Section: New Results

Graph and Combinatorial Algorithms

New Results in Multi-sweep Graph Search

A theoretical model to describe a series of successive graph searches is proposed in [7]. We apply this model to deal with cocomparability graphs (i.e., complement of comparability graphs) in [6] and in [48] or [44]. In this series of papers we provide a general algorithmic framework for many optimization problems on cocomparability graphs, such as Minimum Path Cover, Maximum Independent Set, Maximal interval subgraph, etc.

We also provide a new very simple algorithm for the recognition of cocomparability graphs. This algorithm is also based on a series of successive graph searches in [13].

We mainly use the two well-known Lexicographic graph searches: LBFS and LDFS, but not only. In [48], we also introduced a new graph search LocalMNS which seems to behave nicely on cocomparability graphs.

Studies of Read Networks and Laminar Graphs

In the context of biological networks, in [50] we introduce $k$-laminar graphs — a new class of graphs which extends the idea of Asteroidal-triple-free graphs. A graph is $k$-laminar if it admits a diametral path that is $k$-dominating. This bio-inspired class of graphs was motivated by a biological application dealing with sequence similarity networks of reads. We briefly develop the context of the biological application in which this graph class appeared and then we consider the relationships of this new graph class among known graph classes and then we study its recognition problem. For the recognition of $k$-laminar graphs, we develop polynomial algorithms when k is fixed. For $k=1$, our algorithm improves a Deogun and Krastch's algorithm (1999). We finish by an NP-completeness result when $k$ is unbounded.

Further Studies into Shortest Paths, Eccentricity, and Laminarity

From our recent research on diameter computations on graphs we also investigated some reductions between polynomial problems on graphs [3].

We also extend the well-known multisweep BFS to give a better polynomial-time approximation for the Maximum Eccentricity Shortest Path Problem, in relation with the k-Laminarity Problem [20].

Clique Colourings of Perfect Graphs

A clique-coloring of a graph $G$ is an assignment of colors to the vertices of $G$ in such a way that no inclusion-wise maximal clique of size at least two of $G$ is monochromatic (as usual, a set of vertices is monochromatic if all vertices in the set received the same color). The clique-chromatic number of $G$, denoted by ${\chi }_C\left(G\right)$, is the smallest integer $k$ such that $G$ is admits a clique-coloring using at most $k$ colors. Note that every proper coloring of $G$ is also a clique-coloring of $G$, and so ${\chi }_C\left(G\right)\le \chi \left(G\right)$. Furthermore, if $G$ is triangle-free, then ${\chi }_C\left(G\right)=\chi \left(G\right)$ (since there are triangle-free graphs of arbitrarily large chromatic number, this implies that there are triangle-free graphs of arbitrarily large clique-chromatic number). However, if $G$ contains triangles, ${\chi }_C\left(G\right)$ may be much smaller than $\chi \left(G\right)$. For instance, if $G$ contains a dominating vertex, then ${\chi }_C\left(G\right)\le 2$ (we assign the color 1 to the dominating vertex and the color 2 to all other vertices of $G$), while $\chi \left(G\right)$ may be arbitrarily large. Note that this implies that the clique-chromatic number is not monotone with respect to induced subgraphs, that is, there exist graphs $H$ and $G$ such that $H$ is an induced subgraph of $G$, but ${\chi }_C\left(H\right)>{\chi }_C\left(G\right)$. (In particular, the restriction of a clique-coloring of $G$ to an induced subgraph $H$ of $G$ need not be a clique-coloring of $H$.)

A graph $G$ is perfect if all its induced subgraphs $H$ satisfy $\chi \left(H\right)=\omega \left(H\right)$, where $\omega \left(H\right)$ denotes the size of a maximum clique. It was asked by Duffus, Sands, Sauer, and Woodrow in a paper from 1991 whether perfect graphs have a bounded clique-chromatic number and indeed it has been proven since that for many sublasses of the class of perfect graphs, this holds. Even more, until now it was not known whether there were any perfect graphs of clique-chromatic number greater than three. The main result of [4] is to prove that there exist perfect graphs of arbitrarily large clique-chromatic number, which gives a negative answer for the question of Duffus et al. mentioned above.

