Bounded Degree Graphs Research Articles

Classical and quantum bounded depth approximation algorithms

We consider some classical and quantum approximate optimization algorithms with bounded depth. First, we define a class of "local" classical optimization algorithms and show that a single step version of these algorithms can achieve the same performance as the single step QAOA on MAX-3-LIN-2. Second, we show that this class of classical algorithms generalizes a class previously considered in the literature\cite{hirvonen2014large}, and also that a single step of the classical algorithm will outperform the single-step QAOA on all triangle-free MAX-CUT instances. In fact, for all but 4 choices of degree, existing single-step classical algorithms already outperform the QAOA on these graphs, while for the remaining 4 choices we show that the generalization here outperforms it. Finally, we consider the QAOA and provide strong evidence that, for any fixed number of steps, its performance on MAX-3-LIN-2 on bounded degree graphs cannot achieve the same scaling as can be done by a class of ``global" classical algorithms. These results suggest that such local classical algorithms are likely to be at least as promising as the QAOA for approximate optimization.

Computing the chromatic number using graph decompositions via matrix rank

Computing the smallest number q such that the vertices of a given graph can be properly q-colored, known as the chromatic number, is one of the oldest and most fundamental problems in combinatorial optimization. The q-Coloring problem has been studied intensively using the framework of parameterized algorithmics, resulting in a very good understanding of the best-possible algorithms for several parameterizations based on the structure of the graph. For example, algorithms are known to solve the problem on graphs of treewidth tw in time O⁎(qtw), while a running time of O⁎((q−ε)tw) is impossible assuming the Strong Exponential Time Hypothesis (SETH). While there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators, almost nothing is known about parameterizations based on edge separators. We fill this gap by studying q-Coloring parameterized by cutwidth, and parameterized by pathwidth in bounded-degree graphs. Our research uncovers interesting new ways to exploit small edge separators.We present two algorithms for q-Coloring parameterized by cutwidth ctw: a deterministic one that runs in time O⁎(2ω⋅ctw), where ω is the square matrix multiplication exponent, and a randomized one with runtime O⁎(2ctw). In sharp contrast to earlier work, the running time is independent of q. The dependence on cutwidth is optimal: we prove that even 3-Coloring cannot be solved in O⁎((2−ε)ctw) time assuming SETH. Our algorithms rely on a new rank bound for a matrix that describes compatible colorings. Combined with a simple communication protocol for evaluating a product of two polynomials, this also yields an O⁎((⌊d/2⌋+1)pw) time randomized algorithm for q-Coloring on graphs of pathwidth pw and maximum degree d. Such a runtime was first obtained by Björklund, but only for graphs with few proper colorings. We also prove that this result is optimal in the sense that no O⁎((⌊d/2⌋+1−ε)pw)-time algorithm exists assuming SETH.

Optimal packings of bounded degree trees

We prove that if T_1,\dots, T_n is a sequence of bounded degree trees such that T_i has i vertices, then K_n has a decomposition into T_1,\dots, T_n . This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first o(n) trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions.

Local Algorithms for Sparse Spanning Graphs

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider a relaxed version of this problem in the setting of local algorithms. The relaxation is that the constructed subgraph is a sparse spanning subgraph containing at most \((1+\epsilon )n\) edges (where n is the number of vertices and \(\epsilon \) is a given approximation/sparsity parameter). In the local setting, the goal is to quickly determine whether a given edge e belongs to such a subgraph, without constructing the whole subgraph, but rather by inspecting (querying) the local neighborhood of e. The challenge is to maintain consistency. That is, to provide answers concerning different edges according to the same spanning subgraph. We first show that for general bounded-degree graphs, the query complexity of any such algorithm must be \(\Omega (\sqrt{n})\). This lower bound holds for constant-degree graphs that have high expansion. Next we design an algorithm for (bounded-degree) graphs with high expansion, obtaining a result that roughly matches the lower bound. We then turn to study graphs that exclude a fixed minor (and are hence non-expanding). We design an algorithm for such graphs, which may have an unbounded maximum degree. The query complexity of this algorithm is \(\mathrm{poly}(1/\epsilon , h)\) (independent of n and the maximum degree), where h is the number of vertices in the excluded minor. Though our two algorithms are designed for very different types of graphs (and have very different complexities), on a high-level there are several similarities, and we highlight both the similarities and the differences.

