Subset Sum and k-SAT are two of the most extensively studied problems in computer science, and conjectures about their hardness are among the cornerstones of fine-grained complexity.An important open problem in this area is to base the hardness of one of these problems on the other.Our main result is a tight reduction from k-SAT to Subset Sum on dense instances, proving that Bellman's 1962 pseudo-polynomial O * (T )-time algorithm for Subset Sum on n numbers and target T cannot be improved to time T 1−ε · 2 o(n) for any ε > 0, unless the Strong Exponential Time Hypothesis (SETH) fails.As a corollary, we prove a "Direct-OR" theorem for Subset Sum under SETH, offering a new tool for proving conditional lower bounds: It is now possible to assume that deciding whether one out of N given instances of Subset Sum is a YES instance requires time (N T ) 1−o(1) . As an application of this corollary, we prove a tight SETH-based lower bound for the classical Bicriteria s, t-Path problem, which is extensively studied in Operations Research. We separate its complexity from that of Subset Sum: On graphs with m edges and edge lengths bounded by L, we show that the O(Lm) pseudopolynomial time algorithm by Joksch from 1966 cannot be improved toÕ(L + m), in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017).Subset Sum. Subset Sum is one of the most fundamental problems in computer science. Its most basic form is the following: given n integers x 1 , . . . , x n ∈ N, and a target value T ∈ N, decide whether there is a subset of the numbers that sums to T . The two most classical algorithms for the problem are the pseudo-polynomial O(T n) algorithm using dynamic programming [28], and the O(2 n/2 · poly(n, log T )) algorithm via "meet-in-the-middle" [67]. A central open question in Exact Algorithms [114] is whether faster algorithms exist, e.g., can we combine the two approaches to get a T 1/2 · n O(1) time algorithm? Such a bound was recently found in a Merlin-Arthur setting [94].The status of Subset Sum as a major problem has been established due to many applications, deep connections to other fields, and educational value. The O(T n) algorithm from 1957 is an illuminating example of dynamic programming that is taught in most undergraduate algorithms courses, and the NP-hardness proof (from Karp's original paper [78]) is a prominent example of a reduction to a problem on numbers. Interestingly, one of the earliest cryptosystems by Merkle and Hellman was based on Subset Sum [92], and was later extended to a host of Knapsack-type cryptosystems 4 (see [106,31,97,46,69] and the references therein).The version of Subset Sum where we ask for k numbers that sum to zero (the k-SUM problem) is conjectured to have n ⌈k/2⌉±o(1) time complexity. Most famously, the k = 3 case is the 3-SUM conjecture highlighted in the seminal work of Gajentaan and Overmars [57]. It has been shown that this problem lies at the core and captures the difficulty of dozens of problems in computational geometry. Searching in Google Scholar for "3su...
