49 research outputs found
Complexity and Approximation of the Continuous Network Design Problem
We revisit a classical problem in transportation, known as the continuous
(bilevel) network design problem, CNDP for short. We are given a graph for
which the latency of each edge depends on the ratio of the edge flow and the
capacity installed. The goal is to find an optimal investment in edge
capacities so as to minimize the sum of the routing cost of the induced Wardrop
equilibrium and the investment cost. While this problem is considered as
challenging in the literature, its complexity status was still unknown. We
close this gap showing that CNDP is strongly NP-complete and APX-hard, both on
directed and undirected networks and even for instances with affine latencies.
As for the approximation of the problem, we first provide a detailed analysis
for a heuristic studied by Marcotte for the special case of monomial latency
functions (Mathematical Programming, Vol.~34, 1986). Specifically, we derive a
closed form expression of its approximation guarantee for arbitrary sets S of
allowed latency functions. Second, we propose a different approximation
algorithm and show that it has the same approximation guarantee. As our final
-- and arguably most interesting -- result regarding approximation, we show
that using the better of the two approximation algorithms results in a strictly
improved approximation guarantee for which we give a closed form expression.
For affine latencies, e.g., this algorithm achieves a 1.195-approximation which
improves on the 5/4 that has been shown before by Marcotte. We finally discuss
the case of hard budget constraints on the capacity investment.Comment: 27 page
The Price of Anarchy for Polynomial Social Cost
In this work, we consider an interesting variant
of the well-studied KP model [KP99] for selfish
routing that reflects some influence from the much
older Wardrop [War52]. In the new model, user
traffics are still unsplittable, while social cost
is now the expectation of the sum, over all links,
of a certain polynomial evaluated at the total
latency incurred by all users choosing the link;
we call it polynomial social cost. The polynomials
that we consider have non-negative coefficients.
We are interested in evaluating Nash equilibria in
this model, and we use the Price of Anarchy as our
evaluation measure. We prove the Fully Mixed Nash
Equilibrium Conjecture for identical users and two
links, and establish an approximate version of the
conjecture for arbitrary many links. Moreover, we
give upper bounds on the Price of Anarchy
Quasirandom Load Balancing
We propose a simple distributed algorithm for balancing indivisible tokens on
graphs. The algorithm is completely deterministic, though it tries to imitate
(and enhance) a random algorithm by keeping the accumulated rounding errors as
small as possible.
Our new algorithm surprisingly closely approximates the idealized process
(where the tokens are divisible) on important network topologies. On
d-dimensional torus graphs with n nodes it deviates from the idealized process
only by an additive constant. In contrast to that, the randomized rounding
approach of Friedrich and Sauerwald (2009) can deviate up to Omega(polylog(n))
and the deterministic algorithm of Rabani, Sinclair and Wanka (1998) has a
deviation of Omega(n^{1/d}). This makes our quasirandom algorithm the first
known algorithm for this setting which is optimal both in time and achieved
smoothness. We further show that also on the hypercube our algorithm has a
smaller deviation from the idealized process than the previous algorithms.Comment: 25 page
Existence and Efficiency of Equilibria for Cost-Sharing in Generalized Weighted Congestion Games
This work studies the impact of cost-sharing methods on the existence and efficiency of (pure) Nash equilibria in weighted congestion games. We also study generalized weighted congestion games, where each player may control multiple commodities. Our results are fairly general; we only require that our cost-sharing method and our set of cost functions satisfy certain natural conditions. For general weighted congestion games, we study the existence of pure Nash equilibria in the induced games, and we exhibit a separation from the standard single-commodity per player model by proving that the Shapley value is the only cost-sharing method that guarantees existence of pure Nash equilibria. With respect to efficiency, we present general tight bounds on the price of anarchy, which are robust and apply to general equilibrium concepts. Our analysis provides a tight bound on the price of anarchy, which depends only on the used cost-sharing method and the set of allowable cost functions. Interestingly, the same bound applies to weighted congestion games and generalized weighted congestion games. We then turn to the price of stability and prove an upper bound for the Shapley value cost-sharing method, which holds for general sets of cost functions and which is tight in special cases of interest, such as bounded degree polynomials. Also for bounded degree polynomials, we provide a somewhat surprising result, showing that a slight deviation from the Shapley value has a huge impact on the price of stability. In fact, for this case, the price of stability becomes as bad as the price of anarchy. Again, our bounds on the price of stability are independent on whether players are single or multi-commodity
Reachability Switching Games
We study the problem of deciding the winner of reachability switching games for zero-, one-, and two-player variants. Switching games provide a deterministic analogue of stochastic games. We show that the zero-player case is NL-hard, the one-player case is NP-complete, and that the two-player case is PSPACE-hard and in EXPTIME. For the zero-player case, we also show P-hardness for a succinctly-represented model that maintains the upper bound of NP coNP. For the one- and two-player cases, our results hold in both the natural, explicit model and succinctly-represented model. Our results show that the switching variant of a game is harder in complexity-theoretic terms than the corresponding stochastic version
In Congestion Games, Taxes Achieve Optimal Approximation
In this work, we consider the problem of minimising the social cost in atomic congestion games. For this problem, we provide tight computational lower bounds along with taxation mechanisms yielding polynomial time algorithms with optimal approximation. Perhaps surprisingly, our results show that indirect interventions, in the form of efficiently computed taxation mechanisms, yield the same performance achievable by the best polynomial time algorithm, even when the latter has full control over the agents' actions. It follows that no other tractable approach geared at incentivizing desirable system behavior can improve upon this result, regardless of whether it is based on taxations, coordination mechanisms, information provision, or any other principle. In short: Judiciously chosen taxes achieve optimal approximation. Three technical contributions underpin this conclusion. First, we show that computing the minimum social cost is NP-hard to approximate within a given factor depending solely on the admissible resource costs. Second, we design a tractable taxation mechanism whose efficiency (price of anarchy) matches this hardness factor, and thus is worst-case optimal. As these results extend to coarse correlated equilibria, any no-regret algorithm inherits the same performances, allowing us to devise polynomial time algorithms with optimal approximation