matrix multiplication. As an application, we design an algorithm to evaluate if $X$ is a good expander via the numbers $N_k$ for even $k$.

Recent hardware advances in quantum and quantum-inspired annealers promise substantial speedup for solving NP-hard combinatorial optimization problems compared to general-purpose computers. These special-purpose hardware are built for solving hard instances of Quadratic Unconstrained Binary Optimization (QUBO) problems. In terms of number of variables and precision of these hardware are usually resource-constrained and they work either in Ising space {-1,1} or in Boolean space {0,1}. Many naturally occurring problem instances are higher-order in nature. The known method to reduce the degree of a higher-order optimization problem uses Rosenberg's polynomial. The method works in Boolean space by reducing the degree of one term by introducing one extra variable. In this work, we prove that in Ising space the degree reduction of one term requires the introduction of two variables. Our proposed method of degree reduction works directly in Ising space, as opposed to converting an Ising polynomial to Boolean space and applying previously known Rosenberg's polynomial. For sparse higher-order Ising problems, this results in a more compact representation of the resultant QUBO problem, which is crucial for utilizing resource-constrained QUBO solvers.

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\mathrm{dim}_{COG}(G)$ (resp. $\mathrm{dim}_{TH}(G)$) denotes the minimum number of cographs (resp. threshold graphs) whose intersection gives $G$. We present several new bounds on these parameters for general graphs as well as some special classes of graphs. It is shown that for any graph $G$: (a) $\mathrm{dim}_{COG}(G)\leq\mathrm{tw}(G)+2$, (b) $\mathrm{dim}_{TH}(G)\leq\mathrm{pw}(G)+1$, and (c) $\mathrm{dim}_{TH}(G)\leq\chi(G)\cdot\mathrm{box}(G)$, where $\mathrm{tw}(G)$, $\mathrm{pw}(G)$, $\chi(G)$ and $\mathrm{box}(G)$ denote respectively the treewidth, pathwidth, chromatic number and boxicity of the graph $G$. We also derive the exact values for these parameters for cycles and show that every forest is the intersection of two cographs. These results allow us to derive improved bounds on $\mathrm{dim}_{COG}(G)$ and $\mathrm{dim}_{TH}(G)$ when $G$ belongs to some special graph classes.

This paper is the first from serial papers that provide constructive characterizations for classes of bidirected graphs known as radials and semiradials. In this paper, we provide constructive characterizations for five principle classes of radials and semiradials to be used for characterizing general radials and semiradials. A bidirected graph is a graph in which each end of each edge has a sign $+$ or $-$. Bidirected graphs are a common generalization of digraphs and signed graphs. We define a new concept of radials as a generalization of a classical concept in matching theory, critical graphs. Radials are also a generalization of a class of digraphs known as flowgraphs. We also define semiradials, which are a relaxed concept of radials. We further define special classes of radials and semiradials, that is, absolute semiradials, strong and almost strong radials, linear semiradials, and sublinear radials. We provide constructive characterizations for these five classes of bidirected graphs. Our serial papers are a part of a series of works that establish the strong component decomposition for bidirected graphs.

In this paper, we study a primal and dual relationship about triangles: For any graph $G$, let $\nu(G)$ be the maximum number of edge-disjoint triangles in $G$, and $\tau(G)$ be the minimum subset $F$ of edges such that $G \setminus F$ is triangle-free. It is easy to see that $\nu(G) \leq \tau(G) \leq 3 \nu(G)$, and in fact, this rather obvious inequality holds for a much more general primal-dual relation between $k$-hyper matching and covering in hypergraphs. Tuza conjectured in $1981$ that $\tau(G) \leq 2 \nu(G)$, and this question has received attention from various groups of researchers in discrete mathematics, settling various special cases such as planar graphs and generalized to bounded maximum average degree graphs, some cases of minor-free graphs, and very dense graphs. Despite these efforts, the conjecture in general graphs has remained wide open for almost four decades. In this paper, we provide a proof of a non-trivial consequence of the conjecture; that is, for every $k \geq 2$, there exist a (multi)-set $F \subseteq E(G): |F| \leq 2k \nu(G)$ such that each triangle in $G$ overlaps at least $k$ elements in $F$. Our result can be seen as a strengthened statement of Krivelevich's result on the fractional version of Tuza's conjecture (and we give some examples illustrating this.) The main technical ingredient of our result is a charging argument, that locally identifies edges in $F$ based on a local view of the packing solution. This idea might be useful in further studying the primal-dual relations in general and the Tuza's conjecture in particular.

A hole in a graph is an induced cycle of length at least 4. We give a simple winning strategy for t-3 cops to capture a robber in the game of cops and robbers played in a graph that does not contain a hole of length at least t. This strengthens a theorem of Joret-Kaminski-Theis, who proved that t-2 cops have a winning strategy in such graphs. As a consequence of our bound, we also give an inequality relating the cop number and the Dilworth number of a graph.

In this paper, we present a mixed-integer linear programming formulation of a piecewise, polyhedral relaxation (PPR) of a multilinear term using it's convex hull representation. Based on the solution of the PPR, we also present a MIP-based piecewise formulation which restricts the solutions to be feasible for the multilinear term. We then present computational results showing the effectiveness of proposed formulations on instances from the standard Mixed-Integer Nonlinear Programming Library (MINLPLib) and compare the proposed formulation with a formulation that is built by recursively relaxing bilinear groupings of the multilinear term, typically applied in the literature.

In 1981, Kelly showed that planar posets can have arbitrarily large dimension. However, the posets in Kelly's example have bounded Boolean dimension and bounded local dimension, leading naturally to the questions as to whether either Boolean dimension or local dimension is bounded for the class of planar posets. The question for Boolean dimension was first posed by Ne\v{s}et\v{r}il and Pudl\'ak in 1989 and remains unanswered today. The concept of local dimension is quite new, introduced in 2016 by Ueckerdt. Since that time, researchers have obtained many interesting results concerning Boolean dimension and local dimension, contrasting these parameters with the classic Dushnik-Miller concept of dimension, and establishing links between both parameters and structural graph theory, path-width, and tree-width in particular. Here we show that local dimension is not bounded on the class of planar posets. Our proof also shows that the local dimension of a poset is not bounded in terms of the maximum local dimension of its blocks, and it provides an alternative proof of the fact that the local dimension of a poset cannot be bounded in terms of the tree-width of its cover graph, independent of its height.