Counting and Sampling from Markov Equivalent DAGs Using Clique Trees

A directed acyclic graph (DAG) is the most common graphical model for representing causal relationships among a set of variables. When restricted to using only observational data, the structure of the ground truth DAG is identifiable only up to Markov equivalence, based on conditional independence relations among the variables. Therefore, the number of DAGs equivalent to the ground truth DAG is an indicator of the causal complexity of the underlying structure–roughly speaking, it shows how many interventions or how much additional information is further needed to recover the underlying DAG. In this paper, we propose a new technique for counting the number of DAGs in a Markov equivalence class. Our approach is based on the clique tree representation of chordal graphs. We show that in the case of bounded degree graphs, the proposed algorithm is polynomial time. We further demonstrate that this technique can be utilized for uniform sampling from a Markov equivalence class, which provides a stochastic way to enumerate DAGs in the equivalence class and may be needed for finding the best DAG or for causal inference given the equivalence class as input. We also extend our counting and sampling method to the case where prior knowledge about the underlying DAG is available, and present applications of this extension in causal experiment design and estimating the causal effect of joint interventions.

Counting and Sampling Markov Equivalent Directed Acyclic Graphs

Exploring directed acyclic graphs (DAGs) in a Markov equivalence class is pivotal to infer causal effects or to discover the causal DAG via appropriate interventional data. We consider counting and uniform sampling of DAGs that are Markov equivalent to a given DAG. These problems efficiently reduce to counting the moral acyclic orientations of a given undirected connected chordal graph on n vertices, for which we give two algorithms. Our first algorithm requires O(2nn4) arithmetic operations, improving a previous superexponential upper bound. The second requires O(k!2kk2n) operations, where k is the size of the largest clique in the graph; for bounded-degree graphs this bound is linear in n. After a single run, both algorithms enable uniform sampling from the equivalence class at a computational cost linear in the graph size. Empirical results indicate that our algorithms are superior to previously presented algorithms over a range of inputs; graphs with hundreds of vertices and thousands of edges are processed in a second on a desktop computer.

Approximation Algorithms for Complex-Valued Ising Models on Bounded Degree Graphs

We study the problem of approximating the Ising model partition function with complex parameters on bounded degree graphs. We establish a deterministic polynomial-time approximation scheme for the partition function when the interactions and external fields are absolutely bounded close to zero. Furthermore, we prove that for this class of Ising models the partition function does not vanish. Our algorithm is based on an approach due to Barvinok for approximating evaluations of a polynomial based on the location of the complex zeros and a technique due to Patel and Regts for efficiently computing the leading coefficients of graph polynomials on bounded degree graphs. Finally, we show how our algorithm can be extended to approximate certain output probability amplitudes of quantum circuits.

Learning Graphs from Noisy Epidemic Cascades

We consider the problem of learning the weighted edges of a graph by observing the noisy times of infection for multiple epidemic cascades on this graph. Past work has considered this problem when the cascade information, i.e., infection times, are known exactly. Though the noisy setting is well motivated by many epidemic processes (e.g., most human epidemics), to the best of our knowledge, very little is known about when it is solvable. Previous work on the no-noise setting critically uses the ordering information. If noise can reverse this -- a node's reported (noisy) infection time comes after the reported infection time of some node it infected -- then we are unable to see how previous results can be extended. We therefore tackle two versions of the noisy setting: the limited-noise setting, where we know noisy times of infections, and the extreme-noise setting, in which we only know whether or not a node was infected. We provide a polynomial time algorithm for recovering the structure of bidirectional trees in the extreme-noise setting, and show our algorithm matches lower bounds established in the no-noise setting, and hence is optimal. We extend our results for general degree-bounded graphs, where again we show that our (poly-time) algorithm can recover the structure of the graph with optimal sample complexity. We also provide the first efficient algorithm to learn the weights of the bidirectional tree in the limited-noise setting. Finally, we give a polynomial time algorithm for learning the weights of general bounded-degree graphs in the limited-noise setting. This algorithm extends to general graphs (at the price of exponential running time), proving the problem is solvable in the general case. All our algorithms work for any noise distribution, without any restriction on the variance.

