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 $e$ has a color $c_e$ and a profit $p_e \in \mathbb{Q}^+$, we want to compute a maximum (cardinality or profit) matching in which no more than $w_j \in \mathbb{Z}^+$ edges of color $c_j$ 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 $w_j$ 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

Uniform s-cross-intersecting families

In this paper we study a question related to the classical Erd\H{o}s-Ko-Rado theorem, which states that any family of $k$-element subsets of the set $[n] = \{1,\ldots,n\}$ in which any two sets intersect, has cardinality at most ${n-1\choose k-1}$. We say that two non-empty families are $\mathcal A, \mathcal B\subset {[n]\choose k}$ are {\it $s$-cross-intersecting}, if for any $A\in\mathcal A,B\in \mathcal B$ we have $|A\cap B|\ge s$. In this paper we determine the maximum of $|\mathcal A|+|\mathcal B|$ for all $n$. This generalizes a result of Hilton and Milner, who determined the maximum of $|\mathcal A|+|\mathcal B|$ for nonempty $1$-cross-intersecting families.Comment: This article was previously a portion of arXiv:1603.00938v1, which has been spli

Improved Algorithm for Degree Bounded Survivable Network Design Problem

We consider the Degree-Bounded Survivable Network Design Problem: the objective is to find a minimum cost subgraph satisfying the given connectivity requirements as well as the degree bounds on the vertices. If we denote the upper bound on the degree of a vertex v by b(v), then we present an algorithm that finds a solution whose cost is at most twice the cost of the optimal solution while the degree of a degree constrained vertex v is at most 2b(v) + 2. This improves upon the results of Lau and Singh and that of Lau, Naor, Salavatipour and Singh

Algorithms for Hierarchical and Semi-Partitioned Parallel Scheduling

We propose a model for scheduling jobs in a parallel machine setting that takes into account the cost of migrations by assuming that the processing time of a job may depend on the specific set of machines among which the job is migrated. For the makespan minimization objective, the model generalizes classical scheduling problems such as unrelated parallel machine scheduling, as well as novel ones such as semi-partitioned and clustered scheduling. In the case of a hierarchical family of machines, we derive a compact integer linear programming formulation of the problem and leverage its fractional relaxation to obtain a polynomial-time 2-approximation algorithm. Extensions that incorporate memory capacity constraints are also discussed

ILP-based approaches to partitioning recurrent workloads upon heterogeneous multiprocessors

The problem of partitioning systems of independent constrained-deadline sporadic tasks upon heterogeneous multiprocessor platforms is considered. Several different integer linear program (ILP) formulations of this problem, offering different tradeoffs between effectiveness (as quantified by speedup bound) and running time efficiency, are presented

k-Trails: Recognition, Complexity, and Approximations

The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails. In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve several open questions about k-trails. In particular, we show that one can recognize efficiently whether a graph is a k-trail. Furthermore, we show that deciding whether a graph contains a k-trail is NP-complete; however, every graph that contains a k-trail is a (k+1)-trail. Moreover, further leveraging the connection to matroids, we consider the problem of finding a minimum weight k-trail contained in a graph G. We show that one can efficiently find a (2k-1)-trail contained in G whose weight is no more than the cheapest k-trail contained in G, even when allowing negative weights. The above results settle several open questions raised by Molnar, Newman, and Sebo

Minimizing Flow-Time on Unrelated Machines

We consider some flow-time minimization problems in the unrelated machines setting. In this setting, there is a set of $m$ machines and a set of $n$ jobs, and each job $j$ has a machine dependent processing time of $p_{ij}$ on machine $i$. The flow-time of a job is the total time the job spends in the system (completion time minus its arrival time), and is one of the most natural quality of service measure. We show the following two results: an $O(\min(\log^2 n,\log n \log P))$ approximation algorithm for minimizing the total-flow time, and an $O(\log n)$ approximation for minimizing the maximum flow-time. Here $P$ is the ratio of maximum to minimum job size. These are the first known poly-logarithmic guarantees for both the problems.Comment: The new version fixes some typos in the previous version. The paper is accepted for publication in STOC 201