Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms

We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou's network. We improve upon the value 4/3 by means of Coordination Mechanisms. We increase the latency functions of the edges in the network, i.e., if $\ell_e(x)$ is the latency function of an edge $e$, we replace it by $\hat{\ell}_e(x)$ with $\ell_e(x) \le \hat{\ell}_e(x)$ for all $x$. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if \CM(r) denotes the cost of the worst Nash flow in the modified network for rate $r$ and \Copt(r) denotes the cost of the optimal flow in the original network for the same rate then [\ePoA = \max_{r \ge 0} \frac{\CM(r)}{\Copt(r)}.] We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4 = 1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192.Comment: 17 pages, 2 figures, preliminary version appeared at ESA 201

Contention Resolution under Selfishness

In many communications settings, such as wired and wireless local-area networks, when multiple users attempt to access a communication channel at the same time, a conflict results and none of the communications are successful. Contention resolution is the study of distributed transmission and retransmission protocols designed to maximize notions of utility such as channel utilization in the face of blocking communications. An additional issue to be considered in the design of such protocols is that selfish users may have incentive to deviate from the prescribed behavior, if another transmission strategy increases their utility. The work of Fiat et al. (in SODA ’07, pp. 179–188, SIAM, Philadelphia 2007) addresses this issue by constructing an asymptotically optimal incentive-compatible protocol. However, their protocol assumes the cost of any single transmission is zero, and the protocol completely collapses under non-zero transmission costs. In this paper we treat the case of non-zero transmission cost c. We present asymptotically optimal contention resolution protocols that are robust to selfish users, in two different channel feedback models. Our main result is in the Collision Multiplicity Feedback model, where after each time slot, the number of attempted transmissions is returned as feedback to the users. In this setting, we give a protocol that has expected cost Θ(n+clogn) and is in o(1)-equilibrium, where n is the number of users

Symmetry Breaking in Constraint Satisfaction Problems

Abstract. Two methods performing Symmetry Breaking During Search (SBDS) are presented, as described in [1] and [2]. The first was also the the one that defined SBDS, under the name Symmetry Excluding Search, being the first attempt to use it for symmetries of arbitrary type, in contrast to previous attempts that could only handle certain symmetry types. The basic characteristic is that SBDS does not affect the search procedure, meaning that it does not force it to use paths with a certain order. The method was refined later in [2]. While in [1] mathematical proofs were given in order to obtain the final results, in [2], the use of group theory and the known properties of symmetric groups make the method more compact. Moreover, in the first case the authors had to implement their method by writing explicit code for everything, while in the second case, a combination of two systems, ECL i PS e and GAP, is used, which results in formalizing the use of symmetries, making the separation between the search procedure and symmetry breaking more clear. In addition, the method becomes now easier to use, since the programmer no longer needs to keep in mind how symmetries work. GAP provides all the information needed, leaving the programmer with the need only to implement the search procedure, using directly the results for symmetry exclusion given by GAP. Examples and experimental results are provided for both cases, indicating the significant profit in reducing the search space, and thus search time in solving Constraint Satisfaction Problems.

Efficient Models for Timetable Information in Public Transportation Systems

We consider two approaches that model timetable information in public transportation systems as shortest-path problems in weighted graphs. In the time-expanded approach, every event at a station, e.g., the departure of a train, is modeled as a node in the graph, while in the timedependent approach the graph contains only one node per station. Both approaches have been recently considered for (a simplified version of) the earliest arrival problem, but little is known about their relative performance. Thus far, there are only theoretical arguments in favor of the time-dependent approach. In this paper, we provide the first extensive experimental comparison of the two approaches. Using several real-world data sets, we evaluate the performance of the basic models and of several new extensions towards realistic modeling. Furthermore, new insights on solving bicriteria optimization problems in both models are presented. The time-expanded approach turns out to be more robust for modeling more complex scenarios, whereas the time-dependent approach shows a clearly better performance

Two Approaches for Time-Table Information: A Comparison of Models and Performance

We consider two approaches that model timetable information in public transportation systems as shortest-path problems in weighted graphs. In the time-expanded approach every event at a station, e.g., the departure of a train, is modelled as a node in the graph, while in the time-dependent approach the graph contains only one node per station. Train connections without intermediate stops correspond to edges. There is one edge for each single connection in the time-expanded model; in contrast, a couple of trains belong to the same edge in the time-dependent model. Both approaches have been recently considered for the earliest arrival problem. In this paper, we compare, on the one hand, the approaches with respect to more realistic modelling of real-world requirements. On the other hand, we evaluate their performance in an experimental study using real-world data.The time-expanded approach turns out to be more robust for modelling more complex scenarios, whereas the time-dependent approach shows a clearly better performance.As a conclusion the combination of both approaches seems promising