Project-Mascotte:Algorithms

Inria / Raweb 2009
Presentation of the Project Mascotte

Section: New Results

Algorithms

Participants : Nathann Cohen, Afonso Ferreira, Florian Huc, Gianpiero Monaco, Nicolas Nisse, Stéphane Pérennes, J-P. Perez Seva, Bruce Reed, Hervé Rivano, Ignasi Sau-Valls.

Mascotte is also interested in the algorithmic aspects of Graph Theory. In general we try to find the most efficient algorithms to solve various problems of Graph Theory and telecommunication networks either with exact algorithms or approximation ones. We are mainly focused on four important topics:

The routing problems plays an essential role in communication networks. It involves how to transfer data from some origins to some destinations within a reasonable amount of time. In Mascotte , we consider this problem in centralized and distributed environments. We also consider the routing problem in particular graph classes and in general graphs.
Graph searching gathers an active research community (see, e.g., the first three editions of the "Workshop on Graph Searching, Theory and Applications" took place in Crete (2006), Brazil (2008) and Czech Republic (2009)). In particular, the graph searching problem has been widely studied for its close relationship with graph decompositions (see Section 6.5 ). Note that this problem has also a practical impact in the area of optical network reconfiguration which is dealt with in Section 6.1 .
The field of algorithmic game theory combines computer science concepts of complexity and algorithm design with game theory and economic theory. Algorithmic game theory is considered as the most powerful tool dealing with non-cooperative systems in which the lack of coordination among the players produces inefficient solutions in the optimization of systems requirements. There are a lot of situations occurring in real life in which we seek the maximization of our own benefit and very often the final outcome of our efforts also depends on the behavior of other people we have no control on. In Mascotte , we are interested in considering communication problems arising in networks with non-cooperative users.
Parameterized complexity is a recent approach to deal with intractable computational problems having some parameters that can be relatively small with respect to the input size. This area has been developed extensively during the last decade. More precisely, a parameter P is any function mapping graphs to non-negative integers. The parameterized problem associated with parameter P asks, for some fixed k , whether P(G) $\ge$ k for a given graph G . For decision problems with input size n and parameter k , the goal is to design an algorithm with running time f(k).n , where f depends only on k . Problems having such an algorithm are said to be fixed-parameter tractable (FPT).

Routing

We consider the routing problem through three different approaches: compact routing, the ( $\ell$ , k) -routing problem and the disjoint paths problem. First two studies consider specific classes of graphs while the third one considers general graphs.

Compact routing:

In any distributed communication network it is important to deliver messages between pairs of nodes. Routing schemes consist in the design of a routing table in each node (node's local memory where the routing information is stored) together with a protocol that allow each node to decide toward which port (incident edge) it must transmit a message knowing the message's destination. Compact routing consider the tradeoffs between the length of the computed routes and the size of routing tables (the local knowledge of each node about the network's topology). Antoher issue of interest is the computation of routing table.

In general, it is difficult to establish good tradeoffs between the length of computed routes and the size of the routing tables. Efficient algorithms for computing routing tables should take advantage of the particular properties arising in large scale networks. We consider two properties: low (logarithmic) diameter and high clustering coefficient (implying the existence of few large induced cycles). We propose a routing scheme that computes short routes in the class of k-chordal graphs, i.e., graphs with no chordless cycles of length more than k [81] . This algorithm has been implemented using DRMSim 6.4 .

( $\ell$ , k) -routing problem:

In the ( $\ell$ , k) -routing problem, each node can send at most $\ell$ packets and receive at most k packets. In this setting, the goal is to minimize the number of time steps required to route all packets to their respective destinations, under the constraint that each link can be crossed simultaneously by no more than one packet. Permutation routing is the particular case $\ell$ = k = 1 . In the r -central routing problem, all nodes at distance at most r from a fixed node v want to send a packet to v . In [37] , we study the permutation routing, the r -central routing and the general ( $\ell$ , k) -routing problems on plane grids, that is square grids, triangular grids and hexagonal grids. We use the store-and-forward $\upper_delta$ -port model, and we consider both full and half-duplex networks. The main contributions are the following: (1) tight permutation routing algorithms on full-duplex hexagonal grids, and half duplex triangular and hexagonal grids, (2) tight r -central routing algorithms on triangular and hexagonal grids, (3) tight (k, k) -routing algorithms on square, triangular and hexagonal grids, and (4) good approximation algorithms (in terms of running time) for ( $\ell$ , k) -routing on square, triangular and hexagonal grids, together with new lower bounds on the running time of any algorithm using shortest path routing. All these algorithms are completely distributed, i.e. can be implemented independently at each node. Finally, we also formulate the ( $\ell$ , k) -routing problem as a Weighted Edge Coloring problem on bipartite graphs. We provide a survey on above problems in [93] .

Disjoint paths problems:

Given a number of requests (pair of vertices), the disjoint paths problem asks whether there exist pairwise disjoint paths to be assigned to all requests. We investigate several variants of this widely studied problem.

