Smaller Cuts, Higher Lower Bounds

Abstract

This paper proves strong lower bounds for distributed computing in the congest model, by presenting the bit-gadget: a new technique for constructing graphs with small cuts.The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs.Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortestpaths problem is discussed.Finally, it is shown that graph constructions for congest lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.

Universally-Optimal Distributed Algorithms for Known Topologies

“…Assume, for the sake of contradiction, that a node h is charged more than two times. Then the construction would either seed a crown (if h ∈ L) or grow a crown via h. We conclude that |H| ≥ 1 2 |M out |. We finalize the proof using the two subclaims and M in ⊆ i∈I A i :…”

“…Let {p i } k i=1 be minimal part-paths with endpoints {(s i , t i )} k i=1 . Using Lemma 6.11, we find a crown (T * , A * , U * ) of {p i } i∈A * with |U * | ≥ 1 280 k. Furthermore, Lemma 6.13 guarantees the existence of a relaxed disjoint gadget (P, T ) with endpoints {(s i , t i )} i∈U satisfying |U | ≥ 1 5 |U * | = 1 1400 k.…”

“…In this model Ghaffari [22] gave a polynomial-time preprocessing k-broadcast algorithm which is optimal among routing-based distributed algorithms on the given input. It was noted in [16,17,65] that, while preprocessing intuitively seems very powerful, CONGEST lower bounds [1,8] generally hold even in the supported model. Due to this, [17] asks whether preprocessing offers any benefit at all, and they offer several (specifically-made) tasks that do exhibit a separation.…”

“…Let (T, A, U ) be a crown with respect to {p i } k i=1 with endpoints {(s i , t i )} k i=1 . There exists a subset U ′ ⊆ U of size |U ′ | ≥ 1 5 |U | and a relaxed disjointness gadget (P, T ) with endpoints {(s i , t i )} i∈U ′ .…”

“…These systems implement synchronous message-passing algorithms, in which nodes send (small) messages to their neighbors in each round. 1 This has motivated a recent, broad, concentrated, and highly-successful effort to advance our theoretical understanding of such algorithms for fundamental network optimization problems, such as minimum-spanning trees (MST) [11,14,43,45,60], shortest paths [13,18,39,40,41,47,48,55], flows [25], and cuts [9,23,56]. As a result, many fundamental optimization problems now have worstcase-optimal CONGEST algorithms, running in Θ( √ n + D) rounds on every n-node network with diameter D. 2 In general, these running times cannot be improved due to unconditional lower bounds [8,12,62] showing that there exist pathological n-node topologies with small diameter on which any non-trivial optimization problem requires Ω( √ n) rounds.…”