M-edge Graph Research Articles

The list-coloring function of signed graphs

It is known that, for any k-list assignment L of an m-edge graph G, the number of L-list colorings of G is at least the number of the proper k-colorings of G when k>(m−1)/ln⁡(1+2). In this paper, we extend the Whitney's broken cycle theorem to L-colorings of signed graphs, by which we show that if k>(m3)+(m4)+m−1 then, for any k-assignment L, the number of L-colorings of a signed graph Σ with m edges is at least the number of the proper k-colorings of Σ. Further, if L is 0-free (resp., 0-included) and k is even (resp., odd), then the lower bound (m3)+(m4)+m−1 for k can be improved to (m−1)/ln⁡(1+2).

Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained Complexity

In a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include:Certifying that a list of n integers has no 3-SUM solution can be done in Merlin–Arthur time tilde{O}(n). Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in tilde{O}(n^{1.5}) time (that is, there is a proof system with proofs of length tilde{O}(n^{1.5}) and a deterministic verifier running in tilde{O}(n^{1.5}) time).Counting the number of k-cliques with total edge weight equal to zero in an n-node graph can be done in Merlin–Arthur time {tilde{O}}(n^{lceil k/2rceil }) (where kge 3). For odd k, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in an m-edge graph can be done in Merlin–Arthur time {tilde{O}}(m). Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only count k-cliques in unweighted graphs, and had worse running times for small k.Computing the All-Pairs Shortest Distances matrix for an n-node graph can be done in Merlin–Arthur time tilde{O}(n^2). Note this is optimal, as the matrix can have Omega (n^2) nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an tilde{O}(n^{2.94}) nondeterministic time algorithm.Certifying that an n-variable k-CNF is unsatisfiable can be done in Merlin–Arthur time 2^{n/2 - n/O(k)}. We also observe an algebrization barrier for the previous 2^{n/2}cdot textrm{poly}(n)-time Merlin–Arthur protocol of R. Williams [CCC’16] for #SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol for k-UNSAT running in 2^{n/2}/n^{omega (1)} time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol.Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time 2^{4n/5}cdot textrm{poly}(n). Previously, the only nontrivial result known along these lines was an Arthur–Merlin–Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in 2^{2n/3}cdot textrm{poly}(n) time. Due to the centrality of these problems in fine-grained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution to n integers can be done in Merlin–Arthur time 2^{n/3}cdot textrm{poly}(n), improving on the previous best protocol by Nederlof [IPL 2017] which took 2^{0.49991n}cdot textrm{poly}(n) time.

Optimal Centrality Computations Within Bounded Clique-Width Graphs

Given an n-vertex m-edge graph G of clique-width at most k, and a corresponding k-expression, we present algorithms for computing some well-known centrality indices (eccentricity and closeness) that run in \({\mathcal {O}}(2^{{\mathcal {O}}(k)}(n+m)^{1+\epsilon })\) time for any \(\epsilon > 0\). Doing so, we can solve various distance problems within the same amount of time, including: the diameter, the center, the Wiener index and the median set. Our run-times match conditional lower bounds of Coudert et al. (SODA’18) under the Strong Exponential-Time Hypothesis. On our way, we get a distance-labeling scheme for n-vertex m-edge graphs of clique-width at most k, using \({\mathcal {O}}(k\log ^2{n})\) bits per vertex and constructible in \(\tilde{\mathcal {O}}(k(n+m))\) time from a given k-expression. Doing so, we match the label size obtained by Courcelle and Vanicat (DAM 2016), while we considerably improve the dependency on k in their scheme. As a corollary, we get an \(\tilde{\mathcal {O}}(kn^2)\)-time algorithm for computing All-Pairs Shortest-Paths on n-vertex graphs of clique-width at most k, being given a k-expression. This partially answers an open question of Kratsch and Nelles (STACS’20). Our algorithms work for graphs with non-negative vertex-weights, under two different types of distances studied in the literature. For that, we introduce a new type of orthogonal range query as a side contribution of this work, that might be of independent interest.

Simple Concurrent Connected Components Algorithms

We study a class of simple algorithms for concurrently computing the connected components of an n -vertex, m -edge graph. Our algorithms are easy to implement in either the COMBINING CRCW PRAM or the MPC computing model. For two related algorithms in this class, we obtain Θ (lg n ) step and Θ (m lg n ) work bounds. 1 For two others, we obtain O (lg 2 n ) step and O (m lg 2 n ) work bounds, which are tight for one of them. All our algorithms are simpler than related algorithms in the literature. We also point out some gaps and errors in the analysis of previous algorithms. Our results show that even a basic problem like connected components still has secrets to reveal.

