Theory of Computing Systems

A packet-switching network is stable if the number of packets in the network remains bounded at all times. A very natural question that arises in the context of stability properties of such networks is how network structure precisely affects these properties. In this work we embark on a systematic study of this question in the context of Adversarial Queueing Theory, which assumes that packets are adversarially injected into the network. We consider size, diameter, maximum vertex degree, minimum number of disjoint paths that cover all edges of the network and network subgraphs as crucial structural parameters of the network, and we present a comprehensive collection of structural results, in the form of stability and instability bounds on injection rate of the adversary for various greedy protocols: —Increasing the size of a network may result in dropping its instability bound. This is shown through a novel, yet simple and natural, combinatorial construction of a size-parameterized network on which certain compositions of greedy protocols are running. The convergence of the drop to 0.5 is found to be fast with and proportional to the increase in size. —Maintaining the size of a network small may already suffice to drop its instability bound to a substantially low value. This is shown through a construction of a FIFO network with size 22, which becomes unstable at rate 0.704. This represents the current state-of-the-art trade-off between network size and instability bound. —The diameter, maximum vertex degree and minimum number of edge-disjoint paths that cover a network may be used as control parameters for the stability bound of the network. This is shown through an improved analysis of the stability bound of any arbitrary FIFO network, which takes these parameters into account. —How much can network subgraphs that are forbidden for stability affect the instability bound? Through improved combinatorial constructions of networks and executions, we improve the state-of-the-art instability bound induced by certain known forbidden subgraphs on networks running a certain greedy protocol. —Our results shed more light and contribute significantly to a finer understanding of the impact of structural parameters on stability and instability properties of networks.

Do self-reducible sets inherently lack immunity from deterministic polynomial time? Though this is unlikely to be true in general, in this paper we prove that sufficiently strong self-reducibility precludes sufficiently strong immunity from deterministic polynomial time. In particular, we prove that NT isnot P balanced immune. However, we prove that NT, a class whose sets have very strong self-reducibility properties, is P bi-immune relative to a generic oracle. Thus, the previous result cannot be relativizably extended to bi-immunity. We also prove that NP and ⊕P are both P balanced immune relative to a random oracle; the former provides the strongest known relativized separation of NP from P.

The focus of our work is introducing and constructing probabilistic coresets. A probabilistic coreset can contain probabilistic points, and the number of these points should be polylogarithmic in the input size. However, the overall storage size is also influenced by representation size of the propability distribution of each point. So, our first observation is that the size of probabilistic coresets shall be restricted in the number of points and in the representation size of the points. We propose the first (k, ε)-coreset constructions for the probabilistic k-median problem in the metric and Euclidean case. The coresets are of size poly(ε −1, k, log(W/(p min⋅δ))), where W is the expected total weight of the weighted probabilistic input points when all weights are scaled to be at least one, p min is the probability of a point to be realized at a certain location, and δ is the error probability of the construction. Our coreset for the Euclidean problem can be maintained in data streams.

We introduce the concept of effective dimension for a wide class of metric spaces whose metric is not necessarily based on a measure. Effective dimension was defined by Lutz (Inf. Comput., 187(1), 49–79, 2003) for Cantor space and has also been extended to Euclidean space. Lutz effectivization uses gambling, in particular the concept of gale and supergale, our extension of Hausdorff dimension to other metric spaces is also based on a supergale characterization of dimension, which in practice avoids an extra quantifier present in the classical definition of dimension that is based on Hausdorff measure and therefore allows effectivization for small time-bounds. We present here the concept of constructive dimension and its characterization in terms of Kolmogorov complexity, for which we extend the concept of Kolmogorov complexity to any metric space defining the Kolmogorov complexity of a point at a certain precision. Further research directions are indicated.

According to the proportional allocation mechanism from the network optimization literature, users compete for a divisible resource – such as bandwidth – by submitting bids. The mechanism allocates to each user a fraction of the resource that is proportional to her bid and collects an amount equal to her bid as payment. Since users act as utility-maximizers, this naturally defines a proportional allocation game. Syrgkanis and Tardos (STOC 2013) quantified the inefficiency of equilibria in this game with respect to the social welfare and presented a lower bound of 26.8 % on the price of anarchy over coarse-correlated and Bayes-Nash equilibria in the full and incomplete information settings, respectively. In this paper, we improve this bound to 50 % over both equilibrium concepts. Our analysis is simpler and, furthermore, we argue that it cannot be improved by arguments that do not take the equilibrium structure into account. We also extend it to settings with budget constraints where we show the first constant bound (between 36 and 50 %) on the price of anarchy of the corresponding game with respect to an effective welfare benchmark that takes budgets into account.

We prove that in anyN-node communication network with maximum degreed, any deterministic oblivious algorithm for routing an arbitrary permutation requires Ω(√N/d) parallel communication steps in the worst case. This is an improvement upon the Ω(√N/d 3/2) bound obtained by Borodin and Hopcroft. For theN-node hypercube, in particular, we show a matching upper bound by exhibiting a deterministic oblivious algorithm that routes any permutation in Θ(√N/logN) steps. The best previously known upper bound was Θ(√N). Our algorithm may be practical for smallN (up to about 214 nodes).

We obtain improved approximation ratios for problems of a broad class called weighted hereditary induced-subgraph maximization problems, in particular for the maximum independent set, maximum clique and maximum ℓ-colorable induced subgraph, as well as for the minimum coloring problem. We also study the minimum chromatic sum and show that its weighted version polynomially reduces to the weighted independent set problem in such a way that approximation ratios are preserved (up to a multiplicative constant).

There is a long tradition in understanding graphs by investigating their adjacency matrices by means of linear algebra. Similarly, logic-based graph query languages are commonly used to explore graph properties. In this paper, we bridge these two approaches by regarding linear algebra as a graph query language. More specifically, we consider MATLANG, a matrix query language recently introduced, in which some basic linear algebra functionality is supported. We investigate the problem of characterising the equivalence of graphs, represented by their adjacency matrices, for various fragments of MATLANG. That is, we are interested in understanding when two graphs cannot be distinguished by posing queries in MATLANG on their adjacency matrices. Surprisingly, a complete picture can be painted of the impact of each of the linear algebra operations supported in MATLANG on their ability to distinguish graphs. Interestingly, these characterisations can often be phrased in terms of spectral and combinatorial properties of graphs. Furthermore, we also establish links to logical equivalence of graphs. In particular, we show that MATLANG-equivalence of graphs corresponds to equivalence by means of sentences in the three-variable fragment of first-order logic with counting. Equivalence with regards to a smaller MATLANG fragment is shown to correspond to equivalence by means of sentences in the two-variable fragment of this logic.