4 research outputs found
The Set Cover Conjecture and Subgraph Isomorphism with a Tree Pattern
In the Set Cover problem, the input is a ground set of n elements and a collection of m sets, and the goal is to find the smallest sub-collection of sets whose union is the entire ground set. The fastest algorithm known runs in time O(mn2^n) [Fomin et al., WG 2004], and the Set Cover Conjecture (SeCoCo) [Cygan et al., TALG 2016] asserts that for every fixed epsilon>0, no algorithm can solve Set Cover in time 2^{(1-epsilon)n} poly(m), even if set sizes are bounded by Delta=Delta(epsilon). We show strong connections between this problem and kTree, a special case of Subgraph Isomorphism where the input is an n-node graph G and a k-node tree T, and the goal is to determine whether G has a subgraph isomorphic to T.
First, we propose a weaker conjecture Log-SeCoCo, that allows input sets of size Delta=O(1/epsilon * log n), and show that an algorithm breaking Log-SeCoCo would imply a faster algorithm than the currently known 2^n poly(n)-time algorithm [Koutis and Williams, TALG 2016] for Directed nTree, which is kTree with k=n and arbitrary directions to the edges of G and T. This would also improve the running time for Directed Hamiltonicity, for which no algorithm significantly faster than 2^n poly(n) is known despite extensive research.
Second, we prove that if p-Partial Cover, a parameterized version of Set Cover that requires covering at least p elements, cannot be solved significantly faster than 2^n poly(m) (an assumption even weaker than Log-SeCoCo) then kTree cannot be computed significantly faster than 2^k poly(n), the running time of the Koutis and Williams\u27 algorithm
The Asymptotic Rank Conjecture and the Set Cover Conjecture are not Both True
Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)] claims a
strong submultiplicative upper bound on the rank of a three-tensor obtained as
an iterated Kronecker product of a constant-size base tensor. The conjecture,
if true, most notably would put square matrix multiplication in quadratic time.
We note here that some more-or-less unexpected algorithmic results in the area
of exponential-time algorithms would also follow. Specifically, we study the
so-called set cover conjecture, which states that for any there
exists a positive integer constant such that no algorithm solves the
-Set Cover problem in worst-case time . The -Set Cover problem asks, given as input an
-element universe , a family of size-at-most- subsets of
, and a positive integer , whether there is a subfamily of at most
sets in whose union is . The conjecture was formulated by Cygan
et al. in the monograph Parameterized Algorithms [Springer, 2015] but was
implicit as a hypothesis already in Cygan et al. [CCC 2012, ACM Trans.
Algorithms 2016], there conjectured to follow from the Strong Exponential Time
Hypothesis. We prove that if the asymptotic rank conjecture is true, then the
set cover conjecture is false. Using a reduction by Krauthgamer and Trabelsi
[STACS 2019], in this scenario we would also get a
-time randomized algorithm for some constant
for another well-studied problem for which no such algorithm is
known, namely that of deciding whether a given -vertex directed graph has a
Hamiltonian cycle