A global decomposition theorem for excluding immersions in graphs with no edge-cut of order three

A graph G contains another graph H as an immersion if H can be obtained from a subgraph of G by splitting off edges and removing isolated vertices. There is an obvious necessary degree condition for the immersion containment: if G contains H as an immersion, then for every integer k, the number of vertices of degree at least k in G is at least the number of vertices of degree at least k in H. In this paper, we prove that this obvious necessary condition is “nearly” sufficient for graphs with no edge-cut of order 3: for every graph H, every H-immersion free graph with no edge-cut of order 3 can be obtained by an edge-sum of graphs, where each of the summands is obtained from a graph violating the obvious degree condition by adding a bounded number of edges. The condition for having no edge-cut of order 3 is necessary. A simple application of this theorem shows that for every graph H of maximum degree d≥4, there exists an integer c such that for every positive integer m, there are at most cm unlabeled d-edge-connected H-immersion free m-edge graphs with no isolated vertex, while there are superexponentially many unlabeled (d−1)-edge-connected H-immersion free m-edge graphs with no isolated vertex. Our structure theorem will be applied in a forthcoming paper about determining the clustered chromatic number of the class of H-immersion free graphs.

The Power of Linear-Time Data Reduction for Maximum Matching

Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph primitives. For m-edge and n-vertex graphs, it is well-known to be solvable in O(msqrt{n}) time; however, for several applications this running time is still too slow. We investigate how linear-time (and almost linear-time) data reduction (used as preprocessing) can alleviate the situation. More specifically, we focus on linear-time kernelization. We start a deeper and systematic study both for general graphs and for bipartite graphs. Our data reduction algorithms easily comply (in form of preprocessing) with every solution strategy (exact, approximate, heuristic), thus making them attractive in various settings.

Hypergraph cuts above the average

An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size $$m{\rm/}2 + {\rm\Omega)}(\sqrt m)$$ and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is $${\rm\Omega}(\sqrt m)$$ larger than the expected size of a random r-cut. Moreover, in the case where k = 3 and r = 2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if k ≥ 4 or r ≥ 3), we show that every m-edge k-uniform hypergraph has an r-cut whose size is Ω(m5/9) larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation.

Map graphs having witnesses of large girth

A half-square of a bipartite graph B=(X,Y,EB) has one color class of B as vertex set, say X; two vertices are adjacent whenever they have a common neighbor in Y. If G=(V,EG) is the half-square of a planar bipartite graph B=(V,W,EB), then G is called a map graph, and B is a witness of G. Map graphs generalize planar graphs, and have been introduced and investigated by Chen, Grigni and Papadimitriou [STOC 1998, J. ACM 2002]. They proved that recognizing map graphs is in NP by proving the existence of a witness. Soon later, Thorup [FOCS 1998] claimed that recognizing map graphs is in P, by providing an Ω(n120)-time algorithm for n-vertex input graphs.In this note, we give good characterizations and efficient recognition for half-squares of bipartite graphs with girth at least a given integer g≥8. It turns out that map graphs having witnesses of girth at least g are precisely the graphs whose vertex-clique incidence bipartite graph is planar and of girth at least g. Our structural characterization implies an O(n2m)-time algorithm for recognizing if a given n-vertex m-edge graph G is such a map graph.

Linear-Time Algorithms for Tree Root Problems

Let T be a tree on a set V of nodes. The p-th power T p of T is the graph on V such that any two nodes u and w of V are adjacent in T p if and only if the distance of u and w in T is at most p. Given an n-node m-edge graph G and a positive integer p, the p-th tree root problem asks for a tree T, if any, such that G=T p. Given an n-node m-edge graph G, the tree root problem asks for a positive integer p and a tree T, if any, such that G=T p. Kearney and Corneil gave the best previously known algorithms for both problems. Their algorithm for the former (respectively, latter) problem runs in O(n 3) (respectively, O(n 4)) time. In this paper, we give O(n+m)-time algorithms for both problems.

