Solving kk-SUM using few linear queries

The $k$-SUM problem is given $n$ input real numbers to determine whether any $k$ of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within $P$, and it is in particular open whether it admits an algorithm of complexity $O(n^c)$ with $c<\lceil \frac{k}{2} \rceil$. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth $O(n^3\log^3 n)$ solving $k$-SUM. Furthermore, we show that there exists a randomized algorithm that runs in $\tilde{O}(n^{\lceil \frac{k}{2} \rceil+8})$ time, and performs $O(n^3\log^3 n)$ linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the $+8$) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of $k$. The $O(n^3\log^3 n)$ bound on the number of linear queries is also a tighter bound than any known algorithm solving $k$-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-\`{a}-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-$P$. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist $o(n)$-linear decision trees of depth $o(n^4)$

Solving k-SUM Using Few Linear Queries

The k-SUM problem is given n input real numbers to determine whether any k of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within P, and it is in particular open whether it admits an algorithm of complexity O(n^c) with c<d where d is the ceiling of k/2. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth O(n^3 log^2 n) solving k-SUM. Furthermore, we show that there exists a randomized algorithm that runs in ~O(n^{d+8}) time, and performs O(n^3 log^2 n) linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the +8) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of k. The O(n^3 log^2 n) bound on the number of linear queries is also a tighter bound than any known algorithm solving k-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-a-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-P. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist o(n)-linear decision trees of depth ~O(n^3) for the k-SUM problem

Hardness of Easy Problems: Basing Hardness on Popular Conjectures such as the Strong Exponential Time Hypothesis (Invited Talk)

Algorithmic research strives to develop fast algorithms for fundamental problems. Despite its many successes, however, many problems still do not have very efficient algorithms. For years researchers have explained the hardness for key problems by proving NP-hardness, utilizing polynomial time reductions to base the hardness of key problems on the famous conjecture P != NP. For problems that already have polynomial time algorithms, however, it does not seem that one can show any sort of hardness based on P != NP. Nevertheless, we would like to provide evidence that a problem $A$ with a running time O(n^k) that has not been improved in decades, also requires n^{k-o(1)} time, thus explaining the lack of progress on the problem. Such unconditional time lower bounds seem very difficult to obtain, unfortunately. Recent work has concentrated on an approach mimicking NP-hardness: (1) select a few key problems that are conjectured to require T(n) time to solve, (2) use special, fine-grained reductions to prove time lower bounds for many diverse problems in P based on the conjectured hardness of the key problems. In this abstract we outline the approach, give some examples of hardness results based on the Strong Exponential Time Hypothesis, and present an overview of some of the recent work on the topic

On Nondeterministic Derandomization of Freivalds\u27 Algorithm: Consequences, Avenues and Algorithmic Progress

Motivated by studying the power of randomness, certifying algorithms and barriers for fine-grained reductions, we investigate the question whether the multiplication of two n x n matrices can be performed in near-optimal nondeterministic time O~(n^2). Since a classic algorithm due to Freivalds verifies correctness of matrix products probabilistically in time O(n^2), our question is a relaxation of the open problem of derandomizing Freivalds\u27 algorithm. We discuss consequences of a positive or negative resolution of this problem and provide potential avenues towards resolving it. Particularly, we show that sufficiently fast deterministic verifiers for 3SUM or univariate polynomial identity testing yield faster deterministic verifiers for matrix multiplication. Furthermore, we present the partial algorithmic progress that distinguishing whether an integer matrix product is correct or contains between 1 and n erroneous entries can be performed in time O~(n^2) - interestingly, the difficult case of deterministic matrix product verification is not a problem of "finding a needle in the haystack", but rather cancellation effects in the presence of many errors. Our main technical contribution is a deterministic algorithm that corrects an integer matrix product containing at most t errors in time O~(sqrt{t} n^2 + t^2). To obtain this result, we show how to compute an integer matrix product with at most t nonzeroes in the same running time. This improves upon known deterministic output-sensitive integer matrix multiplication algorithms for t = Omega(n^{2/3}) nonzeroes, which is of independent interest

Fine-Grained Complexity: Exploring Reductions and their Properties

Η σχεδίαση αλγορίθμων αποτελεί ένα απο τα κύρια θέματα ενδιαφέροντος για τον τομέα της Πληροφορικής. Παρά τα πολλά αποτελέσματα σε ορισμένους τομείς, η προσέγγιση αυτή έχει πετύχει κάποια πρακτικά αδιέξοδα που έχουν αποδειχτεί προβληματικά στην πρόοδο του τομέα. Επίσης, οι κλασικές πρακτικές Υπολογιστικής Πολυπλοκότητας δεν ήταν σε θέση να παρακάμψουν αυτά τα εμπόδια. Η κατανόηση της δυσκολίας του κάθε προβλήματος δεν είναι τετριμμένη. Η Ραφιναρισμένη Πολυπλοκότητα παρέχει νέες προ-οπτικές για τα κλασικά προβλήματα, με αποτέλεσμα σταθερούς δεσμούς μεταξύ γνωστών εικασιών στην πολυπλοκότητα και την σχεδίαση αλγορίθμων. Χρησιμεύει επίσης ως εργα-λείο για να αποδείξει τα υπο όρους κατώτατα όρια για προβλήματα πολυωνυμικής χρονικής πολυπλοκότητας, ένα πεδίο που έχει σημειώσει πολύ λίγη πρόοδο μέχρι τώρα. Οι δημοφι-λείς υποθέσεις/παραδοχές όπως το SETH, το OVH, το 3SUM, και το APSP, δίνουν πολλά φράγματα που δεν έχουν ακόμα αποδειχθεί με κλασικές τεχνικές και παρέχουν μια νέα κατανόηση της δομής και της εντροπίας των προβλημάτων γενικά. Σκοπός αυτής της εργασίας είναι να συμβάλει στην εδραίωση του πλαισίου για αναγωγές από κάθε εικασία και να διερευνήσει την διαρθρωτική διαφορά μεταξύ των προβλημάτων σε κάθε περίπτωση.Algorithmic design has been one of the main subjects of interest for Computer science. While very effective in some areas, this approach has been met with some practical dead ends that have been very problematic in the progress of the field. Classical Computational Complexity practices have also not been able to bypass these blocks. Understanding the hardness of each problem is not trivial. Fine-Grained Complexity provides new perspectives on classic problems, resulting to solid links between famous conjectures in Complexity, and Algorithmic design. It serves as a tool to prove conditional lower bounds for problems with polynomial time complexity, a field that had seen very little progress until now. Popular conjectures such as SETH, k-OV, 3SUM, and APSP, imply many bounds that have yet to be proven using classic techniques, and provide a new understanding of the structure and entropy of problems in general. The aim of this thesis is to contribute towards solidifying the framework for reductions from each conjecture, and to explore the structural difference between the problems in each cas