Independent Sets, Matchings, and Occupancy Fractions

We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing that a union of copies of $K_{d,d}$ maximizes the number of independent sets and the independence polynomial of a d-regular graph. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of $K_{d,d}$. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstr\"om. In probabilistic language, our main theorems state that for all d-regular graphs and all $\lambda$, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity $\lambda$ are maximized by $K_{d,d}$. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case.Comment: Typo corrected last equation pg

A stronger bound for the strong chromatic index

We prove $\chi_s'(G)\leq 1.93 \Delta(G)^2$ for graphs of sufficiently large maximum degree where $\chi_s'(G)$ is the strong chromatic index of $G$. This improves an old bound of Molloy and Reed. As a by-product, we present a Talagrand-type inequality where it is allowed to exclude unlikely bad outcomes that would otherwise render the inequality unusable.Comment: 22 page

Generalized full matching and extrapolation of the results from a large-scale voter mobilization experiment

Matching is an important tool in causal inference. The method provides a conceptually straightforward way to make groups of units comparable on observed characteristics. The use of the method is, however, limited to situations where the study design is fairly simple and the sample is moderately sized. We illustrate the issue by revisiting a large-scale voter mobilization experiment that took place in Michigan for the 2006 election. We ask what the causal effects would have been if the treatments in the experiment were scaled up to the full population. Matching could help us answer this question, but no existing matching method can accommodate the six treatment arms and the 6,762,701 observations involved in the study. To offer a solution this and similar empirical problems, we introduce a generalization of the full matching method and an associated algorithm. The method can be used with any number of treatment conditions, and it is shown to produce near-optimal matchings. The worst case maximum within-group dissimilarity is no worse than four times the optimal solution, and simulation results indicate that its performance is considerably closer to the optimal solution on average. Despite its performance, the algorithm is fast and uses little memory. It terminates, on average, in linearithmic time using linear space. This enables investigators to construct well-performing matchings within minutes even in complex studies with samples of several million units

Massively Parallel Symmetry Breaking on Sparse Graphs: MIS and Maximal Matching

The success of modern parallel paradigms such as MapReduce, Hadoop, or Spark, has attracted a significant attention to the Massively Parallel Computation (MPC) model over the past few years, especially on graph problems. In this work, we consider symmetry breaking problems of maximal independent set (MIS) and maximal matching (MM), which are among the most intensively studied problems in distributed/parallel computing, in MPC. These problems are known to admit efficient MPC algorithms if the space per machine is near-linear in $n$, the number of vertices in the graph. This space requirement however, as observed in the literature, is often significantly larger than we can afford; especially when the input graph is sparse. In a sharp contrast, in the truly sublinear regime of $n^{1-\Omega(1)}$ space per machine, all the known algorithms take $\log^{\Omega(1)} n$ rounds which is considered inefficient. Motivated by this shortcoming, we parametrize our algorithms by the arboricity $\alpha$ of the input graph, which is a well-received measure of its sparsity. We show that both MIS and MM admit $O(\sqrt{\log \alpha}\cdot\log\log \alpha + \log^2\log n)$ round algorithms using $O(n^\epsilon)$ space per machine for any constant $\epsilon \in (0, 1)$ and using $\widetilde{O}(m)$ total space. Therefore, for the wide range of sparse graphs with small arboricity---such as minor-free graphs, bounded-genus graphs or bounded treewidth graphs---we get an $O(\log^2 \log n)$ round algorithm which exponentially improves prior algorithms. By known reductions, our results also imply a $(1+\epsilon)$-approximation of maximum cardinality matching, a $(2+\epsilon)$-approximation of maximum weighted matching, and a 2-approximation of minimum vertex cover with essentially the same round complexity and memory requirements.Comment: A merger of this paper and the independent and concurrent paper [arxiv:1807.05374] appeared at PODC 201

Fully dynamic maximal matching in O(log n) update time

We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our data structure is randomized that takes O(log n) expected amortized time for each edge update where n is the number of vertices in the graph. While there is a trivial O(n) algorithm for edge update, the previous best known result for this problem for a graph with n vertices and m edges is O({(n+ m)}^{0.7072})which is sub-linear only for a sparse graph. For the related problem of maximum matching, Onak and Rubinfield designed a randomized data structure that achieves O(log^2 n) amortized time for each update for maintaining a c-approximate maximum matching for some large constant c. In contrast, we can maintain a factor two approximate maximum matching in O(log n) expected time per update as a direct corollary of the maximal matching scheme. This in turn also implies a two approximate vertex cover maintenance scheme that takes O(log n) expected time per update.Comment: 16 pages, 3 figure

"Tri, Tri again": Finding Triangles and Small Subgraphs in a Distributed Setting

Let G = (V,E) be an n-vertex graph and M_d a d-vertex graph, for some constant d. Is M_d a subgraph of G? We consider this problem in a model where all n processes are connected to all other processes, and each message contains up to O(log n) bits. A simple deterministic algorithm that requires O(n^((d-2)/d) / log n) communication rounds is presented. For the special case that M_d is a triangle, we present a probabilistic algorithm that requires an expected O(ceil(n^(1/3) / (t^(2/3) + 1))) rounds of communication, where t is the number of triangles in the graph, and O(min{n^(1/3) log^(2/3) n / (t^(2/3) + 1), n^(1/3)}) with high probability. We also present deterministic algorithms specially suited for sparse graphs. In any graph of maximum degree Delta, we can test for arbitrary subgraphs of diameter D in O(ceil(Delta^(D+1) / n)) rounds. For triangles, we devise an algorithm featuring a round complexity of O(A^2 / n + log_(2+n/A^2) n), where A denotes the arboricity of G.Comment: 22 pages, no figures, extended abstract published at DISC'1

