Subset Sum (The Subset-Sum Problem)

Description

Given a set $S$ of integers and a target sum $t$, determine whether there is a subset of $S$ that sum to $t$.

S: the set of integers

n: the number of integers in the set

n': the number of distinct elements in the set

t: the target sum

σ: sum of elements in the set

Name	Year	Time	Space	Approximation Factor	Model	Reference
Pisinger	2003	$O(nt/logt)$	$O(t/logt)$	Exact	Deterministic	Time & Space
Faaland	1973	$O(n' t)$	$O(t)$	Exact	Deterministic	Time & Space
Pferschy	1999	$O(n' t)$	$O(t)$	Exact	Deterministic	Time & Space
Klinz	1999	$O(σ^{({3}/{2})})$	$O(t)$	Exact	Deterministic	Time & Space
Epstein	1997	$\tilde{O}(n max(S))$	$O(t logt)$	Exact	Deterministic	Time & Space
Serang	2014	$\tilde{O}(n max(S))$	$O(t logt)$	Exact	Deterministic	Time & Space
Serang	2015	$\tilde{O}(n max(S))$	$O(t logt)$	Exact	Deterministic	Time & Space
Lokshtanov	2010	$\tilde{O}(n^{3} t)$	$O(n^{2})$	Exact	Deterministic	Time & Space
Horowitz and Sahni	1974	$O({2}^{(n/{2})})$	$O({2}^{(n/{2})$})	Exact	Deterministic	Time & Space
Bellman dynamic programming algorithm	1956	$O(n t)$	$O(t)$	Exact	Deterministic	Time & Space
Psinger	1999	$O(n max(S))$	$O(t)$	Exact	Deterministic	Time & Space
Koiliaris and Xu	2019	$\tilde{O}(min{\sqrt{n'}t, t^{5/4}, σ})$	$O(t)$	Exact	Deterministic	Time & Space
Bringman	2017	$\tilde{O}(nt^{1+\epsilon})$	\tilde{O}(nt^\epsilon)	(n+t)^{-\Omega(1)} error	Randomized	Time & Space
Naive algorithm	1940	$O({2}^n * n)$	$O(n)$ auxiliary	Exact	Deterministic	Space
Random Split Exponential algorithm	1940	$O({2}^{(n/{2})} * n)$	$O({2}^{(n/{2})})$	Exact	Deterministic	Space

Problem	Implication	Year	Citation	Reduction
k-SAT	assume: SETH then: for any $\epsilon > {0}$ there exists a $\delta > {0}$ such that Subset Sum is not in time $O(T^{1-\epsilon}{2}^{\delta n})$, and $k$-Sum is not in time $O(T^{1-\epsilon}n^{\delta k})$	2022	https://dl.acm.org/doi/full/10.1145/3450524	link