The connected [formula omitted]-center problem on block graphs with forbidden vertices

Let G(V,E,w,l) denote an n-vertex and m-edge graph in which w is a function mapping each vertex v to a positive weight w(v) and l is a function mapping each edge e to a positive length l(e). Given a positive integer p, the p-Center problem involves finding a set Q with p vertices of G to be the locations for building facilities. The objective is to minimize the maximum weighted distance from each vertex in V–Q to its nearest vertex in Q. This paper considers a practical restriction: the induced subgraph of the selected p vertices must be connected. The new variant is called the Connected p-Center problem (the CpC problem). For each fixed integer t≥1, on block graphs with exactly t blocks, we first show that the CpC problem is NP-hard when (1) w(v)=1, for all vertices v, and l(e)∈{1,2}, for all edges e, and (2) w(v)∈{1,2}, for all vertices v, and l(e)=1, for all edges e, respectively. Second, an O(n+m)-time algorithm for solving the CpC problem on block graphs with unit vertex-weights and unit edge-lengths is proposed. Then, the algorithmic result is extended to handle the situation in which some vertices in G cannot be included to form feasible solutions. The complexity of the extended algorithm is also O(n+m).

Undirected forest constraints

We present two constraints that partition the vertices of an undirected n-vertex, m-edge graph $\mathcal {G}=( \mathcal {V}, \mathcal {E})$ into a set of vertex-disjoint trees. The first is the resource-forest constraint, where we assume that a subset $\mathcal {R}\subseteq \mathcal {V}$ of the vertices are resource vertices. The constraint specifies that each tree in the forest must contain at least one resource vertex. This is the natural undirected counterpart of the tree constraint (Beldiceanu et al., CP-AI-OR’05, Springer, Berlin, 2005), which partitions a directed graph into a forest of directed trees where only certain vertices can be tree roots. We describe a hybrid-consistency algorithm that runs in $\mathop {\mathcal {O}}(m+n)$ time for the resource-forest constraint, a sharp improvement over the $\mathop {\mathcal {O}}(mn)$ bound that is known for the directed case. The second constraint is proper-forest. In this variant, we do not have the requirement that each tree contains a resource, but the forest must contain only proper trees, i.e., trees that have at least two vertices each. We develop a hybrid-consistency algorithm for this case whose running time is $\mathop {\mathcal {O}}(mn)$ in the worst case, and $\mathop {\mathcal {O}}(m\sqrt{n})$ in many (typical) cases.

On Compact Encoding of Pagenumber $k$ Graphs

In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;.

An Improved Neural Network Model for the Two-Page Crossing Number Problem

The simplest graph drawing method is that of putting the vertices of a graph on a line and drawing the edges as half-circles either above or below the line. Such drawings are called two-page book drawings. The smallest number of crossings over all two-page drawings of a graph G is called the two-page crossing number of G. Cimikowski and Shope have solved the two-page crossing number problem for an n-vertex and m-edge graph by using a Hopfield network with 2 m neurons. We present here an improved Hopfield model with m neurons. The new model achieves much better performance in the quality of solutions and is more efficient than the model of Cimikowski and Shope for all graphs tested. The parallel time complexity of the algorithm, without considering the crossing number calculations, is O(m) for the new Hopfield model with m processors clearly outperforming the previous algorithm.

An analysis of some linear graph layout heuristics

The worst-case performances of some heuristics for the fixed linear crossing number problem (FLCNP) are analyzed. FLCNP is similar to the 2-page book crossing number problem in which the vertices of a graph are optimally placed on a horizontal "node line" in the plane, each edge is drawn as an arc in one half-plane (page), and the objective is to minimize the number of edge crossings. In FLCNP, the order of the vertices along the node line is predetermined and fixed. FLCNP belongs to the class of NP-hard optimization problems Masuda et al., 1990. In this paper we show that for each of the heuristics described, there exist classes of n-vertex, m-edge graphs which force it to obtain a number of crossings which is a function of n or m when the optimal number is a small constant. This leaves open the problem of finding a heuristic with a constant error bound for the problem.