Total Dominating Sequences in Graphs

A vertex in a graph totally dominates another vertex if they are adjacent. A sequence of vertices in a graph $G$ is called a total dominating sequence if every vertex $v$ in the sequence totally dominates at least one vertex that was not totally dominated by any vertex that precedes $v$ in the sequence, and at the end all vertices of $G$ are totally dominated. While the length of a shortest such sequence is the total domination number of $G$, in this paper we investigate total dominating sequences of maximum length, which we call the Grundy total domination number, $\gamma_{\rm gr}^t(G)$, of $G$. We provide a characterization of the graphs $G$ for which $\gamma_{\rm gr}^t(G)=|V(G)|$ and of those for which $\gamma_{\rm gr}^t(G)=2$. We show that if $T$ is a nontrivial tree of order~$n$ with no vertex with two or more leaf-neighbors, then $\gamma_{\rm gr}^t(T) \ge \frac{2}{3}(n+1)$, and characterize the extremal trees. We also prove that for $k \ge 3$, if $G$ is a connected $k$-regular graph of order~$n$ different from $K_{k,k}$, then $\gamma_{\rm gr}^t(G) \ge (n + \lceil \frac{k}{2} \rceil - 2)/(k-1)$ if $G$ is not bipartite and $\gamma_{\rm gr}^t(G) \ge (n + 2\lceil \frac{k}{2} \rceil - 4)/(k-1)$ if $G$ is bipartite. The Grundy total domination number is proven to be bounded from above by two times the Grundy domination number, while the former invariant can be arbitrarily smaller than the latter. Finally, a natural connection with edge covering sequences in hypergraphs is established, which in particular yields the NP-completeness of the decision version of the Grundy total domination number.Comment: 20 pages, 2 figure

Critical Independent Sets of a Graph

Let $G$ be a simple graph with vertex set $V\left( G\right)$. A set $S\subseteq V\left( G\right)$ is independent if no two vertices from $S$ are adjacent, and by $\mathrm{Ind}(G)$ we mean the family of all independent sets of $G$. The number $d\left( X\right) =$ $\left\vert X\right\vert -\left\vert N(X)\right\vert$ is the difference of $X\subseteq V\left( G\right)$, and a set $A\in\mathrm{Ind}(G)$ is critical if $d(A)=\max \{d\left( I\right) :I\in\mathrm{Ind}(G)\}$ (Zhang, 1990). Let us recall the following definitions: $\mathrm{core}\left( G\right)$ = $\bigcap$ {S : S is a maximum independent set}. $\mathrm{corona}\left( G\right)$ = $\bigcup$ {S :S is a maximum independent set}. $\mathrm{\ker}(G)$ = $\bigcap$ {S : S is a critical independent set}. $\mathrm{diadem}(G)$ = $\bigcup$ {S : S is a critical independent set}. In this paper we present various structural properties of $\mathrm{\ker}(G)$, in relation with $\mathrm{core}\left( G\right)$, $\mathrm{corona}\left( G\right)$, and $\mathrm{diadem}(G)$.Comment: 15 pages; 12 figures. arXiv admin note: substantial text overlap with arXiv:1102.113

Matching and MIS for Uniformly Sparse Graphs in the Low-Memory MPC Model

The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale parallel computation frameworks and has recently gained a lot of importance, especially in the context of classic graph problems. Unsatisfactorily, all current $\text{poly} (\log \log n)$-round MPC algorithms seem to get fundamentally stuck at the linear-memory barrier: their efficiency crucially relies on each machine having space at least linear in the number $n$ of nodes. As this might not only be prohibitively large, but also allows for easy if not trivial solutions for sparse graphs, we are interested in the low-memory MPC model, where the space per machine is restricted to be strongly sublinear, that is, $n^{\delta}$ for any $0<\delta<1$. We devise a degree reduction technique that reduces maximal matching and maximal independent set in graphs with arboricity $\lambda$ to the corresponding problems in graphs with maximum degree $\text{poly}(\lambda)$ in $O(\log^2 \log n)$ rounds. This gives rise to $O\left(\log^2\log n + T(\text{poly} \lambda)\right)$-round algorithms, where $T(\Delta)$ is the $\Delta$-dependency in the round complexity of maximal matching and maximal independent set in graphs with maximum degree $\Delta$. A concurrent work by Ghaffari and Uitto shows that $T(\Delta)=O(\sqrt{\log \Delta})$. For graphs with arboricity $\lambda=\text{poly}(\log n)$, this almost exponentially improves over Luby's $O(\log n)$-round PRAM algorithm [STOC'85, JALG'86], and constitutes the first $\text{poly} (\log \log n)$-round maximal matching algorithm in the low-memory MPC model, thus breaking the linear-memory barrier. Previously, the only known subpolylogarithmic algorithm, due to Lattanzi et al. [SPAA'11], required strongly superlinear, that is, $n^{1+\Omega(1)}$, memory per machine

On Distance-dd Independent Set and other problems in graphs with few minimal separators

Fomin and Villanger (STACS 2010) proved that Maximum Independent Set, Feedback Vertex Set, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant $t$, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let \Gpoly be the class of graphs with at most \poly(n) minimal separators, for some polynomial \poly. We show that the odd powers of a graph $G$ have at most as many minimal separators as $G$. Consequently, \textsc{Distance-$d$ Independent Set}, which consists in finding maximum set of vertices at pairwise distance at least $d$, is polynomial on \Gpoly, for any even $d$. The problem is NP-hard on chordal graphs for any odd $d \geq 3$. We also provide polynomial algorithms for Connected Vertex Cover and Connected Feedback Vertex Set on subclasses of \Gpoly including chordal and circular-arc graphs, and we discuss variants of independent domination problems