We study competitive resource allocation problems in which players distribute their demands integrally on a set of resources subject to player-specific submodular capacity constraints. Each player has to pay for each unit of demand a cost that is a nondecreasing and convex function of the total allocation of that resource. This general model of resource allocation generalizes both singleton congestion games with integer-splittable demands and matroid congestion games with playerspecific costs. As our main result, we show that in such general resource allocation problems a pure Nash equilibrium is guaranteed to exist by giving a pseudo-polynomial algorithm computing a pure Nash equilibrium.
In competitive packet routing games, packets are routed selfishly through a network and scheduling policies at edges determine which packages are forwarded first if there is not enough capacity on an edge to forward all packages at once. We analyze the impact of priority lists on the worst-case quality of pure Nash equilibria. A priority list is an ordered list of players that may or may not depend on the edge. Whenever the number of packets entering an edge exceeds the inflow capacity, packets are processed in list order. We derive several new bounds on the price of anarchy and stability for global and local priority policies. We also consider the question of the complexity of computing an optimal priority list. It turns out that even for very restricted cases, i.e., for routing on a tree, the computation of an optimal priority list is APX-hard.
Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely laminar crossing spanning tree), and (2) by incorporating 'degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.• Our main result is a (1, b+O(log n))-approximation algorithm for the minimum crossing spanning tree (MCST) problem with laminar degree constraints. The laminar MCST problem is a natural generalization of the well-studied bounded-degree MST, and is a special case of general crossing spanning tree. We also give an additive Ω(log α m) hardness of approximation for general MCST, even in the absence of costs (α > 0 is a fixed constant, and m is the number of degree constraints).• We then consider the crossing contra-polymatroid intersection problem and obtain a (2, 2b + ∆−1)-approximation algorithm, where ∆ is the maximum element frequency. This models for example the degree-bounded spanning-set intersection in two matroids. Finally, we introduce the crossing lattice polyhedra problem, and obtain a (1, b + 2∆ − 1) approximation under certain condition. This result provides a unified framework and common generalization of various problems studied previously, such as degree bounded matroids.
Store-and-forward packet routing belongs to the most fundamental tasks in network optimization. Limited bandwidth requires that some packets cannot move to their destination directly but need to wait at intermediate nodes on their path or take detours. In particular, for time critical applications, it is desirable to find schedules that ensure fast delivery of the packets. It is thus a natural objective to minimize the makespan, i.e., the time at which the last packet arrives at its destination. In this paper we present several new ideas and techniques that lead to novel algorithms and hardness results.
Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely laminar crossing spanning tree), and (2) by incorporating 'degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.• Our main result is a (1, b+O(log n))-approximation algorithm for the minimum crossing spanning tree (MCST) problem with laminar degree constraints. The laminar MCST problem is a natural generalization of the well-studied bounded-degree MST, and is a special case of general crossing spanning tree. We also give an additive Ω(log α m) hardness of approximation for general MCST, even in the absence of costs (α > 0 is a fixed constant, and m is the number of degree constraints).• We then consider the crossing contra-polymatroid intersection problem and obtain a (2, 2b + ∆−1)-approximation algorithm, where ∆ is the maximum element frequency. This models for example the degree-bounded spanning-set intersection in two matroids. Finally, we introduce the crossing lattice polyhedra problem, and obtain a (1, b + 2∆ − 1) approximation under certain condition. This result provides a unified framework and common generalization of various problems studied previously, such as degree bounded matroids.
We study the sensitivity of optimal solutions of convex separable optimization problems over an integral polymatroid base polytope with respect to parameters determining both the cost of each element and the polytope. Under convexity and a regularity assumption on the functional dependency of the cost function with respect to the parameters, we show that reoptimization after a change in parameters can be done by elementary local operations. Applying this result, we derive that starting from any optimal solution there is a new optimal solution to new parameters such that the L 1 -norm of the difference of the two solutions is at most two times the L 1 -norm of the difference of the parameters.We apply these sensitivity results to a class of non-cooperative games with a finite set of players where a strategy of a player is to choose a vector in a player-specific integral polymatroid base polytope defined on a common set of elements. The players' private cost functions are regular, convex-separable and the cost of each element is a non-decreasing function of the own usage of that element and the overall usage of the other players. Under these assumptions, we establish the existence of a pure Nash equilibrium. The existence is proven by an algorithm computing a pure Nash equilibrium that runs in polynomial time whenever the rank of the polymatroid base-polytope is polynomially bounded. Both the existence result and the algorithm generalize and unify previous results appearing in the literature.We finally complement our results by showing that polymatroids are the maximal combinatorial structure enabling these results. For any non-polymatroid region, there is a corresponding optimization problem for which the sensitivity results do not hold. In addition, there is a game where the players' strategies are isomorphic to the non-polymatroid region and that does not admit a pure Nash equilibrium.
A fundamental problem in communication networks is store-andforward packet routing. In a celebrated paper Leighton, Maggs, and Rao [12] proved that the length of an optimal schedule is linear in the trivial lower bounds congestion and dilation. However, there has been no improvement on the actual bounds in more than 10 years. Also, commonly the problem is studied only in the setting of unit bandwidths and unit transit times. In this paper, we prove bounds on the length of optimal schedules for packet routing in the setting of arbitrary bandwidths and arbitrary transit times. Our results generalize the existing work to a much broader class of instances and also improve the known bounds significantly. For the case of unit transit times and bandwidths, we improve the best known bound of 39(C + D) to 23.4(C + D), where C and D denote the congestion and dilation, respectively. If every link in the network has a certain minimum transit time or capacity we improve this bounds to up to 4.25(C + D). Key to our results is a framework which employs tight bounds for instances where each packet travels along only a small number of edges. Further insights for such instances would reduce our constants even more.
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