In [39] , we propose a polynomial-time algorithm that, given $\ell$ requests, finds $\ell$ disjoint paths in a symmetric directed graph. It is known that the problem of finding $\ell$ $\ge$ 2 disjoint paths in a directed graph is NP-hard [S. Fortune, J. Hopcroft, J. Wyllie, Journal of Theoretical Computer Science 10 (2) (1980) 111-121]. However, by studying minimal solutions it turns out that only a finite number of configurations are possible in a symmetric digraph. We use Robertson and Seymour's polynomial-time algorithm [N. Robertson, P. D. Seymour, Graph minors xiii, Journal of Combinatorial Theory B (63) (1995) 65-110] to check the feasibility of each configuration.

A graph G is k -linked if G has at least 2k vertices $Im16 ${{(x_1,y_1)},\#8943 ,{(x_k,y_k)}}$$ , such that G contains k pairwise disjoint paths between x_i and y_i (i = 1 to k ). We say that G is parity-k -linked if G is k -linked and, in addition, the paths can be chosen such that the parities of their length are prescribed. Thomassen was the first to prove the existence of a function f(k) such that every f(k) -connected graph is parity-k -linked if the deletion of any 4k-3 vertices leaves a nonbipartite graph. In [41] , we show that the above statement is still valid for 50k -connected graphs. This is the first result that connectivity which is a linear function of k guarantees the Erdös-Pósa type result for parity-k -linked graphs. In [72] , we consider a similar problem where each vertex is on at most two of these paths. We present an O(m $\alpha$ (m, n)logn) algorithm for fixed k , where n , m are the number of vertices and the number of edges, respectively, and the function $\alpha$ (m, n) is the inverse of the Ackermann function. This is the first polynomial time algorithm for this problem, and generalizes polynomial time algorithms by Kleinberg, and Kawarabayashi and Reed, respectively, for the half integral disjoint paths packing problem, i.e., without the parity requirement. We also have algorithms running in O(m(1 + $\epsilon$ )) time for any $\epsilon$ >0 for this problem, if k is up to o(logloglogn) for general graphs, up to o(loglogn ) for planar graphs, and up to o(loglogn/g) for graphs on the surface, where g is Euler genus. Furthermore, if k is fixed, then we have linear time algorithms for the planar case and for the bounded genus case.

Graph Searching

Graph Searching encompasses a wide variety of combinatorial problems related to the capture of an arbitrary fast fugitive residing in a network by a team of searchers. The goal consists in minimizing the number of searchers required to capture the fugitive in a network and in computing the corresponding capture strategy. We mainly investigate three variants of graph searching: visible graph searching, connected graph searching and distributed graph searching. The first study is motivated by the relationship between graph searching and graph decompositions, while the main motivation for both other studies is the design of distributed protocols allowing searchers to compute a capture strategy.

Roughly, if the fugitive is visible (i.e., the searchers are permanently aware of the position of the fugitive) then graph searching is equivalent to treewidth, while it is equivalent to pathwidth otherwise. In [30] , we introduce non-deterministic graph searching where the fugitive is visible a limited number of steps. We prove the NP-hardness of this problem and design an exponential exact algorithm for solving it. This new variant leads to the unified view of graph decompositions in terms of partition functions and partitioning-trees [23] .

A strategy is called connected if the clear part (the part where the fugitive cannot stand) always induces a connected subgraph. In particular, when the strategy has to be computed online, this property ensures safe communications between the searchers during the whole strategy. In [44] , we investigate the cost of the connectedness of a strategy. We design an algorithm that computes a connected capture strategy using at most O(tw(G)*k) times the search number of G , in any k -chordal graph G with treewidth tw(G) .

We then propose a polynomial-time distributed algorithm for clearing any network using the optimal number of searchers assuming that the searchers have some knowledge about the network they are clearing. More precisely, we prove that the amount of information necessary to clear any n -node network in a monotone distributed way is $\upper_theta$ (nlogn) bits [45] . When the network is unknown a priori, we propose a polynomial-time distributed algorithm for clearing any n -node network using $Im17 ${O(\mfrac n{logn})}$$ times the optimal number of searchers and we prove this is optimal [38] .

Algorithmic Game Theory

In highly distributed systems, it might be too strong or unrealistic to assume that the resources of the system are directly accessible and controllable by a centralized authority. Therefore, we consider communication problems arising in networks with autonomous or non-cooperative users. In such a scenario, users pursue an own often selfish strategy and the system evolves as a consequence of the interactions among them. The interesting arising scenario is thus characterized by the conflicting needs of the users aiming to maximize their personal profit and of the system wishing to compute a socially efficient solution.

The uncoordinated users' behavior, addressing communication primitives in an individualistic and selfish manner, poses several intriguing questions ranging from the definition of reasonable and practical models, to the quantification of the efficiency loss due to the lack of users' cooperation. We survey several results lately achievied in this research area and propose interesting future research directions [91] .

