Amir Abboud

Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems

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(n2-ε) time for some ε > 0? 2) Can one determine the satisfiability of a CNF formula on n variables and poly n clauses in O((2 - ε)npoly n) time for some ε > 0? 3) Is the All Pairs Shortest Paths problem for graphs on n vertices in O(n3-ε) time for some ε > 0? 4) Is there a linear time algorithm that detects whether a given graph contains a triangle? 5) Is there an O(n3-ε) time combinatorial algorithm for n×n Boolean matrix multiplication? The problems we consider include dynamic versions of bipartite perfect matching, bipartite maximum weight matching, single source reachability, single source shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems such as Paghs problem defined in a recent paper by Patrascu[STOC 2010].

Consequences of Faster Alignment of Sequences

The Local Alignment problem is a classical problem with applications in biology. Given two input strings and a scoring function on pairs of letters, one is asked to find the substrings of the two input strings that are most similar under the scoring function. The best algorithms for Local Alignment run in time that is roughly quadratic in the string length. It is a big open problem whether substantially subquadratic algorithms exist. In this paper we show that for all e > 0, an O(n 2 − e ) time algorithm for Local Alignment on strings of length n would imply breakthroughs on three longstanding open problems: it would imply that for some δ > 0, 3SUM on n numbers is in O(n 2 − δ ) time, CNF-SAT on n variables is in O((2 − δ) n ) time, and Max Weight 4-Clique is in O(n 4 − δ ) time. Our result for CNF-SAT also applies to the easier problem of finding the longest common substring of binary strings with don’t cares. We also give strong conditional lower bounds for the more general Multiple Local Alignment problem on k strings, under both k-wise and SP scoring, and for other string similarity problems such as Global Alignment with gap penalties and normalized Longest Common Subsequence.

Tight Hardness Results for LCS and Other Sequence Similarity Measures

Two important similarity measures between sequences are the longest common subsequence (LCS) and the dynamic time warping distance (DTWD). The computations of these measures for two given sequences are central tasks in a variety of applications. Simple dynamic programming algorithms solve these tasks in O(n2) time, and despite an extensive amount of research, no algorithms with significantly better worst case upper bounds are known. In this paper, we show that for any constant ε >0, an O(n2-ε) time algorithm for computing the LCS or the DTWD of two sequences of length n over a constant size alphabet, refutes the popular Strong Exponential Time Hypothesis (SETH).

Simulating branching programs with edit distance and friends: or: a polylog shaved is a lower bound made

A recent, active line of work achieves tight lower bounds for fundamental problems under the Strong Exponential Time Hypothesis (SETH). A celebrated result of Backurs and Indyk (STOC’15) proves that computing the Edit Distance of two sequences of length n in truly subquadratic O(n2−ε) time, for some ε>0, is impossible under SETH. The result was extended by follow-up works to simpler looking problems like finding the Longest Common Subsequence (LCS). SETH is a very strong assumption, asserting that even linear size CNF formulas cannot be analyzed for satisfiability with an exponential speedup over exhaustive search. We consider much safer assumptions, e.g. that such a speedup is impossible for SAT on more expressive representations, like subexponential-size NC circuits. Intuitively, this assumption is much more plausible: NC circuits can implement linear algebra and complex cryptographic primitives, while CNFs cannot even approximately compute an XOR of bits. Our main result is a surprising reduction from SAT on Branching Programs to fundamental problems in P like Edit Distance, LCS, and many others. Truly subquadratic algorithms for these problems therefore have far more remarkable consequences than merely faster CNF-SAT algorithms. For example, SAT on arbitrary o(n)-depth bounded fan-in circuits (and therefore also NC-Circuit-SAT) can be solved in (2−ε)n time. An interesting feature of our work is that we get major consequences even from mildly subquadratic algorithms for Edit Distance or LCS. For example, we show that if an arbitrarily large polylog factor is shaved from n2 for Edit Distance then NEXP does not have non-uniform NC1 circuits.

The 4/3 additive spanner exponent is tight

A spanner is a sparse subgraph that approximately preserves the pairwise distances of the original graph. It is well known that there is a smooth tradeoff between the sparsity of a spanner and the quality of its approximation, so long as distance error is measured multiplicatively. A central open question in the field is to prove or disprove whether such a tradeoff exists also in the regime of additive error. That is, is it true that for all ε>0, there is a constant kε such that every graph has a spanner on O(n1+ε) edges that preserves its pairwise distances up to +kε? Previous lower bounds are consistent with a positive resolution to this question, while previous upper bounds exhibit the beginning of a tradeoff curve: all graphs have +2 spanners on O(n3/2) edges, +4 spanners on Õ(n7/5) edges, and +6 spanners on O(n4/3) edges. However, progress has mysteriously halted at the n4/3 bound, and despite significant effort from the community, the question has remained open for all 0 < ε < 1/3. Our main result is a surprising negative resolution of the open question, even in a highly generalized setting. We show a new information theoretic incompressibility bound: there is no function that compresses graphs into O(n4/3 − ε) bits so that distance information can be recovered within +no(1) error. As a special case of our theorem, we get a tight lower bound on the sparsity of additive spanners: the +6 spanner on O(n4/3) edges cannot be improved in the exponent, even if any subpolynomial amount of additive error is allowed. Our theorem implies new lower bounds for related objects as well; for example, the twenty-year-old +4 emulator on O(n4/3) edges also cannot be improved in the exponent unless the error allowance is polynomial. Central to our construction is a new type of graph product, which we call the Obstacle Product. Intuitively, it takes two graphs G,H and produces a new graph G H whose shortest paths structure looks locally like H but globally like G.

Exact weight subgraphs and the k -sum conjecture

We consider the Exact-Weight-H problem of finding a (not necessarily induced) subgraph H of weight 0 in an edge-weighted graph G. We show that for every H, the complexity of this problem is strongly related to that of the infamous k-sum problem. In particular, we show that under the k-sum Conjecture, we can achieve tight upper and lower bounds for the Exact-Weight-H problem for various subgraphs H such as matching, star, path, and cycle. One interesting consequence is that improving on the O(n3) upper bound for Exact-Weight-4-path or Exact-Weight-5-path will imply improved algorithms for 3-sum, 5-sum, All-Pairs Shortest Paths and other fundamental problems. This is in sharp contrast to the minimum-weight and (unweighted) detection versions, which can be solved easily in time O(n2). We also show that a faster algorithm for any of the following three problems would yield faster algorithms for the others: 3-sum, Exact-Weight-3-matching, and Exact-Weight-3-star.

Near-Linear Lower Bounds for Distributed Distance Computations, Even in Sparse Networks

We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an

Distributed PCP Theorems for Hardness of Approximation in P

\widetilde{\Omega}(n)

Subtree isomorphism revisited

lower bound for computing the diameter in sparse networks, which was previously known only for dense networks [Frishknecht et al., SODA 2012]. In fact, we can even modify our construction to obtain graphs with constant degree, using a simple but powerful degree-reduction technique which we define. Moreover, our technique allows us to show

Losing Weight by Gaining Edges

\widetilde{\Omega}(n)