This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and study models of computing that are weaker versions of the widely-studied port-numbering model. In the port-numbering model, a node of degree @math d receives messages through @math d input ports and sends messages through @math d output ports, both numbered with @math 1 , 2 , ? , d . In this work, @math VV c is the class of all graph problems that can be solved in the standard port-numbering model. We study the following subclasses of @math VV c : Now we have many trivial containment relations, such as @math SB ⊆ MB ⊆ VB ⊆ VV ⊆ VV c , but it is not obvious if, for example, either of @math VB ⊆ SV or @math SV ⊆ VB should hold. Nevertheless, it turns out that we can identify a linear order on these classes. We prove that @math SB ? MB = VB ? SV = MV = VV ? VV c . The same holds for the constant-time versions of these classes. We also show that the constant-time variants of these classes can be characterised by a corresponding modal logic. Hence the linear order identified in this work has direct implications in the study of the expressibility of modal logic. Conversely, one can use tools from modal logic to study these classes.
We are by no means the first to investigate the weak models. Computation in models that are strictly weaker than the standard port-numbering model has been studied since the 1990s, under various terms---see Table for a summary of terminology, and Table for an overview of the main differences in the research directions. Questions related to specific problems, models, and graph families have been studied previously, and indeed many of the techniques and ideas that we use are now standard---this includes the use of symmetry and isomorphisms, local views, covering graphs (lifts) and universal covering graphs, and factors and factorisations. Mayer, Naor, and Stockmeyer @cite_47 @cite_68 made it explicit that the parity of node degrees makes a huge difference in the port-numbering model, and Yamashita and Kameda @cite_45 discussed factors and factorisations in this context; the underlying graph-theoretic observations can be traced back to as far as Petersen's 1891 work @cite_35 . Some equivalences and separations between the classes are already known, or at least implicit in prior work---see, in particular, @cite_72 and Yamashita and Kameda @cite_10 .
@cite_35 @cite_72 @cite_45 @cite_47 @cite_68 @cite_10
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This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and study models of computing that are weaker versions of the widely-studied port-numbering model. In the port-numbering model, a node of degree @math d receives messages through @math d input ports and sends messages through @math d output ports, both numbered with @math 1 , 2 , ? , d . In this work, @math VV c is the class of all graph problems that can be solved in the standard port-numbering model. We study the following subclasses of @math VV c : Now we have many trivial containment relations, such as @math SB ⊆ MB ⊆ VB ⊆ VV ⊆ VV c , but it is not obvious if, for example, either of @math VB ⊆ SV or @math SV ⊆ VB should hold. Nevertheless, it turns out that we can identify a linear order on these classes. We prove that @math SB ? MB = VB ? SV = MV = VV ? VV c . The same holds for the constant-time versions of these classes. We also show that the constant-time variants of these classes can be characterised by a corresponding modal logic. Hence the linear order identified in this work has direct implications in the study of the expressibility of modal logic. Conversely, one can use tools from modal logic to study these classes. In the 0-extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge's weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of [SIAM J. Comput. , 23 (1994), pp. 864--894] and is closely related to the metric labeling problem introduced by Kleinberg and Tardos [Proceedings of the 40th IEEE Annual Symposium on Foundations of Computer Science, New York, 1999, pp. 14--23]. We present approximation algorithms for 0-Extension . In arbitrary graphs, we present a O(log k)-approximation algorithm, k being the number of terminals. We also give O(1)-approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem previously considered by Karzanov [European J. Combin., 19 (1998), pp. 71--101]. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis [SIAM J. Comput. , 25 (1996), pp. 235--251] for the multicut problem, and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least @math for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos, and they further our understanding on how to use metric relaxations. Mulmuley [Mul12a] recently gave an explicit version of Noether’s Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley’s reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors ([FS12]), who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors. As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation. We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model. Finally, we consider the depth-3 diagonal circuit model as defined by Saxena [Sax08], as PIT algorithms for this model also have implications in Mulmuley’s work. Previous work (such as [ASS12] and [FS12]) have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich [SV09]. We initiate a study of when the value of mathematical relaxations such as linear and semi-definite programs for constraint satisfaction problems (CSPs) is approximately preserved when restricting the instance to a sub-instance induced by a small random subsample of the variables. Let C be a family of CSPs such as 3SAT, Max-Cut, etc., and let Π be a mathematical program that is a relaxation for C, in the sense that for every instance P ∈ C, Π(P) is a number in [0, 1] upper bounding the maximum fraction of satisfiable constraints of P. Loosely speaking, we say that subsampling holds for C and Π if for every sufficiently dense instance P ∈ C and every e > 0, if we let P' be the instance obtained by restricting P to a sufficiently large constant number of variables, then Π(P') ∈ (1 ± e)Π(P). We say that weak subsampling holds if the above guarantee is replaced with Π(P') = 1 − θ(γ) whenever Π(P) = 1 − γ, where θ hides only absolute constants. We obtain both positive and negative results, showing that: 1. Subsampling holds for the BasicLP and BasicSDP programs. BasicSDP is a variant of the semi-definite program considered by Raghavendra (2008), who showed it gives an optimal approximation factor for every constraint-satisfaction problem under the unique games conjecture. BasicLP is the linear programming analog of BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of unique games type. 3. There are non-unique CSPs for which even weak subsampling fails for the above tighter semi-definite programs. Also there are unique CSPs for which (even weak) subsampling fails for the Sherali-Adams linear programming hierarchy. As a corollary of our weak subsampling for strong semi-definite programs, we obtain a polynomial-time algorithm to certify that random geometric graphs (of the type considered by Feige and Schechtman, 2002) of max-cut value 1 − γ have a cut value at most 1 − γ 10. More generally, our results give an approach to obtaining average-case algorithms for CSPs using semi-definite programming hierarchies.