Local Search Breaks 1.75 for Graph Balancing

Graph Balancing is the problem of orienting the edges of a weighted multigraph so as to minimize the maximum weighted in-degree. Since the introduction of the problem the best algorithm known achieves an approximation ratio of 1.75 and it is based on rounding a linear program with this exact integrality gap. It is also known that there is no (1.5 - epsilon)-approximation algorithm, unless P=NP. Can we do better than 1.75? We prove that a different LP formulation, the configuration LP, has a strictly smaller integrality gap. Graph Balancing was the last one in a group of related problems from literature, for which it was open whether the configuration LP is stronger than previous, simple LP relaxations. We base our proof on a local search approach that has been applied successfully to the more general Restricted Assignment problem, which in turn is a prominent special case of makespan minimization on unrelated machines. With a number of technical novelties we are able to obtain a bound of 1.749 for the case of Graph Balancing. It is not clear whether the local search algorithm we present terminates in polynomial time, which means that the bound is non-constructive. However, it is a strong evidence that a better approximation algorithm is possible using the configuration LP and it allows the optimum to be estimated within a factor better than 1.75. A particularly interesting aspect of our techniques is the way we handle small edges in the local search. We manage to exploit the configuration constraints enforced on small edges in the LP. This may be of interest to other problems such as Restricted Assignment as well

On the Configuration LP for Maximum Budgeted Allocation

We study the Maximum Budgeted Allocation problem, i.e., the problem of selling a set of $m$ indivisible goods to $n$ players, each with a separate budget, such that we maximize the collected revenue. Since the natural assignment LP is known to have an integrality gap of $\frac{3}{4}$, which matches the best known approximation algorithms, our main focus is to improve our understanding of the stronger configuration LP relaxation. In this direction, we prove that the integrality gap of the configuration LP is strictly better than $\frac{3}{4}$, and provide corresponding polynomial time roundings, in the following restrictions of the problem: (i) the Restricted Budgeted Allocation problem, in which all the players have the same budget and every item has the same value for any player it can be sold to, and (ii) the graph MBA problem, in which an item can be assigned to at most 2 players. Finally, we improve the best known upper bound on the integrality gap for the general case from $\frac{5}{6}$ to $2\sqrt{2}-2\approx 0.828$ and also prove hardness of approximation results for both cases.Comment: 29 pages, 4 figures. To appear in the 17th Conference on Integer Programming and Combinatorial Optimization (IPCO), 201

Estimating The Makespan of The Two-Valued Restricted Assignment Problem

We consider a special case of the scheduling problem on unrelated machines,namely the Restricted Assignment Problem with two different processing times.We show that the configuration LP has an integrality gap of at most~$\frac{5}{3} + \frac{1}{156} \approx 1.6731$ for this problem. This allows us to estimate the optimal makespan within a factor of~$\frac{5}{3} + \frac{1}{156}$,improving upon the previously best known estimation algorithm with ratio~\frac{11}{6} \approx \numprint{1.833} due to Chakrabarty, Khanna, and Li \cite{CKL15}

On the Configuration-LP for Scheduling on Unrelated Machines

One of the most important open problems in machine scheduling is the problem of scheduling a set of jobs on unrelated machines to minimize the makespan. The best known approximation algorithm for this problem guarantees an approximation factor of 2. It is known to be NP-hard to approximate with a better ratio than 3/2. Closing this gap has been open for over 20 years. The best known approximation factors are achieved by LP-based algorithms. The strongest known linear program formulation for the problem is the configuration-LP. We show that the configuration-LP has an integrality gap of 2 even for the special case of unrelated graph balancing, where each job can be assigned to at most two machines. In particular, our result implies that a large family of cuts does not help to diminish the integrality gap of the canonical assignment-LP. Also, we present cases of the problem which can be approximated with a better factor than 2. They constitute valuable insights for constructing an NP-hardness reduction which improves the known lowerbound. Very recently Svensson [22] studied the restricted assignment case, where each job can only be assigned to a given set of machines on which it has the same processing time. He shows that in this setting the configuration-LP has an integrality gap of 33/17≈1.94. Hence, our result imply that the unrelated graph balancing case is significantly more complex than the restricted assignment case. Then we turn to another objective function: maximizing the minimum machine load. For the case that every job can be assigned to at most two machines we give a purely combinatorial 2-approximation which is best possible, unless P=NP. This improves on the computationally costly LP-based (2 +ε)-approximation algorithm by Chakrabarty et al. [7]