Algorithmic Aspects of Switch Cographs

Cographs are the graphs totally decomposable using series and parallel operations, in [5] we introduced an interesting generalization, namely the class of switch cographs. These are the class of graphs that are totally decomposable w.r.t involution modular decomposition — a generalization of the modular decomposition of 2-structure, which has a unique linear-sized decomposition tree. We use our new decomposition tool to design three practical algorithms for the maximum cut, vertex cover and vertex separator problems. The complexity of these problems was previously unknown for this class of graphs.

Shrinking Maxima, Decreasing Costs: New Online Packing and Covering Problems

In [16], we consider two new variants of online integer programs that are duals. In the packing problem we are given a set of items and a collection of knapsack constraints over these items that are revealed over time in an online fashion. Upon arrival of a constraint we may need to remove several items (irrevocably) so as to maintain feasibility of the solution. Hence, the set of packed items becomes smaller over time. The goal is to maximize the number, or value, of packed items. The problem originates from a buffer-overflow model in communication networks, where items represent information units broken into multiple packets. The other problem considered is online covering: there is a universe to be covered. Sets arrive online, and we must decide for each set whether we add it to the cover or give it up. The cost of a solution is the total cost of sets taken, plus a penalty for each uncovered element. The number of sets in the solution grows over time, but its cost goes down. This problem is motivated by team formation, where the universe consists of skills, and sets represent candidates we may hire. The packing problem was introduced in Emek et al. (SIAM J Comput 41(4):728-746, 2012) for the special case where the matrix is binary; in this paper we extend the solution to general matrices with non-negative integer entries. The covering problem is introduced in this paper; we present matching upper and lower bounds on its competitive ratio.

The Complexity of the Shortest-path Broadcast Problem

In [8], we study the shortest-path broadcast problem in graphs and digraphs, where a message has to be transmitted from a source node $s$ to all the nodes along shortest paths, in the classical telephone model. For both graphs and digraphs, we show that the problem is equivalent to the broadcast problem in layered directed graphs. We then prove that this latter problem is NP-hard, and therefore that the shortest-path broadcast problem is NP-hard in graphs as well as in digraphs. Nevertheless, we prove that a simple polynomial-time algorithm, called MDST-broadcast, based on min-degree spanning trees, approximates the optimal broadcast time within a multiplicative factor $3/2$ in 3-layer digraphs, and $O(logn/loglogn)$ in arbitrary multi-layer digraphs. As a consequence, one can approximate the optimal shortest-path broadcast time in polynomial time within a multiplicative factor $3/2$ whenever the source has eccentricity at most 2, and within a multiplicative factor $O(logn/loglogn)$ in the general case, for both graphs and digraphs. The analysis of MDST-broadcast is tight, as we prove that this algorithm cannot approximate the optimal broadcast time within a factor smaller than $\Omega (logn/loglogn)$.

Setting Ports in an Anonymous Network: How to Reduce the Level of Symmetry

A fundamental question in the setting of anonymous graphs concerns the ability of nodes to spontaneously break symmetries, based on their local perception of the network. In contrast to previous work, which focuses on symmetry breaking under arbitrary port labelings, in [37] we study the following design question: Given an anonymous graph $G$ without port labels, how to assign labels to the ports of $G$, in interval form at each vertex, so that symmetry breaking can be achieved using a message-passing protocol requiring as few rounds of synchronous communication as possible?

More formally, for an integer $l>0$, the truncated view ${𝒱}_l\left(v\right)$ of a node $v$ of a port-labeled graph is defined as a tree encoding labels encountered along all walks in the network which originate from node $v$ and have length at most $l$, and we ask about an assignment of labels to the ports of $G$ so that the views ${𝒱}_l\left(v\right)$ are distinct for all nodes $v\in V$, with the goal being to minimize $l$.

We present such efficient port labelings for any graph $G$, and we exhibit examples of graphs showing that the derived bounds are asymptotically optimal in general. More precisely, our results imply the following statements.

1. For any graph $G$ with $n$ nodes and diameter $D$, a uniformly random port labeling achieves $l=O(min(D,logn\left)\right)$, w.h.p.

2. For any graph $G$ with $n$ nodes and diameter $D$, it is possible to construct in polynomial time a labeling that satisfies $l=O(min(D,logn\left)\right)$.

