Tree Projections and Constraint Optimization Problems: Fixed-Parameter Tractability and Parallel Algorithms

Tree projections provide a unifying framework to deal with most structural decomposition methods of constraint satisfaction problems (CSPs). Within this framework, a CSP instance is decomposed into a number of sub-problems, called views, whose solutions are either already available or can be computed efficiently. The goal is to arrange portions of these views in a tree-like structure, called tree projection, which determines an efficiently solvable CSP instance equivalent to the original one. Deciding whether a tree projection exists is NP-hard. Solution methods have therefore been proposed in the literature that do not require a tree projection to be given, and that either correctly decide whether the given CSP instance is satisfiable, or return that a tree projection actually does not exist. These approaches had not been generalized so far on CSP extensions for optimization problems, where the goal is to compute a solution of maximum value/minimum cost. The paper fills the gap, by exhibiting a fixed-parameter polynomial-time algorithm that either disproves the existence of tree projections or computes an optimal solution, with the parameter being the size of the expression of the objective function to be optimized over all possible solutions (and not the size of the whole constraint formula, used in related works). Tractability results are also established for the problem of returning the best K solutions. Finally, parallel algorithms for such optimization problems are proposed and analyzed. Given that the classes of acyclic hypergraphs, hypergraphs of bounded treewidth, and hypergraphs of bounded generalized hypertree width are all covered as special cases of the tree projection framework, the results in this paper directly apply to these classes. These classes are extensively considered in the CSP setting, as well as in conjunctive database query evaluation and optimization

Finding Optimal Triangulations Parameterized by Edge Clique Cover

Publisher Copyright: © 2022, The Author(s).We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover (cc) to such problems. This parameterization occurs naturally in many problems in this setting, e.g., in the perfect phylogeny problem cc is at most the number of taxa, in fractional hypertreewidth cc is at most the number of hyperedges, and in treewidth of Bayesian networks cc is at most the number of non-root nodes. We show that the number of minimal separators of graphs is at most 2 cc, the number of potential maximal cliques is at most 3 cc, and these objects can be listed in times O∗(2 cc) and O∗(3 cc) , respectively, even when no edge clique cover is given as input; the O∗(·) notation omits factors polynomial in the input size. These enumeration algorithms imply O∗(3 cc) time algorithms for problems such as treewidth, weighted minimum fill-in, and feedback vertex set. For generalized and fractional hypertreewidth we give O∗(4 m) time and O∗(3 m) time algorithms, respectively, where m is the number of hyperedges. When an edge clique cover of size cc′ is given as a part of the input we give O∗(2cc′) time algorithms for treewidth, minimum fill-in, and chordal sandwich. This implies an O∗(2 n) time algorithm for perfect phylogeny, where n is the number of taxa. We also give polynomial space algorithms with time complexities O∗(9cc′) and O∗(9cc+O(log2cc)) for problems in this framework.Peer reviewe

Finding Optimal Triangulations Parameterized by Edge Clique Cover

Peer reviewe

The Complexity of Surjective Homomorphism Problems -- a Survey

We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of which remain of open complexity

Counting hypergraph matchings up to uniqueness threshold

We study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges. With an activity parameter $\lambda$, each matching $M$ is assigned a weight $\lambda^{|M|}$. The counting problem is formulated as computing a partition function that gives the sum of the weights of all matchings in a hypergraph. This problem unifies two extensively studied statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer-dimer model (graph matchings). For this model, the critical activity $\lambda_c= \frac{d^d}{k (d-1)^{d+1}}$ is the threshold for the uniqueness of Gibbs measures on the infinite $(d+1)$-uniform $(k+1)$-regular hypertree. Consider hypergraphs of maximum degree at most $k+1$ and maximum size of hyperedges at most $d+1$. We show that when $\lambda < \lambda_c$, there is an FPTAS for computing the partition function; and when $\lambda = \lambda_c$, there is a PTAS for computing the log-partition function. These algorithms are based on the decay of correlation (strong spatial mixing) property of Gibbs distributions. When $\lambda > 2\lambda_c$, there is no PRAS for the partition function or the log-partition function unless NP$=$RP. Towards obtaining a sharp transition of computational complexity of approximate counting, we study the local convergence from a sequence of finite hypergraphs to the infinite lattice with specified symmetry. We show a surprising connection between the local convergence and the reversibility of a natural random walk. This leads us to a barrier for the hardness result: The non-uniqueness of infinite Gibbs measure is not realizable by any finite gadgets

