A bond in a graph is a minimal nonempty edge-cut. A connected graph $G$ is dual Hamiltonian if the vertex set can be partitioned into two subsets $X$ and $Y$ such that the subgraphs induced by $X$ and $Y$ are both trees. There is much interest in studying the longest cycles and largest bonds in graphs. H. Wu conjectured that any longest cycle must meet any largest bond in a simple 3-connected graph. In this paper, the author proves that the above conjecture is true for certain classes of 3-connected graphs: Let $G$ be a simple 3-connected graph with $n$ vertices and $m$ edges. Suppose $c(G)$ is the size of a longest cycle, and $c^*(G)$ is the size of a largest bond. Then each longest cycle meets each largest bond if either $c(G) \geq n - 3$ or $c^*(G) \geq m - n - 1$. Sanford determined in her Ph.D. thesis the cycle spectrum of the well-known generalized Petersen graph $P(n, 2)$ ($n$ is odd) and $P(n, 3)$ ($n$ is even). Flynn proved in her honors thesis that any generalized Petersen graph $P(n, k)$ is dual Hamiltonian. The author studies the bond spectrum (called the co-spectrum) of the generalized Petersen graphs and extends Flynn's result by proving that in any generalized Petersen graph $P(n, k)$, $1 \leq k < \frac{n}{2}$, the co-spectrum of $P(n, k)$ is $\{3, 4, 5, ..., n+2\}$.
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We present a novel method to compute $\textit{assume-guarantee contracts}$ in non-zerosum two-player games over finite graphs where each player has a different $ \omega $-regular winning condition. Given a game graph $G$ and two parity winning conditions $\Phi_0$ and $\Phi_1$ over $G$, we compute $\textit{contracted strategy-masks}$ ($\texttt{csm}$) $(\Psi_{i},\Phi_{i})$ for each Player $i$. Within a $\texttt{csm}$, $\Phi_{i}$ is a $\textit{permissive strategy template}$ which collects an infinite number of winning strategies for Player $i$ under the assumption that Player $1-i$ chooses any strategy from the $\textit{permissive assumption template}$ $\Psi_{i}$. The main feature of $\texttt{csm}$'s is their power to $\textit{fully decentralize all remaining strategy choices}$ -- if the two player's $\texttt{csm}$'s are compatible, they provide a pair of new local specifications $\Phi_0^\bullet$ and $\Phi_1^\bullet$ such that Player $i$ can locally and fully independently choose any strategy satisfying $\Phi_i^\bullet$ and the resulting strategy profile is ensured to be winning in the original two-objective game $(G,\Phi_0,\Phi_1)$. In addition, the new specifications $\Phi_i^\bullet$ are $\textit{maximally cooperative}$, i.e., allow for the distributed synthesis of any cooperative solution. Further, our algorithmic computation of $\texttt{csm}$'s is complete and ensured to terminate. We illustrate how the unique features of our synthesis framework effectively address multiple challenges in the context of \enquote{correct-by-design} logical control software synthesis for cyber-physical systems and provide empirical evidence that our approach possess desirable structural and computational properties compared to state-of-the-art techniques.
We consider the randomized communication complexity of the distributed $\ell_p$-regression problem in the coordinator model, for $p\in (0,2]$. In this problem, there is a coordinator and $s$ servers. The $i$-th server receives $A^i\in\{-M, -M+1, \ldots, M\}^{n\times d}$ and $b^i\in\{-M, -M+1, \ldots, M\}^n$ and the coordinator would like to find a $(1+\epsilon)$-approximate solution to $\min_{x\in\mathbb{R}^n} \|(\sum_i A^i)x - (\sum_i b^i)\|_p$. Here $M \leq \mathrm{poly}(nd)$ for convenience. This model, where the data is additively shared across servers, is commonly referred to as the arbitrary partition model. We obtain significantly improved bounds for this problem. For $p = 2$, i.e., least squares regression, we give the first optimal bound of $\tilde{\Theta}(sd^2 + sd/\epsilon)$ bits. For $p \in (1,2)$,we obtain an $\tilde{O}(sd^2/\epsilon + sd/\mathrm{poly}(\epsilon))$ upper bound. Notably, for $d$ sufficiently large, our leading order term only depends linearly on $1/\epsilon$ rather than quadratically. We also show communication lower bounds of $\Omega(sd^2 + sd/\epsilon^2)$ for $p\in (0,1]$ and $\Omega(sd^2 + sd/\epsilon)$ for $p\in (1,2]$. Our bounds considerably improve previous bounds due to (Woodruff et al. COLT, 2013) and (Vempala et al., SODA, 2020).
