Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems

Abstract

We consider several well-studied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms:1. Is the 3SUM problem on n numbers in O(n 2−ε ) time for some ε > 0?Conjecture 1 (No truly subquadratic 3SUM). In the Word RAM model with words of O(log n) bits, 1 any algorithm requires n 2−o(1) time in expectation to determine whether a set S ⊂ {−n 3 , . . . , n 3 } of |S| = n integers contains three distinct elements a, b, c ∈ S with a + b = c.Conjecture 2 (No truly subcubic APSP). There is a constant c, such that in the Word RAM model with words of O(log n) bits, any algorithm requires n 3−o(1) time in expectation to compute the distances between every pair of vertices in an n node graph with edge weights in {1, . . . , n c }.Conjecture 3 (Strong Exponential Time Hypothesis (SETH)). For every ε > 0, there exists a k, such that SAT on k-CNF formulas on n variables cannot be solved in O * (2 (1−ε)n ) time 1 .Conjecture 4 (No almost linear time triangle). There is a constant δ > 0, such that in the Word RAM model with words of O(log n) bits, any algorithm requires m 1+δ−o(1) time in expectation to detect whether an m edge graph contains a triangle."Conjecture" 5 (No truly subcubic combinatorial BMM). In the Word RAM model with words of O(log n) bits, any combinatorial algorithm requires n 3−o(1) time in expectation to compute the Boolean product of two n × n matrices. 2 This paper is the first study that relates the complexity of any dynamic problem to the exact complexity of Boolean Satisfiability (via the SETH). Our lower bounds hold even for randomized fully dynamic algorithms with (expected) amortized update times. Most of our results also hold for partially dynamic (incremental and decremental) algorithms with worst-case time bounds.Interestingly, many of our lower bounds (those based on the SETH) hold even when one allows arbitrary polynomial preprocessing time, and achieve essentially optimal guarantees. These are the first lower bounds of this nature.Most of our lower bounds also hold in the setting when one knows the list of updates and queries in advance, i.e. in the lookahead model. This is of interest since many dynamic problems can be solved faster given sufficient lookahead, e.g. graph transitive closure [83] and matrix rank [59].Organization. In Section 2 we discuss our results and the prior work on the problems we address. In Section 3 we describe our techniques. In Section 4 we give an overview of the prior work on the conjectures. In Section 5 we give a formal statement of the theorems we prove. The problems we consider are summarized in Table 1 and the results are summarized in Table 2. In section 6 we define some useful notation and prove reductions between dynamic problems. In section 7 we prove lower bounds based on Conjecture 3 (SETH). In section 8 we prove lower bounds based on Conjectures 4 and 5 (Triangle and BMM). In section 9 we prove lower bounds based on Conjecture 2 (APSP). And fi...

Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture

“…Our approach results in stronger bounds which are tight for some problems. Additionally, we show that a number of previous proofs can be unified as they can now start from only one problem, that is OMv, and can be done in a much simpler way (compare, e.g., the hardness proof for Pagh's problem in this paper and in [1]). Thus proving the hardness of a problem via OMv should be a simpler task.…”

“…Thus, breaking Conjecture 1.1 is arguably at least as hard as making a breakthrough for Boolean matrix multiplication and other long-standing open problems (e.g., [12,52,1,37,21]). This conjecture is also supported by an algebraic lower bound [8].…”

Parameterized Aspects of Triangle Enumeration

“…Since X / ∈ T 1 , it follows X / ∈ f ({a, b, c}, A). Next we show that for any two triangles p = {u 1 , u 2 , u 3 } and q = {v 1 …”

“…Hence, one cannot use the parameter minimum degree to design faster algorithms for △-Detect. Upon this trivial example, we study which parameters for △-Detect cannot be used to design linear-time FPT algorithms under the hypothesis that △-Detect is not linear-time solvable [1]. To this end we reduce in linear time an arbitrary instance of △-Detect to a new equivalent (and not too large) instance of the problem with the parameter upper-bounded by a constant.…”

“…As to counting the number of triangles in a graph, the best known algorithm takes O(n ω ) ⊂ O(n 2.373 ) time [40] and is based on fast matrix multiplication. 1 Finally, detecting a triangle in a graph is doable in O(n ω ) time [40] and it is conjectured that every algorithm for detecting a triangle in a graph takes at least Θ(n ω−o(1) ) time [1]. We mention that for m-edge graphs there is also an O(m 1.5 )-time algorithm [34] which is interesting in case of sparse graphs.…”