Algorithms for Integer Programming and Allocation

The first part of the thesis contains pseudo-polynomial algorithms for integer linear programs (ILP). When certain parameters of an ILP are fixed, that is, they are treated as constants in the running time, it is possible to obtain algorithms with a running time that is pseudo-polynomial in the entries of the ILP’s matrix. We present a tight pseudo-polynomial running time for ILPs with a constant number of constraints. Furthermore, we study an extension of this model to MILPs (linear programs that contain both fractional and integer variables). Then we move to n-fold ILPs, a class of ILPs with block structured matrices. We present the first algorithm for n-folds, which is near-linear in the dimensions of the ILP. The second part is about scheduling in non-identical machine models, more precisely, restricted allocation problems. Here a set of jobs has to be allocated to a set of machines. However, every job has a subset of machines and may only be assigned to a machine from this subset. We consider the objectives of minimizing the makespan or maximizing the minimum load. We study the integrality gap of a particularly strong linear programming relaxation, the configuration LP, for variations of this problem. The integrality gap can be seen as a measure of strength of an LP relaxation. A local search technique can be used to bound this value. However, the proofs are generally non-constructive, i.e., they do not give an efficient approximation algorithm right away. We derive better upper bounds on the integrality gap of the problems Restricted Assignment, Restricted Santa Claus, and Graph Balancing. Furthermore, we give the first (constructive) quasi-polynomial time approximation algorithm for Restricted Assignment with an approximation ratio strictly less than 2.Der erste Teil der Thesis umfasst pseudopolynomielle Algorithmen für ganzzahlige lineare Programme (ILP). Wenn bestimmte Parameter eines ILPs fixiert sind, d.h. sie werden in der Laufzeit als Konstanten betrachtet, dann ist es möglich Algorithmen zu entwerfen, deren Laufzeit pseudopolynomiell in dem größten absoluten Wert eines Eintrags der Matrix des ILPs ist. Ein Ergebnis, das wir präsentieren, ist eine scharfe Schranke für die pseudopolynomielle Laufzeit, die nötig ist um ein ILP mit konstant vielen Bedingungen zu lösen. Danach befassen wir uns mit n-fold ILPs, eine Klasse von ILPs, deren matrix eine Blockstruktur besitzt. Wir geben den ersten Algorithmus für n-folds an, dessen Laufzeit gleichzeitig nahezu linear in der Dimension des ILPs ist. Der zweite Teil handelt von nicht-identischen (heterogenen) Maschinen Modellen, genauer gesagt restricted allocation problems. Hier soll eine Menge von Jobs auf eine Menge von Maschinen verteilt werden. Jeder Job darf aber nur auf bestimmte Maschinen zugewiesen werden. Wir betrachten als Zielfunktionen sowohl die Minimierung des Makespans als auch die Maximierung der minimalen Last einer Maschine. Wir untersuchen den integrality gap einer besonders starken LP Relaxierung, dem Konfigurations LP, für Variationen dieses Problems. Der integrality gap kann als Maß für die Stärke einer LP Relaxierung gesehen werden. Über ein Argument mittels einer lokalen Suche wird dieser Wert beschränkt. Jedoch sind die Beweise typischerweise nicht konstruktiv, d.h. sie implizieren nicht direkt effiziente Approximationsalgorithmen. Wir beweisen neue obere Schranken an den integrality gap für die Probleme Restricted Assignment, Restricted Santa Claus und Graph Balancing. Desweiteren präsentieren wir den ersten (konstruktiven) Quasipolynomialzeit Approximationsalgorithmus für das Restricted Assignment Problem mit Approximationsrate echt kleiner als 2