3. For any integers $n\ge 2$ and $D\le {log}_{2}n-{log}_2{log}_{2}n$, there exists a graph $G$ with $n$ nodes and diameter $D$ which satisfies $l\ge \frac{1}{2}D-\frac{5}{2}$.

Robustness of the Rotor-Router Mechanism

The rotor-router model, also called the Propp machine, was first considered as a deterministic alternative to the random walk. The edges adjacent to each node $v$ (or equivalently, the exit ports at $v$) are arranged in a fixed cyclic order, which does not change during the exploration. Each node $v$ maintains a port pointer ${\pi }_v$ which indicates the exit port to be adopted by an agent on the conclusion of the next visit to this node (the “next exit port”). The rotor-router mechanism guarantees that after each consecutive visit at the same node, the pointer at this node is moved to the next port in the cyclic order. It is known that, in an undirected graph $G$ with $m$ edges, the route adopted by an agent controlled by the rotor-router mechanism forms eventually an Euler tour based on arcs obtained via replacing each edge in $G$ by two arcs with opposite direction. The process of ushering the agent to an Euler tour is referred to as the lock-in problem. In [Yanovski et al., Algorithmica 37(3), 165–186 (2003)], it was proved that, independently of the initial configuration of the rotor-router mechanism in $G$, the agent locks-in in time bounded by $2mD$, where $D$ is the diameter of $G$.

In [2], we examine the dependence of the lock-in time on the initial configuration of the rotor-router mechanism. Our analysis is performed in the form of a game between a player $p$ intending to lock-in the agent in an Euler tour as quickly as possible and its adversary $a$ with the counter objective. We consider all cases of who decides the initial cyclic orders and the initial values ${\pi }_v$. We show, for example, that if $a$ provides its own port numbering after the initial setup of pointers by $p$, the complexity of the lock-in problem is $O(m·min\{logm,D\left\}\right)$.

We also investigate the robustness of the rotor-router graph exploration in presence of faults in the pointers ${\pi }_v$ or dynamic changes in the graph. We show, for example, that after the exploration establishes an Eulerian cycle, if $k$ edges are added to the graph, then a new Eulerian cycle is established within $O\left(km\right)$ steps.

The Multi-Agent Rotor-Router on the Ring: A Deterministic Alternative to Parallel Random Walks

Continuing the line of research on the rotor-router model, in [18] we consider the setting in which multiple, indistinguishable agents are deployed in parallel in the nodes of the graph, and move around the graph in synchronous rounds, interacting with a single rotor-router system. We propose new techniques which allow us to perform a theoretical analysis of the multi-agent rotor-router model, and to compare it to the scenario of parallel independent random walks in a graph. Our main results concern the $n$-node ring, and suggest a strong similarity between the performance characteristics of this deterministic model and random walks.

We show that on the ring the rotor-router with $k$ agents admits a cover time of between $\Theta (n^2/k^2)$ in the best case and $\Theta (n^2/logk)$ in the worst case, depending on the initial locations of the agents, and that both these bounds are tight. The corresponding expected value of the cover time for $k$ random walks, depending on the initial locations of the walkers, is proven to belong to a similar range, namely between $\Theta (n^2/(k^2/{log}^{2}k))$ and $\Theta (n^2/logk)$.

Finally, we study the limit behavior of the rotor-router system. We show that, once the rotor-router system has stabilized, all the nodes of the ring are always visited by some agent every $\Theta (n/k)$ steps, regardless of how the system was initialized. This asymptotic bound corresponds to the expected time between successive visits to a node in the case of $k$ random walks. All our results hold up to a polynomially large number of agents ($1\le k<n^{1/11}$).

Bounds on the Cover Time of Parallel Rotor Walks

In [12], we study the parallel rotor-router model in the case of general graphs. We consider the cover time of such a system, i.e., the number of steps after which each node has been visited by at least one walk, regardless of the initialization of the walks. We show that for any graph with $m$ edges and diameter $D$, this cover time is at most $\Theta (mD/logk)$ and at least $\Theta (mD/k)$, which corresponds to a speedup of between $\Theta (logk)$ and $\Theta \left(k\right)$ with respect to the cover time of a single walk.