When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks

Let $G=(V,E)$ be a supply graph and $H=(V,F)$ a demand graph defined on the same set of vertices. An assignment of capacities to the edges of $G$ and demands to the edges of $H$ is said to satisfy the \emph{cut condition} if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair $(G,H)$ is called \emph{cut-sufficient} if for any assignment of capacities and demands that satisfy the cut condition, there is a multiflow routing the demands defined on $H$ within the network with capacities defined on $G$. We prove a previous conjecture, which states that when the supply graph $G$ is series-parallel, the pair $(G,H)$ is cut-sufficient if and only if $(G,H)$ does not contain an \emph{odd spindle} as a minor; that is, if it is impossible to contract edges of $G$ and delete edges of $G$ and $H$ so that $G$ becomes the complete bipartite graph $K_{2,p}$, with $p\geq 3$ odd, and $H$ is composed of a cycle connecting the $p$ vertices of degree 2, and an edge connecting the two vertices of degree $p$. We further prove that if the instance is \emph{Eulerian} --- that is, the demands and capacities are integers and the total of demands and capacities incident to each vertex is even --- then the multiflow problem has an integral solution. We provide a polynomial-time algorithm to find an integral solution in this case. In order to prove these results, we formulate properties of tight cuts (cuts for which the cut condition inequality is tight) in cut-sufficient pairs. We believe these properties might be useful in extending our results to planar graphs.Comment: An extended abstract of this paper will be published at the 44th Symposium on Theory of Computing (STOC 2012

Routing with Guaranteed Delivery on Virtual Coordinates

We propose four simple algorithms for routing on planar graphs using virtual coordinates. These algorithms are superior to existing algorithms in that they are oblivious, work also for non-triangular graphs, and their virtual coordinates are easy to construct.Engineering and Applied Science

Walking Through Waypoints

We initiate the study of a fundamental combinatorial problem: Given a capacitated graph $G=(V,E)$, find a shortest walk ("route") from a source $s\in V$ to a destination $t\in V$ that includes all vertices specified by a set $\mathscr{W}\subseteq V$: the \emph{waypoints}. This waypoint routing problem finds immediate applications in the context of modern networked distributed systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial-time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable

A Quasi-Polynomial-Time Approximation Scheme for Vehicle Routing on Planar and Bounded-Genus Graphs

The Capacitated Vehicle Routing problem is a generalization of the Traveling Salesman problem in which a set of clients must be visited by a collection of capacitated tours. Each tour can visit at most Q clients and must start and end at a specified depot. We present the first approximation scheme for Capacitated Vehicle Routing for non-Euclidean metrics. Specifically we give a quasi-polynomial-time approximation scheme for Capacitated Vehicle Routing with fixed capacities on planar graphs. We also show how this result can be extended to bounded-genus graphs and polylogarithmic capacities, as well as to variations of the problem that include multiple depots and charging penalties for unvisited clients

MFACE: A Multicast Backbone-Assisted Face Traversal Algorithm for Arbitrary Planar Ad Hoc and Sensor Network Topologies

Face is a well-known localized routing protocol for ad hoc and sensor networks which guarantees delivery of the message as long as a path exists between the source and the destination. This is achieved by employing a left/right hand rule to route the message along the faces of a planar topology. Although face was developed for the unicast case, it has recently been used in combination with multicasting protocols, where there are multiple destinations. Some of the proposed solutions handle each destination separately and lead thus to increased energy consumption. Extensions of face recovery to the multicast case described so far are either limited to certain planar graphs or do not provide delivery guarantees. A recently described scheme employs multicast face recovery based on a so called multicast backbone. A multicast backbone is a Euclidean spanning tree which contains at least the source and the destination nodes. The idea of backbone assisted routing it to follow the edges of the backbone in order to deliver a multicast message to all spanned destination nodes. The existing backbone face routing scheme is however limited to a certain planar graph type and a certain backbone construction. One of the key aspects of the multicast face algorithm MFACE we propose in this work is that it may be applied on top of any planar topology. Moreover, our solution may be used as a generic framework since it is able to work with any arbitrary multicast backbone. In MFACE, any edge of the backbone originated at the source node will generate a new copy of the message which will be routed toward the set of destination nodes spanned by the corresponding edge. Whenever the message arrives at a face edge intersected by a backbone edge different from the initial edge, the message is split into two copies, both handling a disjoint subset of the multicast destinations which are defined by splitting the multicast backbone at that intersection point

NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs

In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in $K_{3,3}$-minor-free graphs by building on Kasteleyn's scheme for planar graphs, and stated that this "opens up the possibility of obtaining an NC algorithm for finding a perfect matching in $K_{3,3}$-free graphs." In this paper, we finally settle this 30-year-old open problem. Building on recent NC algorithms for planar and bounded-genus perfect matching by Anari and Vazirani and later by Sankowski, we obtain NC algorithms for perfect matching in any minor-closed graph family that forbids a one-crossing graph. This family includes several well-studied graph families including the $K_{3,3}$-minor-free graphs and $K_5$-minor-free graphs. Graphs in these families not only have unbounded genus, but can have genus as high as $O(n)$. Our method applies as well to several other problems related to perfect matching. In particular, we obtain NC algorithms for the following problems in any family of graphs (or networks) with a one-crossing forbidden minor: $\bullet$ Determining whether a given graph has a perfect matching and if so, finding one. $\bullet$ Finding a minimum weight perfect matching in the graph, assuming that the edge weights are polynomially bounded. $\bullet$ Finding a maximum $st$-flow in the network, with arbitrary capacities. The main new idea enabling our results is the definition and use of matching-mimicking networks, small replacement networks that behave the same, with respect to matching problems involving a fixed set of terminals, as the larger network they replace.Comment: 21 pages, 6 figure

Engineering Planar-Separator and Shortest-Path Algorithms

"Algorithm engineering" denotes the process of designing, implementing, testing, analyzing, and refining computational proceedings to improve their performance. We consider three graph problems -- planar separation, single-pair shortest-path routing, and multimodal shortest-path routing -- and conduct a systematic study in order to: classify different kinds of input; draw concrete recommendations for choosing the parameters involved; and identify and tune crucial parts of the algorithm

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application