We consider the pure Nash equilibrium as the outcome of the game and in turn as the concept capturing the notion of stable solution of the system. We make different progresses on the understanding of a variety of problems in communication networks. We study the performances of Nash equilibria in isolation games, a class of competitive location games recently introduced by Zhao et al. For all the cases in which the existence of Nash equilibria has been shown, we give tight or asymptotically tight bounds on the prices of anarchy and stability under the two classical social functions mostly investigated in the scientific literature, namely, the minimum utility per player and the sum of the players' utilities. Moreover, we prove that the convergence to Nash equilibria is not guaranteed in some of the not yet analyzed cases [53] .

Parameterized complexity

We design FPT-algorithms for four NP-complete problems: the Bounded-Degree Connected Sub-graph Problems on Planar Graphs, the Bounded leaves Sub-tree Problem, the Fractional Path Coloring Problem and the Spanning Galaxies Problem.

In [18] , [83] , we present subexponential parameterized algorithms on planar graphs for a family of problems that consist in, given a graph G , finding a connected subgraph H with bounded maximum degree, while maximizing the number of edges (or vertices) of H . These problems are natural generalisations of the Longest Path problem. Our approach uses bidimensionality theory to obtain combinatorial bounds, combined with dynamic programming techniques over a branch decomposition of the input graph. These techniques need to be able to keep track of the connected components of the partial solutions over the branch decomposition, and can be seen as an algorithmic tensor that can be applied to a wide family of problems that deal with finding connected subgraphs under certain constraints.

An out-tree T is an oriented tree with exactly one vertex of in-degree zero and a vertex x of T is called internal if its out-degree is positive. In [55] , we design randomized and deterministic algorithms for deciding whether an input digraph contains a subgraph isomorphic to a given out-tree with k vertices. Both algorithms run in O^*(5.704^k) time. We apply the deterministic algorithm to obtain an algorithm of runtime O^*(c^k) , where c is a constant, for deciding whether an input digraph contains a spanning out-tree with at least k internal vertices. This answers in the affirmative a question of Gutin, Razgon and Kim (Proc. AAIM'08).

In [27] , we study the natural linear programming relaxation of the path coloring problem. We prove constructively that finding an optimal fractional path coloring is FPT with the degree of the tree as parameter: the fractional coloring of paths in a bounded degree trees can be done in a time which is linear in the size of the tree, quadratic in the load of the set of paths, while exponential in the degree of the tree. We give an algorithm based on the generation of an efficient polynomial size linear program. Our algorithm is able to explore in polynomial time the exponential number of different fractional colorings, thanks to the notion of trace of a coloring that we introduce. We further give an upper bound on the cost of such a coloring in binary trees and extend this algorithm to bounded degree graphs with bounded treewidth. Finally, we also show some relationships between the integral and fractional problems, and derive a (1 + 5/3e) -approximation algorithm for the path coloring problem in bounded degree trees, improving on existing results. This classic combinatorial problem finds applications in the minimization of the number of wavelengths in wavelength division multiplexing (WDM) optical networks. In [74] , we describe a linear time algorithm for fractionally edge colouring simple graphs with maximum degree at least |V|/c (c>1 ).

In [69] , we prove that the parametrized version of the Spanning Galaxies Problem (see Section 6.5 ) has a linear kernel.

Miscellaneous

In [78] Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This class of graphs, which generalizes in a natural way both interval and permutation graphs, has attracted many research efforts since their introduction, as it finds many important applications in constraint-based temporal reasoning, resource allocation and scheduling problems, among others. In this article we propose the first non-trivial intersection model for general tolerance graphs, given by three-dimensional parallelepipeds, which extends the widely known intersection model of parallelograms in the plane that characterizes the class of bounded tolerance graphs. Apart from being important on its own, this new representation also enables us to improve the time complexity of three problems on tolerance graphs. Namely, we present optimal O(nlogn) algorithms for computing a minimum coloring and a maximum clique, and an O(n²) algorithm for computing a maximum weight independent set in a tolerance graph with n vertices, thus improving the best known running times O(n²) and O(n³) for these problems, respectively.

In [42] , we consider the following problem in a n -node graph G = (V, E) . Place m = n points on the vertices of G independently and uniformly at random. Once the points are placed, relocate them using a bijection from the points to the vertices that minimizes the maximum distance between the random place of the points and their target vertices. We look for an upper bound on this maximum relocation distance that holds with high probability (over the initial placements of the points). For general graphs and in the case m $\le$ n , we prove the #P -hardness of the problem and that the maximum relocation distance is $Im18 ${O(\sqrt n)}$$ with high probability. We present a Fully Polynomial Randomized Approximation Scheme when the input graph admits a polynomial-size family of witness cuts while for trees we provide a 2-approximation algorithm. Many applications concern the variation in which m = (1-q)n for some 0<q<1 . We provide several bounds for the maximum relocation distance according to different graph topologies.

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