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On Approximating Target Set Selection
We study the Target Set Selection (TSS) problem introduced by Kempe, Kleinberg, and Tardos (2003). This problem models the propagation of influence in a network, in a sequence of rounds. A set of nodes is made "active" initially. In each subsequent round, a vertex is activated if at least a certain number of its neighbors are (already) active. In the minimization version, the goal is to activate a small set of vertices initially - a seed, or target, set - so that activation spreads to the entire graph. In the absence of a sublinear-factor algorithm for the general version, we provide a (sublinear) approximation algorithm for the bounded-round version, where the goal is to activate all the vertices in r rounds. Assuming a known conjecture on the hardness of Planted Dense Subgraph, we establish hardness-of-approximation results for the bounded-round version. We show that they translate to general Target Set Selection, leading to a hardness factor of n^(1/2-epsilon) for all epsilon > 0. This is the first polynomial hardness result for Target Set Selection, and the strongest conditional result known for a large class of monotone satisfiability problems. In the maximization version of TSS, the goal is to pick a target set of size k so as to maximize the number of nodes eventually active. We show an n^(1-epsilon) hardness result for the undirected maximization version of the problem, thus establishing that the undirected case is as hard as the directed case. Finally, we demonstrate an SETH lower bound for the exact computation of the optimal seed set
Unique key Horn functions
Given a relational database, a key is a set of attributes such that a value
assignment to this set uniquely determines the values of all other attributes.
The database uniquely defines a pure Horn function , representing the
functional dependencies. If the knowledge of the attribute values in set
determines the value for attribute , then is an implicate
of . If is a key of the database, then is an implicate
of for all attributes .
Keys of small sizes play a crucial role in various problems. We present
structural and complexity results on the set of minimal keys of pure Horn
functions. We characterize Sperner hypergraphs for which there is a unique pure
Horn function with the given hypergraph as the set of minimal keys.
Furthermore, we show that recognizing such hypergraphs is co-NP-complete
already when every hyperedge has size two. On the positive side, we identify
several classes of graphs for which the recognition problem can be decided in
polynomial time.
We also present an algorithm that generates the minimal keys of a pure Horn
function with polynomial delay. By establishing a connection between keys and
target sets, our approach can be used to generate all minimal target sets with
polynomial delay when the thresholds are bounded by a constant. As a byproduct,
our proof shows that the Minimum Key problem is at least as hard as the Minimum
Target Set Selection problem with bounded thresholds.Comment: 12 pages, 5 figure
Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment
We provide new approximation algorithms for the Red-Blue Set Cover and
Circuit Minimum Monotone Satisfying Assignment (MMSA) problems. Our algorithm
for Red-Blue Set Cover achieves -approximation improving on
the -approximation due to Elkin and Peleg (where is the
number of sets). Our approximation algorithm for MMSA (for circuits of
depth ) gives an approximation for , where is the number of gates and
variables. No non-trivial approximation algorithms for MMSA with
were previously known.
We complement these results with lower bounds for these problems: For
Red-Blue Set Cover, we provide a nearly approximation preserving reduction from
Min -Union that gives an hardness
under the Dense-vs-Random conjecture, while for MMSA we sketch a proof that an
SDP relaxation strengthened by Sherali--Adams has an integrality gap of
where as the circuit depth .Comment: APPROX 202
On approximating the rank of graph divisors
Baker and Norine initiated the study of graph divisors as a graph-theoretic analogue of
the Riemann-Roch theory for Riemann surfaces. One of the key concepts of graph divisor
theory is the rank of a divisor on a graph. The importance of the rank is well illustrated by
Bakerâs Specialization lemma, stating that the dimension of a linear system can only go up
under specialization from curves to graphs, leading to a fruitful interaction between divisors
on graphs and curves.
Due to its decisive role, determining the rank is a central problem in graph divisor theory.
Kiss and TÂŽothmÂŽeresz reformulated the problem using chip-firing games, and showed that
computing the rank of a divisor on a graph is NP-hard via reduction from the Minimum
Feedback Arc Set problem.
In this paper, we strengthen their result by establishing a connection between chip-firing
games and the Minimum Target Set Selection problem. As a corollary, we show that the rank
is difficult to approximate to within a factor of O(2log1âΔ n) for any Δ > 0 unless P = N P .
Furthermore, assuming the Planted Dense Subgraph Conjecture, the rank is difficult to
approximate to within a factor of O(n1/4âΔ) for any Δ > 0