Irith Ben-Arroyo Hartman

Theory and Practice in Large Carpooling Problems

Abstract We address the carpooling problem as a graph-theoretic problem. If the set of drivers is known in advance, then for any car capacity, the problem is equivalent to the assignment problem in bipartite graphs. Otherwise, when we do not know in advance who will drive their vehicle and who will be a passenger, the problem is NP-hard. We devise and implement quick heuristics for both cases, based on graph algorithms, as well as parallel algorithms based on geometric/algebraic approach. We compare between the algorithms on random graphs, as well as on real, very large, data.

Long Cycles Generate the Cycle Space of a Graph

Abstract Let G be a 2-connected graph in which the degree of every vertex is at least d. We prove that the cycles of length at least d + 1 generate the cycle space of G, unless G ≌ Kd+1 and d is odd. As a corollary, we deduce that the cycles of length at least d + 1 generate the subspace of even cycles in G. We also establish the existence of odd cycles of length at least d + 1 in the case when G is not bipartite. A second result states: if G is 2-connected with chromatic number at least k, then the cycles of length at least k generate the cycle space of G, unless G ≌ Kk and k is even. Similar corollaries follow, among them a stronger version of a theorem of Erdos and Hajnal.

Scalability issues in optimal assignment for carpooling

Carpooling for commuting can save cost and helps in reducing pollution. An automatic Web based Global CarPooling Matching Service (GCPMS) for matching commuting trips has been designed. The service supports carpooling candidates by supplying advice during their exploration for potential partners. Such services collect data about the candidates, and base their advice for each pair of trips to be combined, on an estimate of the probability for successful negotiation between the candidates to carpool. The probability values are calculated by a learning mechanism using, on one hand, the registered person and trip characteristics, and on the other hand, the negotiation feedback. The problem of maximizing the expected value of carpooling negotiation success was formulated and was proved to be NP-hard. In addition, the network characteristics for a realistic case have been analyzed. The carpooling network was established using results predicted by the operational FEATHERS activity based model for Flanders (Belgium).

Berge's conjecture on directed path partitions-a survey

Berges conjecture from 1982 on path partitions in directed graphs generalizes and extends Dilworths theorem and the Greene-Kleitman theorem which are well known for partially ordered sets. The conjecture relates path partitions to a collection of k independent sets, for each k>=1. The conjecture is still open and intriguing for all k>1. In this paper, we will survey partial results on the conjecture, look into different proof techniques for these results, and relate the conjecture to other theorems, conjectures and open problems of Berge and other mathematicians.

On Greene-Kleitman's theorem for general digraphs

Abstract Linial conjectured that Greene—Kleitmans theorem can be extended to general digraphs. We prove a stronger conjecture of Berge for digraphs having k-optimal path partitions consisting of ‘long’ paths. The same method yields known results for acyclic digraphs, and extensions of various theorems of Greene and Frank to acyclic digraphs.

On greene's theorem for digraphs

Greenes Theorem states that the maximum cardinality of an optimal k-path in a poset is equal to the minimum k-norm of a k-optimal coloring. This result was extended to all acyclic digraphs, and is conjectured to hold for general digraphs. We prove the result for general digraphs in which an optimal k-path contains a path of cardinality one. This implies the validity of the conjecture for all bipartite digraphs. We also extend Greenes Theorem to all split graphs.

Group planning with time constraints

Embedding planning systems in real-world domains has led to the necessity of Distributed Continual Planning (DCP) systems where planning activities are distributed across multiple agents and plan generation may occur concurrently with plan execution. A key challenge in DCP systems is how to coordinate activities for a group of planning agents. This problem is compounded when these agents are situated in a real-world dynamic domain where the agents often encounter differing, incomplete, and possibly inconsistent views of their environment. To date, DCP systems have only focused on cases where agents’ behavior is designed to optimize a global plan. In contrast, this paper presents a temporal reasoning mechanism for self-interested planning agents. To do so, we model agents’ behavior based on the Belief-Desire-Intention (BDI) theoretical model of cooperation, while modeling dynamic joint plans with group time constraints through creating hierarchical abstraction plans integrated with temporal constraints network. The contribution of this paper is threefold: (i) the BDI model specifies a behavior for self interested agents working in a group, permitting an individual agent to schedule its activities in an autonomous fashion, while taking into consideration temporal constraints of its group members; (ii) abstract plans allow the group to plan a joint action without explicitly describing all possible states in advance, making it possible to reduce the number of states which need to be considered in a BDI-based approach; and (iii) a temporal constraints network enables each agent to reason by itself about the best time for scheduling activities, making it possible to reduce coordination messages among a group. The mechanism ensures temporal consistency of a cooperative plan, enables the interleaving of planning and execution at both individual and group levels. We report on how the mechanism was implemented within a commercial training and simulation application, and present empirical evidence of its effectiveness in real-life scenarios and in reducing communication to coordinate group members’ activities.

Optimal k -colouring and k -nesting of intervals

We describe and solve two problems motivated from routing in CMOS cells layed out in the style of one-dimensional transistor arrays, as well as from channel routing. In the first problem we find an optimal subset of intervals to be layed out on k tracks, for any given k. In the second problem we find an optimal set of nested intervals, to be layed out on any given number of tracks. Both solutions are polynomial time, and have applications in many routing problems.

Enumerating minimum path decompositions to support route choice set generation

Abstract: This paper concerns the structure of movements as were recorded by GPS traces and converted to routes by map matching. Each route in a transportation network corresponds to a collection of directed paths or cycles in a digraph. When considering only directed paths, corresponding to utilitarian trips, the path is not necessarily a shortest path between its origin and destination, and can be split up into a small number of segments, each of which is a shortest or least cost path. Two consecutive segments are separated by split vertices . Split vertices act as intermediate destinations in the mind of travellers who try to hop between them using minimum cost paths. Hence they provide useful information to build route choice models. In this paper we identify and enumerate all possible decompositions of a path into a minimum number of shortest segments. This gives us an indication of the importance of split vertices occurring in particular sets of revealed routes that belong either to a single traveller or to a specific group. The proposed technique allows for automatic extraction of frequently used intermediate destinations (way-points) from revealed preference data.

A unified approach to known and unknown cases of Berge's conjecture

Berges elegant dipath partition conjecture from 1982 states that in a dipath partition P of the vertex set of a digraph minimizing , there exists a collection Ck of k disjoint independent sets, where each dipath P∈P meets exactly min{|P|, k} of the independent sets in C. This conjecture extends Linials conjecture, the Greene–Kleitman Theorem and Dilworths Theorem for all digraphs. The conjecture is known to be true for acyclic digraphs. For general digraphs, it is known for k=1 by the Gallai–Milgram Theorem, for k⩾λ (where λis the number of vertices in the longest dipath in the graph), by the Gallai–Roy Theorem, and when the optimal path partition P contains only dipaths P with |P|⩾k. Recently, it was proved (Eur J Combin (2007)) for k=2. There was no proof that covers all the known cases of Berges conjecture. In this article, we give an algorithmic proof of a stronger version of the conjecture for acyclic digraphs, using network flows, which covers all the known cases, except the case k=2, and the new, unknown case, of k=λ−1 for all digraphs. So far, there has been no proof that unified all these cases. This proof gives hope for finding a proof for all k.