Online Multi-Commodity Flow with High Demands

This paper deals with the problem of computing, in an online fashion, a maximum benefit multi-commodity flow (\ONMCF), where the flow demands may be bigger than the edge capacities of the network. We present an online, deterministic, centralized, all-or-nothing, bi-criteria algorithm. The competitive ratio of the algorithm is constant, and the algorithm augments the capacities by at most a logarithmic factor. The algorithm can handle two types of flow requests: (i) low demand requests that must be routed along a path, and (ii) high demand requests that may be routed using a multi-path flow. Two extensions are discussed: requests with known durations and machine scheduling.

💡 Research Summary

The paper tackles the online multi‑commodity flow (ON‑MCF) problem in a setting where individual flow requests may demand more bandwidth than any single edge’s capacity. Unlike prior work that restricts each request to a single path and assumes demands are bounded by edge capacities, this study allows high‑demand requests to be split across multiple paths while still requiring an all‑or‑nothing service model (a request is either fully satisfied or rejected).

The authors present a deterministic, centralized algorithm called ALG that operates in an online fashion: requests arrive one by one, and the algorithm must decide immediately whether to accept or reject each request. The algorithm is built on the primal‑dual framework originally introduced by Buchbinder and Naor, but it extends the framework by incorporating a tri‑criteria oracle. This oracle, when given a request, returns a unit‑flow that satisfies three properties: (i) λ‑approximation of the minimum‑cost flow, (ii) µ‑augmentation of edge capacities (the flow may use up to µ times the original capacities), and (iii) ε‑granularity (every edge used by the flow carries at least ε units of flow). The concrete implementation achieves λ = µ = 2 and ε = 1/(2m²), where m is the number of edges.

ALG proceeds as follows. For each incoming request r_j = (s_j, t_j, d_j, b_j), it first checks feasibility under the µ‑augmented capacities; if infeasible, the request is rejected outright (the same decision any optimal offline algorithm would make). If feasible, the oracle supplies a unit‑flow f_j. The algorithm computes the cost of scaling this flow to the required demand, i.e., d_j·cost_j(f_j). If this scaled cost is less than λ·b_j, the request is accepted: the flow d_j·f_j is added to the current multi‑commodity flow, and the primal variables x_e (edge cost potentials) and z_j (request‑specific slack) are updated using exponential rules that keep the potentials bounded. If the cost exceeds the threshold, the request is rejected without any updates.

The analysis shows two key guarantees. First, the total benefit collected by ALG is at least 1/α of the benefit obtained by an optimal offline all‑or‑nothing solution, where α ≤ 2 (a constant). Second, for every edge e, the cumulative flow never exceeds β·c_e, where β = O(log n). The logarithmic factor arises from the exponential update of the edge potentials and the fact that the oracle’s µ‑augmentation is bounded by a constant. Consequently, ALG is (O(1), O(log n))‑competitive: it achieves a constant competitive ratio while only augmenting capacities by a logarithmic factor.

The paper also distinguishes between low‑demand and high‑demand requests. Low‑demand requests (d_j ≤ min_e c_e) are forced to use a single path, which can be handled by a standard min‑cost path oracle. High‑demand requests may use the multi‑path flow returned by the tri‑criteria oracle, thereby allowing demands larger than any single edge capacity.

Two extensions are discussed. When request durations are known in advance, the time dimension can be treated as an additional resource, and the same primal‑dual/tri‑criteria approach yields a (constant, O(log n))‑competitive algorithm for the time‑expanded network. The second extension maps the problem to machine scheduling: each job becomes a flow request, machines correspond to edges, and processing capacities become edge capacities. The algorithm again provides a constant‑factor competitive schedule with only logarithmic over‑provisioning of machine capacity.

In summary, the contribution of the paper is threefold: (1) it introduces a novel online algorithm that handles arbitrarily large flow demands while preserving an all‑or‑nothing service model; (2) it extends the primal‑dual method with a tri‑criteria oracle that simultaneously controls approximation quality, capacity augmentation, and granularity; and (3) it demonstrates that a constant competitive ratio is achievable with only O(log n) capacity augmentation, improving upon prior results that required either polynomial augmentation or incurred logarithmic competitive loss. The work opens avenues for practical deployment in data‑center networks, virtual network embedding, and other settings where large, bursty traffic must be accommodated online.