Hierarchy Theorems for Testing Properties in Size-Oblivious Query Complexity

Focusing on property testing tasks that have query complexity that is independent of the size of the tested object (i.e., depends on the proximity parameter only), we prove the existence of a rich hierarchy of the corresponding complexity classes. That is, for essentially any function $$q : (0, 1] \rightarrow \mathbb{N}$$ , we prove the existence of properties for which $$\epsilon$$ -testing has query complexity $$\Theta(q(\Theta(\epsilon)))$$ . Such results are proved in three standard domains that are often considered in property testing: generic functions, adjacency predicates describing (dense) graphs, and incidence functions describing bounded-degree graphs. These results complement hierarchy theorems of Goldreich, Krivelevich, Newman, and Rozenberg (Computational Complexity, 2012), which refer to the dependence of the query complexity on the size of the tested object, and focus on the case that the proximity parameter is set to some small positive constant. We actually combine both flavors and get tight results on the query complexity of testing when allowing the query complexity to depend on both the size of the object and the proximity parameter.

Filling the complexity gaps for colouring planar and bounded degree graphs

AbstractA colouring of a graph is a function such that for every . A ‐regular list assignment of is a function with domain such that for every , is a subset of of size . A colouring of respects a ‐regular list assignment of if for every . A graph is ‐choosable if for every ‐regular list assignment of , there exists a colouring of that respects . We may also ask if for a given ‐regular list assignment of a given graph , there exists a colouring of that respects . This yields the ‐Regular List Colouring problem. For , we determine a family of classes of planar graphs, such that either ‐Regular List Colouring is ‐complete for instances with , or every is ‐choosable. By using known examples of non‐‐choosable and non‐‐choosable graphs, this enables us to classify the complexity of ‐Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle‐free graphs, and planar graphs with no ‐cycles and no ‐cycles. We also classify the complexity of ‐Regular List Colouring and a number of related colouring problems for graphs with bounded maximum degree.

Structural sparsity of complex networks: Bounded expansion in random models and real-world graphs

This research establishes that many real-world networks exhibit bounded expansion2, a strong notion of structural sparsity, and demonstrates that it can be leveraged to design efficient algorithms for network analysis. Specifically, we give a new linear-time fpt algorithm for motif counting and linear time algorithms to compute localized variants of several centrality measures. To establish structural sparsity in real-world networks, we analyze several common network models regarding their structural sparsity. We show that, with high probability, (1) graphs sampled with a prescribed sparse degree sequence; (2) perturbed bounded-degree graphs; (3) stochastic block models with small probabilities; result in graphs of bounded expansion. In contrast, we show that the Kleinberg and the Barabási–Albert model have unbounded expansion. We support our findings with empirical measurements on a corpus of real-world networks.

Non-clairvoyant scheduling with conflicts for unit-size jobs

We consider the problem of scheduling unit-size jobs with conflicts in a non-clairvoyant setting. In this problem, all jobs are released at the same time but the conflict graph is unknown to the scheduling algorithm beforehand. Instead, the conflicts between the jobs are revealed online by an adversary during the jobs' execution. There is an unlimited number of processors so that all jobs could be executed simultaneously. However, two jobs with a conflict cannot be both completed in the same time step; when their conflict is revealed, one of them must be aborted and re-executed later. The objective is to minimize the total number of time steps required to complete all jobs, or the makespan. We present an online non-clairvoyant algorithm and analyze its performance for three types of conflict graphs. For bipartite graphs, we show that it is O(log⁡n)-competitive, where n denotes the total number of jobs. For unit-interval graphs, we show that it is O(log⁡χ)-competitive, where χ denotes the chromatic number of the graph. For bounded-degree graphs, we show that it is O(Δ)-competitive, where Δ denotes the maximum degree of any node in the graph. For bipartite and bounded-degree graphs, we also provide matching lower bounds and show that the obtained competitive ratios are asymptotically tight.