Geometric thickness in a grid

The geometric thickness of a graph is the minimum number of layers such that the graph can be drawn in the plane with edges as straight-line segments, and with each edge assigned to a layer so that no two edges in the same layer cross. We consider a variation on this theme in which each edge is allowed one bend. We prove that the vertices of an n-vertex m-edge graph can be positioned in a ⌈ n ⌉×⌈ n ⌉ grid and the edges assigned to O( m ) layers, so that each edge is drawn with at most one bend and no two edges on the same layer cross. The proof is a 2-dimensional generalisation of a theorem of Malitz (J. Algorithms 17(1) (1994) 71–84) on book embeddings. We obtain a Las Vegas algorithm to compute the drawing in O(m log 3 n log log n) time (with high probability).

An efficient parallel algorithm for the efficient domination problem on distance-hereditary graphs

In the literature, there are quite a few sequential and parallel algorithms for solving problems on distance-hereditary graphs. With an n-vertex and m-edge distance-hereditary graph G, we show that the efficient domination problem on G can be solved in O(log/sup 2/ n) time using O(n + m) processors on a CREW PRAM. Moreover, if a binary tree representation of G is given, the problem can be optimally solved in O(log n) time using O(n/log n) processors on an EREW PRAM.

Augmenting Undirected Edge Connectivity in Õ(n2) Time

We give improved randomized (Monte Carlo) algorithms for undirected edge splitting and edge connectivity augmentation problems. Our algorithms run in time Õ(n2) on n-vertex graphs, making them an Ω̃(m/n) factor faster than the best known deterministic ones on m-edge graphs.

Hammock-on-ears decomposition: A technique for the efficient parallel solution of shortest paths and other problems

We show how to decompose efficiently in parallel any graph into a number, \\ ̃ gg, of outerplanar subgraphs (called hammocks) satisfying certain separator properties. Our work combines and extends the sequential hammock decomposition technique introduced by Frederickson and the parallel ear decomposition technique, thus we call it the hammock-on-ears decomposition. We mention that hammock-on-ears decomposition also draws from techniques in computational geometry and that an embedding of the graph does not need to be provided with the input. We achieve this decomposition in O( log n log log n) time using O( n + m) CREW PRAM processors, for an n-vertex, m-edge graph or digraph. The hammock-on-ears decomposition implies a general framework for solving graph problems efficiently. Its value is demonstrated by a variety of applications on a significant class of graphs, namely that of sparse (di)graphs. This class consists of all (di)graphs which have a \\ ̃ gg between 1 and Θ( n), and includes planar graphs and graphs with genus o( n). We improve previous bounds for certain instances of shortest paths and related problems, in this class of graphs. These problems include all pairs shortest paths, all pairs reachability, and detection of a negative cycle.

A fast and efficient NC algorithm for maximal matching

The fastest known NC algorithm for maximal matching is a simple reduction to Luby's NC algorithm for maximal independent sets. It runs in O( log 2 n) time using m 3 Δ processors on an EREW PRAM. In this paper, we present an algorithm that finds a maximal matching of a given n-vertex m-edge graph with maximum degree Δ in O(( Δ 2 p ) · log 2 n) time using p · (m + n) log n processors on an EREW PRAM for any 1 ⩽ p ⩽ Δ 2. In particular for p = Δ 2, our algorithm has the same running time as the algorithm of Luby but uses a much smaller number of processors ( Δ 2 · (m + n) log n instead of m 3 Δ).

An efficient parallel algorithm for the minimal elimination ordering (MEO) of an arbitrary graph

We design the first efficient parallel algorithm for computing the minimal elimination ordering (MEO) of an arbitrary graph. The algorithm works in O(log 3 n) parallel time and O( nm) processors on CREW PRAM, for an n-vertex, m-edge graph, and is optimal up to a polylogarithmic factor with respect to the best sequential algorithm of Rose, et al. (1976). The MEO problem for arbitrary graphs arises in a number of combinatorial optimization problems, as well as in database applications, scheduling problems, and the sparse Gaussian elimination on symmetric matrices. It was believed before to be inherently sequential, and strongly resisting sublinear parallel time (sublinear sequential storage) algorithms. As an application, this paper gives the first efficient parallel solutions to the problem of minimal fill-in for arbitrary graphs and connected combinatorial optimization problems (see e.g. Rose et al., 1976; Tarjan, 1985), and to the problem of the Gaussian elimination of sparse symmetric matrices (Rose, 1970; 1973) (The problem of computing a minimum fill-in is known to be NP-complete (Yannakakis, 1981). The method of solution involves a development of new techniques for solving connected minimal set system problem, and combining it with some new divide-and-conquer methods.