81 research outputs found
The Feedback Arc Set Problem with Triangle Inequality is a Vertex Cover Problem
We consider the (precedence constrained) Minimum Feedback Arc Set problem
with triangle inequalities on the weights, which finds important applications
in problems of ranking with inconsistent information. We present a surprising
structural insight showing that the problem is a special case of the minimum
vertex cover in hypergraphs with edges of size at most 3. This result leads to
combinatorial approximation algorithms for the problem and opens the road to
studying the problem as a vertex cover problem
Notes on Max Flow Time Minimization with Controllable Processing Times
In a scheduling problem with controllable processing times the job processing time can be compressed through incurring an additional cost. We consider the identical parallel machines max flow time minimization problem with controllable processing times. We address the preemptive and non-preemptive version of the problem. For the preemptive case, a linear programming formulation is presented which solves the problem optimally in polynomial time. For the non-preemptive problem it is shown that the First In First Out (FIFO) heuristic has a tight worst-case performance of 3−2/m, when jobs processing times and costs are set as in some optimal preemptive schedul
Tight Sum-of-Squares lower bounds for binary polynomial optimization problems
We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre
hierarchy. For binary polynomial optimization problems of degree and an
odd number of variables , we prove that levels of the
SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This
matches the recent upper bound result by Sakaue, Takeda, Kim and Ito.
Additionally, we study a conjecture by Laurent, who considered the linear
representation of a set with no integral points. She showed that the
Sherali-Adams hierarchy requires levels to detect the empty integer hull,
and conjectured that the SoS/Lasserre rank for the same problem is . We
disprove this conjecture and derive lower and upper bounds for the rank
The Feedback Arc Set Problem with Triangle Inequality Is a Vertex Cover Problem
We consider the (precedence constrained) Minimum Feedback Arc Set problem with triangle inequalities on the weights, which finds important applications in problems of ranking with inconsistent information. We present a surprising structural insight showing that the problem is a special case of the minimum vertex cover in hypergraphs with edges of size at most 3
Bi-Criteria and Approximation Algorithms for Restricted Matchings
In this work we study approximation algorithms for the \textit{Bounded Color
Matching} problem (a.k.a. Restricted Matching problem) which is defined as
follows: given a graph in which each edge has a color and a profit
, we want to compute a maximum (cardinality or profit)
matching in which no more than edges of color are
present. This kind of problems, beside the theoretical interest on its own
right, emerges in multi-fiber optical networking systems, where we interpret
each unique wavelength that can travel through the fiber as a color class and
we would like to establish communication between pairs of systems. We study
approximation and bi-criteria algorithms for this problem which are based on
linear programming techniques and, in particular, on polyhedral
characterizations of the natural linear formulation of the problem. In our
setting, we allow violations of the bounds and we model our problem as a
bi-criteria problem: we have two objectives to optimize namely (a) to maximize
the profit (maximum matching) while (b) minimizing the violation of the color
bounds. We prove how we can "beat" the integrality gap of the natural linear
programming formulation of the problem by allowing only a slight violation of
the color bounds. In particular, our main result is \textit{constant}
approximation bounds for both criteria of the corresponding bi-criteria
optimization problem
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