The Ising Partition Function: Zeros and Deterministic Approximation

We study the problem of approximating the partition function of the ferromagnetic Ising model with both pairwise as well as higher order interactions (equivalently, in graphs as well as hypergraphs). Our approach is based on the classical Lee–Yang theory of phase transitions, along with a new Lee–Yang theorem for the Ising model with higher order interactions, and on an extension of ideas developed recently by Barvinok, and Patel and Regts that can be seen as an algorithmic realization of the Lee–Yang theory. Our first result is a deterministic polynomial time approximation scheme (an FPTAS) for the partition function in bounded degree graphs that is valid over the entire range of parameters \(\beta \) (the interaction) and \(\lambda \) (the external field), except for the case \(\left| \lambda \right| =1\) (the “zero-field” case). A polynomial time randomized approximation scheme (FPRAS) for all graphs and all \(\beta ,\lambda \), based on Markov chain Monte Carlo simulation, has long been known. Unlike most other deterministic approximation algorithms for problems in statistical physics and counting, our algorithm does not rely on the “decay of correlations” property, but, as pointed out above, on Lee–Yang theory. This approach extends to the more general setting of the Ising model on hypergraphs of bounded degree and edge size, where no previous algorithms (even randomized) were known for a wide range of parameters. In order to achieve this latter extension, we establish a tight version of the Lee–Yang theorem for the Ising model on hypergraphs, improving a classical result of Suzuki and Fisher.

On the Number of Connected Sets in Bounded Degree Graphs

A set of vertices in a graph is connected if it induces a connected subgraph. Using Shearer's entropy lemma and a computer search, we show that the number of connected sets in a graph with $n$ vertices and maximum vertex degree $d$ is at most $O(1.9351^n)$ for $d=3$, $O(1.9812^n)$ for $d=4$, and $O(1.9940^n)$ for $d=5$. Dually, we construct infinite families of graphs where the number of connected sets is at least $\Omega(1.7651^n)$ for $d=3$, $\Omega(1.8925^n)$ for $d=4$, and $\Omega(1.9375^n)$ for $d=5$.

Online Dominating Set

This paper is devoted to the online dominating set problem and its variants. We believe the paper represents the first systematic study of the effect of two limitations of online algorithms: making irrevocable decisions while not knowing the future, and being incremental, i.e., having to maintain solutions to all prefixes of the input. This is quantified through competitive analyses of online algorithms against two optimal algorithms, both knowing the entire input, but only one having to be incremental. We also consider the competitive ratio of the weaker of the two optimal algorithms against the other. We consider important graph classes, distinguishing between connected and not necessarily connected graphs. For the classic graph classes of trees, bipartite, planar, and general graphs, we obtain tight results in almost all cases. We also derive upper and lower bounds for the class of bounded-degree graphs. From these analyses, we get detailed information regarding the significance of the necessary requirement that online algorithms be incremental. In some cases, having to be incremental fully accounts for the online algorithm’s disadvantage.

Computing the Number of Induced Copies of a Fixed Graph in a Bounded Degree Graph

In this paper we show that for any graph H of order m and any graph G of order n and maximum degree Delta one can compute the number of subsets S of V(G) that induces a graph isomorphic to H in time O(c^m cdot n ) for some constant c = c(Delta ) >0. This is essentially best possible (in the sense that there is no c^{o(m)}poly(n)-time algorithm under the exponential time hypothesis).

