We give time lower bounds for the distributed approximation of minimum vertex cover (MVC) and related problems such as minimum dominating set (MDS). In k communication rounds, MVC and MDS can only be approximated by factors# /k) and # /k) for some constant c, where n and # denote the number of nodes and the largest degree in the graph. The number of rounds required in order to achieve a constant or even only a polylogarithmic approximation ratio is at log n/ log log n) and#1 #/ log log #). By a simple reduction, the latter lower bounds also hold for the construction of maximal matchings and maximal independent sets.

Achieving a global goal based on local information is challenging, especially in complex and large-scale networks such as the Internet or even the human brain. In this paper, we provide an almost tight classification of the possible trade-off between the amount of local information and the quality of the global solution for general covering and packing problems. Specifically, we give a distributed algorithm using only small messages which obtains an (ρ∆) 1/k-approximation for general covering and packing problems in time O(k 2), where ρ depends on the LP’s coefficients. If message size is unbounded, we present a second algorithm that achieves an O(n 1/k) approximation in O(k) rounds. Finally, we prove that these algorithms are close to optimal by giving a lower bound on the approximability of packing problems given that each node has to base its decision on information from its k-neighborhood. 1

Abstract This paper proves \Omega (n) lower bounds on the time to perform a single instance of an operationin any implementation of a large class of data structures shared by n processes. For standarddata structures such as counters, stacks, and queues, the bound is tight. The implementations considered may apply any deterministic primitives to a base object. No bounds are assumedon either the number of base objects or their size. Time is measured as the number of steps a process performs on base objects and the number of stalls it incurs as a result of contentionwith other processes. 1

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We study the problem of the amount of information (advice) about a graph that must be given to its nodes in order to achieve fast distributed computations. The required size of the advice enables to measure the information sensitivity of a network problem. A problem is information sensitive if little advice is enough to solve the problem rapidly (i.e., much faster than in the absence of any advice), whereas it is information insensitive if it requires giving a lot of information to the nodes in order to ensure fast computation of the solution. In this paper, we study the information sensitivity of distributed graph coloring.

the date of receipt and acceptance should be inserted later Abstract We study the power of reliable anonymous distributed systems, where processes do not fail, do not have identifiers, and run identical programmes. We are interested specifically in the relative powers of systems with different communication mechanisms: anonymous broadcast, read-write registers, or read-write registers plus additional shared-memory objects. We show that a system with anonymous broadcast can simulate a system of shared-memory objects if and only if the objects satisfy a property we call idemdicence; this result holds regardless of whether either system is synchronous or asynchronous. Conversely, the key to simulating anonymous broadcast in anonymous shared memory is the ability to count: broadcast can be simulated by an asynchronous shared-memory system that uses only counters, but readwrite registers by themselves are not enough. We further examine the relative power of different types and sizes of bounded counters and conclude with a non-robustness result.

We make a step towards understanding the distributed complexity of global optimization problems. We give bounds on the trade-off between locality and achievable approximation ratio of distributed algorithms for packing and covering problems. Extending a result of [9], we show that in k communication rounds, maximum matching and therefore packing problems cannot be approximated better than Ω(nc/k2/k) and Ω(∆1/k /k) where c is a small constant and n and ∆ denote the number of nodes and the maximum degree of the network graph, respectively. This means that in order to obtain a constant or polylogarithmic approximation, there are graphs with n nodes and graphs with maximum degree ∆ on which Ω ( √ log n / log log n) and Ω(log ∆ / log log ∆) rounds are needed, respectively. On the positive side, we prove that maximum matching and minimum vertex cover (the dual problem) can be approximated by O(∆1/k) in O(k) rounds, showing that the given lower bound is almost tight. We also give a distributed algorithm which approximates any packing or covering LP by O(n1/k) in O(k) rounds. 1

We motivate and study a generic relaxation of correctness of reactive and concurrent systems with respect to a temporal specification. We define a system to be fairly correct if there exists a fairness assumption under which it satisfies its specification. Equivalently, a system is fairly correct if the set of runs satisfying the specification is large from a topological point of view, i.e., it is a co-meager set. We compare topological largeness with its more popular sibling, probabilistic largeness, where a specification is probabilistically large if the set of runs satisfying the specification has probability 1. We show that topological and probabilistic largeness of ω-regular specifications coincide for bounded Borel measures on finite-state systems. As a corollary, we show that, for specifications expressed in LTL or by Büchi automata, checking that a finite-state system is fairly correct has the same complexity as checking that it is correct. Finally we study variants of the logics CTL and CTL*, where the ‘for all runs ’ quantifier is replaced by a ‘for a large set of runs ’ quantifier. We show that the model checking complexity for these variants is the same as for the original logics. 1

The “wait-free hierarchy ” classifies multiprocessor synchronization primitives according to their power to solve consensus. The classification is based on assigning a number n to each synchronization primitive, where n is the maximal number of processes for which deterministic wait-free consensus can be solved using instances of the primitive and read write registers. Conditional synchronization primitives, such as compare-and-swap and load-linked/store-conditional, can implement deterministic wait-free consensus for any number of processes (they have consensus number ∞), and are thus considered to be among the strongest synchronization primitives. To some extent because of that, compare-and-swap and load-linked/store-conditional have became the synchronization primitives of choice, and have been implemented in hardware in many multiprocessor architectures. This paper shows that, though they are strong in the context of consensus, conditional synchronization primitives are not efficient in terms of memory space for implementing many key objects. Our results hold for starvation-free implementations of mutual exclusion, and for wait-free implementations of a large class of concurrent objects, that we call Visible(n). Roughly, Visible(n) is a class that includes all objects that support some operation that must perform a “visible”

An adaptive algorithm, whose step complexity adjusts to the number of active processes, is attractive for situations in which the number of participating processes is highly variable. This paper studies the number and type of multiwriter registers that are needed for adaptive algorithms. We prove that if a collect algorithm is f-adaptive to total contention, namely, its step complexity is f(k), where k is the number of processes that ever tooka step, then it uses Ω(f −1 (n)) multi-writer registers, where n is the total number of processes in the system. Furthermore, we show that competition for the underlying registers is inherent for adaptive collect algorithms. We consider c-write registers, to which at most c processes can be concurrently about to write. Special attention is given to exclusive-write registers, the case c = 1 where no competition is allowed, and concurrent-write registers, the case c = n where any amount of competition is allowed. A collect algorithm is f-adaptive to point contention, if its step complexity is f(k), where k is the maximum number of simultaneously active processes. Such an algorithm is shown to require Ω(f −1 ( n c)) concurrent-write registers, even if an un-limited number of c-write registers are available. A smaller lower bound is also obtained in this situation for collect algorithms that are f-adaptive to total contention. The lower bounds also hold for nondeterministic implementations of sensitive objects from historyless objects. Finally, we present lower bounds on the step complexity in solo executions (i.e., without any contention), when only c-write registers are used: For weaktest&set objects, we log n present an Ω() lower bound. Our lower bound log c+log log n for collect and sensitive objects is Ω ( n−1 c).