Semi-algebraic Ramsey numbers

Given a finite point set $P \subset \mathbb{R}^d$, a $k$-ary semi-algebraic relation $E$ on $P$ is the set of $k$-tuples of points in $P$, which is determined by a finite number of polynomial equations and inequalities in $kd$ real variables. The description complexity of such a relation is at most $t$ if the number of polynomials and their degrees are all bounded by $t$. The Ramsey number $R^{d,t}_k(s,n)$ is the minimum $N$ such that any $N$-element point set $P$ in $\mathbb{R}^d$ equipped with a $k$-ary semi-algebraic relation $E$, such that $E$ has complexity at most $t$, contains $s$ members such that every $k$-tuple induced by them is in $E$, or $n$ members such that every $k$-tuple induced by them is not in $E$. We give a new upper bound for $R^{d,t}_k(s,n)$ for $k\geq 3$ and $s$ fixed. In particular, we show that for fixed integers $d,t,s$, $R^{d,t}_3(s,n) \leq 2^{n^{o(1)}},$ establishing a subexponential upper bound on $R^{d,t}_3(s,n)$. This improves the previous bound of $2^{n^C}$ due to Conlon, Fox, Pach, Sudakov, and Suk, where $C$ is a very large constant depending on $d,t,$ and $s$. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in $\mathbb{R}^d$. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results

Geometric Graph Theory and Wireless Sensor Networks

In this work, we apply geometric and combinatorial methods to explore a variety of problems motivated by wireless sensor networks. Imagine sensors capable of communicating along straight lines except through obstacles like buildings or barriers, such that the communication network topology of the sensors is their visibility graph. Using a standard distributed algorithm, the sensors can build common knowledge of their network topology. We first study the following inverse visibility problem: What positions of sensors and obstacles define the computed visibility graph, with fewest obstacles? This is the problem of finding a minimum obstacle representation of a graph. This minimum number is the obstacle number of the graph. Using tools from extremal graph theory and discrete geometry, we obtain for every constant h that the number of n-vertex graphs that admit representations with h obstacles is 2o(n2). We improve this bound to show that graphs requiring Ω(n / log2 n) obstacles exist. We also study restrictions to convex obstacles, and to obstacles that are line segments. For example, we show that every outerplanar graph admits a representation with five convex obstacles, and that allowing obstacles to intersect sometimes decreases their required number. Finally, we study the corresponding problem for sensors equipped with GPS. Positional information allows sensors to establish common knowledge of their communication network geometry, hence we wish to compute a minimum obstacle representation of a given straight-line graph drawing. We prove that this problem is NP-complete, and provide a O(logOPT)-factor approximation algorithm by showing that the corresponding hypergraph family has bounded Vapnik-Chervonenkis dimension

A Combinatorial Approach to Nonlocality and Contextuality

So far, most of the literature on (quantum) contextuality and the Kochen-Specker theorem seems either to concern particular examples of contextuality, or be considered as quantum logic. Here, we develop a general formalism for contextuality scenarios based on the combinatorics of hypergraphs which significantly refines a similar recent approach by Cabello, Severini and Winter (CSW). In contrast to CSW, we explicitly include the normalization of probabilities, which gives us a much finer control over the various sets of probabilistic models like classical, quantum and generalized probabilistic. In particular, our framework specializes to (quantum) nonlocality in the case of Bell scenarios, which arise very naturally from a certain product of contextuality scenarios due to Foulis and Randall. In the spirit of CSW, we find close relationships to several graph invariants. The recently proposed Local Orthogonality principle turns out to be a special case of a general principle for contextuality scenarios related to the Shannon capacity of graphs. Our results imply that it is strictly dominated by a low level of the Navascu\'es-Pironio-Ac\'in hierarchy of semidefinite programs, which we also apply to contextuality scenarios. We derive a wealth of results in our framework, many of these relating to quantum and supraquantum contextuality and nonlocality, and state numerous open problems. For example, we show that the set of quantum models on a contextuality scenario can in general not be characterized in terms of a graph invariant. In terms of graph theory, our main result is this: there exist two graphs $G_1$ and $G_2$ with the properties \begin{align*} \alpha(G_1) &= \Theta(G_1), & \alpha(G_2) &= \vartheta(G_2), \\[6pt] \Theta(G_1\boxtimes G_2) & > \Theta(G_1)\cdot \Theta(G_2),& \Theta(G_1 + G_2) & > \Theta(G_1) + \Theta(G_2). \end{align*}Comment: minor revision, same results as in v2, to appear in Comm. Math. Phy