New Tools and Connections for Exponential-Time Approximation

In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer r>1, and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio ofr for maximum independent set in O^*(exp ({tilde{O}}(n/r log ^2 r+rlog ^2r))) time,r for chromatic number in O^*(exp (tilde{O}(n/r log r+rlog ^2r))) time,(2-1/r) for minimum vertex cover in O^*(exp (n/r^{varOmega (r)})) time, and(k-1/r) for minimum k-hypergraph vertex cover in O^*(exp (n/ (kr)^{varOmega (kr)})) time.(Throughout, {tilde{O}} and O^* omit hbox {polyloglog} (r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were O^*(2^{n/r}) (Bourgeois et al. in Discret Appl Math 159(17):1954–1970, 2011; Cygan et al. in Exponential-time approximation of hard problems, 2008). For maximum independent set and chromatic number, these bounds were complemented by exp (n^{1-o(1)}/r^{1+o(1)}) lower bounds (under the Exponential Time Hypothesis (ETH)) (Chalermsook et al. in Foundations of computer science, FOCS, pp. 370–379, 2013; Laekhanukit in Inapproximability of combinatorial problems in subexponential-time. Ph.D. thesis, 2014). Our results show that the naturally-looking O^*(2^{n/r}) bounds are not tight for all these problems. The key to these results is a sparsification procedure that reduces a problem to a bounded-degree variant, allowing the use of approximation algorithms for bounded-degree graphs. To obtain the first two results, we introduce a new randomized branching rule. Finally, we show a connection between PCP parameters and exponential-time approximation algorithms. This connection together with our independent set algorithm refute the possibility to overly reduce the size of Chan’s PCP (Chan in J. ACM 63(3):27:1–27:32, 2016). It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture (Dinur in Electron Colloq Comput Complex (ECCC) 23:128, 2016; Manurangsi and Raghavendra in A birthday repetition theorem and complexity of approximating dense CSPs, 2016).

Excluding subdivisions of bounded degree graphs

Let H be a fixed graph. What can be said about graphs G that have no subgraph isomorphic to a subdivision of H? Grohe and Marx proved that such graphs G satisfy a certain structure theorem that is not satisfied by graphs that contain a subdivision of a (larger) graph H1. Dvořák found a clever strengthening—his structure is not satisfied by graphs that contain a subdivision of a graph H2, where H2 has “similar embedding properties” as H. Building upon Dvořák's theorem, we prove that said graphs G satisfy a similar structure theorem. Our structure is not satisfied by graphs that contain a subdivision of a graph H3 that has similar embedding properties as H and has the same maximum degree as H. This will be important in a forthcoming application to well-quasi-ordering.

Best of two local models: Centralized local and distributed local algorithms

We consider two models of computation: centralized local algorithms and local distributed algorithms. Algorithms in one model are adapted to the other model to obtain improved algorithms.Distributed vertex coloring is employed to design improved centralized local algorithms for: maximal independent set, maximal matching, and an approximation scheme for maximum (weighted) matching over bounded degree graphs. The improvement is threefold: the algorithms are deterministic, stateless, and the number of probes grows polynomially in log⁎⁡n, where n is the number of vertices of the input graph.The recursive centralized local improvement technique by Nguyen and Onak (FOCS 2008) is employed to obtain a distributed approximation scheme for maximum (weighted) matching.

Parameterized Algorithms and Kernels for Rainbow Matching

In this paper, we study the NP-complete colorful variant of the classical Matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, this problem asks whether there exists a matching of size at least k such that all the edges in the matching have distinct colors. We first develop a deterministic algorithm that solves Rainbow Matching on paths in time $$\mathcal{O}^\star \left( \left( \frac{1+\sqrt{5}}{2}\right) ^k\right) $$ and polynomial space. This algorithm is based on a curious combination of the method of bounded search trees and a “divide-and-conquer-like” approach, where the branching process is guided by the maintenance of an auxiliary bipartite graph where one side captures “divided-and-conquered” pieces of the path. Our second result is a randomized algorithm that solves Rainbow Matching on general graphs in time $$\mathcal {O} ^\star (2^k)$$ and polynomial-space. Here, we show how a result by Bjorklund et al. (J Comput Syst Sci 87:119–139, 2017) can be invoked as a black box, wrapped by a probability-based analysis tailored to our problem. We also complement our two main results by designing kernels for Rainbow Matching on general and bounded-degree graphs.