The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. In this paper, we introduce a natural generalization of this game in which $k$ random vertices $u_1, \ldots, u_k$ are presented to the player in each round. She needs to select one of the presented vertices and connect to any vertex she wants. We focus on the following three monotone properties: minimum degree at least $\ell$, the existence of a perfect matching, and the existence of a Hamiltonian cycle.
Given a graph $G=(V,E)$, for a vertex set $S\subseteq V$, let $N(S)$ denote the set of vertices in $V$ that have a neighbor in $S$. Extending the concept of binding number of graphs by Woodall~(1973), for a vertex set $X \subseteq V$, we define the binding number of $X$, denoted by $\bind(X)$, as the maximum number $b$ such that for every $S \subseteq X$ where $N(S)\neq V(G)$ it holds that $|N(S)|\ge b {|S|}$. Given this definition, we prove that if a graph $V(G)$ contains a subset $X$ with $\bind(X)= 1/k$ where $k$ is an integer, then $G$ possesses a matching of size at least $|X|/(k+1)$. Using this statement, we derive tight bounds for the estimators of the matching size in planar graphs. These estimators are previously used in designing sublinear space algorithms for approximating the maching size in the data stream model of computation. In particular, we show that the number of locally superior vertices is a $3$ factor approximation of the matching size in planar graphs. The previous analysis by Jowhari (2023) proved a $3.5$ approximation factor. As another application, we show a simple variant of an estimator by Esfandiari \etal (2015) achieves $3$ factor approximation of the matching size in planar graphs. Namely, let $s$ be the number of edges with both endpoints having degree at most $2$ and let $h$ be the number of vertices with degree at least $3$. We prove that when the graph is planar, the size of matching is at least $(s+h)/3$. This result generalizes a known fact that every planar graph on $n$ vertices with minimum degree $3$ has a matching of size at least $n/3$.
Directed acyclic graphs (DAGs) are directed graphs in which there is no path from a vertex to itself. DAGs are an omnipresent data structure in computer science and the problem of counting the DAGs of given number of vertices and to sample them uniformly at random has been solved respectively in the 70's and the 00's. In this paper, we propose to explore a new variation of this model where DAGs are endowed with an independent ordering of the out-edges of each vertex, thus allowing to model a wide range of existing data structures. We provide efficient algorithms for sampling objects of this new class, both with or without control on the number of edges, and obtain an asymptotic equivalent of their number. We also show the applicability of our method by providing an effective algorithm for the random generation of classical labelled DAGs with a prescribed number of vertices and edges, based on a similar approach. This is the first known algorithm for sampling labelled DAGs with full control on the number of edges, and it meets a need in terms of applications, that had already been acknowledged in the literature.
A long-standing conjecture for the traveling salesman problem (TSP) states that the integrality gap of the standard linear programming relaxation of the TSP is at most 4/3. Despite significant efforts, the conjecture remains open. We consider the half-integral case, in which the LP has solution values in $\{0, 1/2, 1\}$. Such instances have been conjectured to be the most difficult instances for the overall four-thirds conjecture. Karlin, Klein, and Oveis Gharan, in a breakthrough result, were able to show that in the half-integral case, the integrality gap is at most 1.49993. This result led to the first significant progress on the overall conjecture in decades; the same authors showed the integrality gap is at most $1.5- 10^{-36}$ in the non-half-integral case. For the half-integral case, the current best-known ratio is 1.4983, a result by Gupta et al. With the improvements on the 3/2 bound remaining very incremental even in the half-integral case, we turn the question around and look for a large class of half-integral instances for which we can prove that the 4/3 conjecture is correct. The previous works on the half-integral case perform induction on a hierarchy of critical tight sets in the support graph of the LP solution, in which some of the sets correspond to "cycle cuts" and the others to "degree cuts". We show that if all the sets in the hierarchy correspond to cycle cuts, then we can find a distribution of tours whose expected cost is at most 4/3 times the value of the half-integral LP solution; sampling from the distribution gives us a randomized 4/3-approximation algorithm. We note that the known bad cases for the integrality gap have a gap of 4/3 and have a half-integral LP solution in which all the critical tight sets in the hierarchy are cycle cuts; thus our result is tight.
