A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constant-size neighbourhood of the node. We study the applicability of local algorithms to max-min LPs where the objective is to maximise $\min_k \sum_v c_{kv} x_v$ subject to $\sum_v a_{iv} x_v \le 1$ for each $i$ and $x_v \ge 0$ for each $v$. Here $c_{kv} \ge 0$, $a_{iv} \ge 0$, and the support sets $V_i = \{v : a_{iv} > 0 \}$, $V_k = \{v : c_{kv}>0 \}$, $I_v = \{i : a_{iv} > 0 \}$ and $K_v = \{k : c_{kv} > 0 \}$ have bounded size. In the distributed setting, each agent $v$ is responsible for choosing the value of $x_v$, and the communication network is a hypergraph $\mathcal{H}$ where the sets $V_k$ and $V_i$ constitute the hyperedges. We present inapproximability results for a wide range of structural assumptions; for example, even if $|V_i|$ and $|V_k|$ are bounded by some constants larger than 2, there is no local approximation scheme. To contrast the negative results, we present a local approximation algorithm which achieves good approximation ratios if we can bound the relative growth of the vertex neighbourhoods in $\mathcal{H}$.
We study the limits of what can and what cannot be achieved by local algorithms [13]. We focus on the approximability of a certain class of linear optimisation problems, which generalises beyond widely studied packing LPs; the emphasis is on deterministic algorithms and worst-case analysis.
1.1. Local algorithms. A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constant-size neighbourhood of the node. We focus on problems where the size of the input per node is bounded by a constant; in such problems, local algorithms provide an extreme form of scalability: the communication, space and time complexity of a local algorithm is constant per node, and a local algorithm scales to an arbitrarily large or even infinite network.
The study of local algorithms has several uses beyond providing highly scalable distributed algorithms. The existence of a local algorithm shows that the function can be computed by bounded-fan-in, constant-depth Boolean circuits; we can say that the function is in the class NC 0 . A local algorithm is also an efficient centralised algorithm: the time complexity of the centralised algorithm is linear in the number of nodes; furthermore, due to spatial locality in memory accesses, we may be able to achieve a low I/O complexity in the external memory [18] model of computation. In certain problems, a local approximation algorithm can be used to construct a sublinear time algorithm which approximates the size of the optimal solution, assuming that we tolerate an additive error and some probability of failure [16]. A local algorithm can be turned into an efficient self-stabilising algorithm [3]; the time to stabilise is constant [1]. Finally, the existence and nonexistence of local algorithms gives us insight into the algorithmic value of information in distributed decision-making [14].
1.2. Max-min packing problem. In this section, we define the optimisation problem that we study in this work. Let V , I and K be index sets with I ∩K = ∅; we say that each v ∈ V is an agent, each k ∈ K is a beneficiary party, and each i ∈ I is a resource (constraint). We assume that one unit of activity by v benefits the party k by c kv ≥ 0 units and consumes a iv ≥ 0 units of the resource i; the objective is to set the activities to provide a fair share of benefit for each party. In notation, assuming that the activity of agent v is x v units, the objective is to (1) maximise ω = min k∈K v∈V
Throughout this work we assume that the support sets defined for all i ∈ I, k ∈ K, and v ∈ V by
To avoid uninteresting degenerate cases, we furthermore assume that I v , V i and V k are nonempty. 1.3. LP formulation. If the sets V , I and K are finite, the problem can be represented using matrix notation. Let A be the nonnegative |I| × |V | matrix where the entry at row i, column v is a iv ; define C analogously. We write a i for the row i of A and c k for the row k of C. Let x be a column vector of length |V |. The goal is to maximise ω = min k∈K c k x subject to Ax ≤ 1 and x ≥ 0.
In the special case |K| = 1, this is the widely studied fractional packing problem: maximise cx subject to Ax ≤ 1 and x ≥ 0. This simple linear program (LP) has nonnegative coefficients in c and A. We refer to a problem of this form as a packing LP ; the dual is a covering LP. Naturally the case of any finite K can also be written as a linear program, but the constraint matrix is no longer nonnegative: maximise ω subject to Ax ≤ 1, ω1 -Cx ≤ 0 and x ≥ 0.
1.4. Distributed setting. We construct the hypergraph H = (V, E) where the hyperedges are E = {V i : i ∈ I} ∪ {V k : k ∈ K}. This is the communication graph in our distributed optimisation problem. The variable x v is controlled by the agent v ∈ V , and two agents u, v ∈ V can communicate directly with each other if they are adjacent in H. We write d H (u, v) for the shortest-path distance between u and v in H. The agents are cooperating, not selfish; the difficulty arises from the fact that the agents have to make decisions based on incomplete information.
Initially, each agent v ∈ V knows only the following local information: the identity of its neighbours in the graph H; the sets I v and K v ; the values a iv for each i ∈ I v ; and the values c kv for each k ∈ K v . That is, v knows with whom it is competing on which resources, and with whom it is working together to benefit which parties.
When we compare the present work with previous work, we often mention the special case |K| = 1, as this corresponds to the widely studied packing LP. However, in this case the size of V k is not bounded by a constant ∆ V K : we have V k = V for the sole k ∈ K. Therefore we introduce a restricted variant of the distributed setting, which we call collaboration-oblivious. In this variant, the hyperedges are E = {V i : i ∈ I}. Whenever we study related work on the packing LP, we focus on the collaboratio
This content is AI-processed based on open access ArXiv data.