SETH-based Lower Bounds for Subset Sum and Bicriteria Path

Abstract

Subset Sumand 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 ) pseudo-polynomial time algorithm by Joksch from 1966 cannot be improved to Õ( L + m ), in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017).

Current Algorithms for Detecting Subgraphs of Bounded Treewidth are Probably Optimal

“…To do the latter, we need to encode certain equality constraints only via weights. This can be done using so-called k-average free sets We use the following construction for k-average free sets, originally proven in [16], modified into a more useful version in [7] and formulated in this form in [4].…”

“…The homomorphism f simulates the colors, with all the vertices in a preimage of f being of equal color (which is unique over all preimages). The Exact Weight Colored Subgraph Isomorphism is defined analogously 4 . The weight function is always be denoted by w.…”

“…Partial sketch of the 3-wide Twin Water Lily of order (6,4). The dashed edges are not actual edges, they just represent which other vertices are in the set {v1, .…”

“…We use the following construction for k-average free sets, originally proven in [16], modified into a more useful version in [7] and formulated in this form in [4].…”

“…In [4], a slightly better lower bound for Subset Sum is proven under SETH: They prove that unless SETH fails, Subset Sum cannot have an algorithm running in time O(T 1−ε 2 o(n) ).…”