An obstacle representation of a graph $G$ consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of $G$ to points such that two vertices are adjacent in $G$ if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each $n$-vertex graph is $O(n \log n)$ [Balko, Cibulka, and Valtr, 2018] and that there are $n$-vertex graphs whose obstacle number is $\Omega(n/(\log\log n)^2)$ [Dujmovi\'c and Morin, 2015]. We improve this lower bound to $\Omega(n/\log\log n)$ for simple polygons and to $\Omega(n)$ for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of $n$-vertex graphs with bounded obstacle number, solving a conjecture by Dujmovi\'c and Morin. We also show that if the drawing of some $n$-vertex graph is given as part of the input, then for some drawings $\Omega(n^2)$ obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph $G$ is fixed-parameter tractable in the vertex cover number of $G$. Second, we show that, given a graph $G$ and a simple polygon $P$, it is NP-hard to decide whether $G$ admits an obstacle representation using $P$ as the only obstacle.
This paper presents a novel Importance Sampling (IS) scheme for estimating distribution tails of performance measures modeled with a rich set of tools such as linear programs, integer linear programs, piecewise linear/quadratic objectives, feature maps specified with deep neural networks, etc. The conventional approach of explicitly identifying efficient changes of measure suffers from feasibility and scalability concerns beyond highly stylized models, due to their need to be tailored intricately to the objective and the underlying probability distribution. This bottleneck is overcome in the proposed scheme with an elementary transformation which is capable of implicitly inducing an effective IS distribution in a variety of models by replicating the concentration properties observed in less rare samples. This novel approach is guided by developing a large deviations principle that brings out the phenomenon of self-similarity of optimal IS distributions. The proposed sampler is the first to attain asymptotically optimal variance reduction across a spectrum of multivariate distributions despite being oblivious to the specifics of the underlying model. Its applicability is illustrated with contextual shortest path and portfolio credit risk models informed by neural networks
Maximum weight independent set (MWIS) admits a $\frac1k$-approximation in inductively $k$-independent graphs and a $\frac{1}{2k}$-approximation in $k$-perfectly orientable graphs. These are a a parameterized class of graphs that generalize $k$-degenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudo-disks, and several others. We consider a generalization of MWIS to a submodular objective. Given a graph $G=(V,E)$ and a non-negative submodular function $f: 2^V \rightarrow \mathbb{R}_+$, the goal is to approximately solve $\max_{S \in \mathcal{I}_G} f(S)$ where $\mathcal{I}_G$ is the set of independent sets of $G$. We obtain an $\Omega(\frac1k)$-approximation for this problem in the two mentioned graph classes. The first approach is via the multilinear relaxation framework and a simple contention resolution scheme, and this results in a randomized algorithm with approximation ratio at least $\frac{1}{e(k+1)}$. This approach also yields parallel (or low-adaptivity) approximations. Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively $k$-independent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primal-dual algorithm. In addition to being simpler and faster, these algorithms, in the monotone submodular case, yield the first deterministic constant factor approximations for various special cases that have been previously considered such as intersection graphs of intervals, disks and pseudo-disks.
In structure learning, the output is generally a structure that is used as supervision information to achieve good performance. Considering the interpretation of deep learning models has raised extended attention these years, it will be beneficial if we can learn an interpretable structure from deep learning models. In this paper, we focus on Recurrent Neural Networks (RNNs) whose inner mechanism is still not clearly understood. We find that Finite State Automaton (FSA) that processes sequential data has more interpretable inner mechanism and can be learned from RNNs as the interpretable structure. We propose two methods to learn FSA from RNN based on two different clustering methods. We first give the graphical illustration of FSA for human beings to follow, which shows the interpretability. From the FSA's point of view, we then analyze how the performance of RNNs are affected by the number of gates, as well as the semantic meaning behind the transition of numerical hidden states. Our results suggest that RNNs with simple gated structure such as Minimal Gated Unit (MGU) is more desirable and the transitions in FSA leading to specific classification result are associated with corresponding words which are understandable by human beings.
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