345 research outputs found
Online Bipartite Matching with Decomposable Weights
We study a weighted online bipartite matching problem: G(V1, V2, E) is a weighted bipartite graph where V1 is known beforehand and the vertices of V2 arrive online. The goal is to match vertices of V2 as they arrive to vertices in V1, so as to maximize the sum of weights of edges in the matching. If assignments to V1 cannot be changed, no bounded competitive ratio is achievable. We study the weighted online matching problem with free disposal, where vertices in V1 can be assigned multiple times, but only get credit for the maximum weight edge assigned to them over the course of the algorithm. For this problem, the greedy algorithm is 0.5-competitive and determining whether a better competitive ratio is achievable is a well known open problem. We identify an interesting special case where the edge weights are decomposable as the product of two factors, one corresponding to each end point of the edge. This is analogous to the well studied related machines model in the scheduling literature, although the objective functions are different. For this case of decomposable edge weights, we design a 0.5664 competitive randomized algorithm in complete bipartite graphs. We show that such instances with decomposable weights are non-trivial by establishing upper bounds of 0.618 for deterministic and 0.8 for randomized algorithms. A tight competitive ratio of 1 − 1/e ≈ 0.632 was known previously for both the 0-1 case as well as the case where edge weights depend on the offline vertices only, but for these cases, reassignments cannot change the quality of the solution. Beating 0.5 for weighted matching where reassignments are necessary has been a significant challenge. We thus give the first online algorithm with competitive ratio strictly better than 0.5 for a non-trivial case of weighted matching with free disposal.
Online Matching with Stochastic Rewards: Optimal Competitive Ratio via Path Based Formulation
The problem of online matching with stochastic rewards is a generalization of
the online bipartite matching problem where each edge has a probability of
success. When a match is made it succeeds with the probability of the
corresponding edge. Introducing this model, Mehta and Panigrahi (FOCS 2012)
focused on the special case of identical edge probabilities. Comparing against
a deterministic offline LP, they showed that the Ranking algorithm of Karp et
al. (STOC 1990) is 0.534 competitive and proposed a new online algorithm with
an improved guarantee of for vanishingly small probabilities. For the
case of vanishingly small but heterogeneous probabilities Mehta et al. (SODA
2015), gave a 0.534 competitive algorithm against the same LP benchmark. For
the more general vertex-weighted version of the problem, to the best of our
knowledge, no results being were previously known even for identical
probabilities.
We focus on the vertex-weighted version and give two improvements. First, we
show that a natural generalization of the Perturbed-Greedy algorithm of
Aggarwal et al. (SODA 2011), is competitive when probabilities
decompose as a product of two factors, one corresponding to each vertex of the
edge. This is the best achievable guarantee as it includes the case of
identical probabilities and in particular, the classical online bipartite
matching problem. Second, we give a deterministic competitive algorithm
for the previously well studied case of fully heterogeneous but vanishingly
small edge probabilities. A key contribution of our approach is the use of
novel path-based analysis. This allows us to compare against the natural
benchmarks of adaptive offline algorithms that know the sequence of arrivals
and the edge probabilities in advance, but not the outcomes of potential
matches.Comment: Preliminary version in EC 202
On Correcting Inputs: Inverse Optimization for Online Structured Prediction
Algorithm designers typically assume that the input data is correct, and then
proceed to find "optimal" or "sub-optimal" solutions using this input data.
However this assumption of correct data does not always hold in practice,
especially in the context of online learning systems where the objective is to
learn appropriate feature weights given some training samples. Such scenarios
necessitate the study of inverse optimization problems where one is given an
input instance as well as a desired output and the task is to adjust the input
data so that the given output is indeed optimal. Motivated by learning
structured prediction models, in this paper we consider inverse optimization
with a margin, i.e., we require the given output to be better than all other
feasible outputs by a desired margin. We consider such inverse optimization
problems for maximum weight matroid basis, matroid intersection, perfect
matchings, minimum cost maximum flows, and shortest paths and derive the first
known results for such problems with a non-zero margin. The effectiveness of
these algorithmic approaches to online learning for structured prediction is
also discussed.Comment: Conference version to appear in FSTTCS, 201
Approximating the multi-level bottleneck assignment problem.
We consider the multi-level bottleneck assignment problem (MBA). This problem is described in the recent book 'Assignment Problems' by Burkard et al. (2009) on pages 188-189. One of the applications described there concerns bus driver scheduling.We view the problem as a special case of a bottleneck m-dimensional multi-index assignment problem. We give approximation algorithms and inapproximability results, depending upon the completeness of the underlying graph. Keywords: bottleneck problem; multidimensional assignment; approximation; computational complexity; efficient algorithm.Bottleneck problem; Multidimensional assignment; Approximation; Computational complexity; Efficient algorithm;
The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems
We study the matroid secretary problems with submodular valuation functions.
In these problems, the elements arrive in random order. When one element
arrives, we have to make an immediate and irrevocable decision on whether to
accept it or not. The set of accepted elements must form an {\em independent
set} in a predefined matroid. Our objective is to maximize the value of the
accepted elements. In this paper, we focus on the case that the valuation
function is a non-negative and monotonically non-decreasing submodular
function.
We introduce a general algorithm for such {\em submodular matroid secretary
problems}. In particular, we obtain constant competitive algorithms for the
cases of laminar matroids and transversal matroids. Our algorithms can be
further applied to any independent set system defined by the intersection of a
{\em constant} number of laminar matroids, while still achieving constant
competitive ratios. Notice that laminar matroids generalize uniform matroids
and partition matroids.
On the other hand, when the underlying valuation function is linear, our
algorithm achieves a competitive ratio of 9.6 for laminar matroids, which
significantly improves the previous result.Comment: preliminary version appeared in STACS 201
Algorithmic Graph Theory
The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions
Follow Your Star: New Frameworks for Online Stochastic Matching with Known and Unknown Patience
We study several generalizations of the Online Bipartite Matching problem. We
consider settings with stochastic rewards, patience constraints, and weights
(both vertex- and edge-weighted variants). We introduce a stochastic variant of
the patience-constrained problem, where the patience is chosen randomly
according to some known distribution and is not known until the point at which
patience has been exhausted. We also consider stochastic arrival settings
(i.e., online vertex arrival is determined by a known random process), which
are natural settings that are able to beat the hard worst-case bounds of more
pessimistic adversarial arrivals.
Our approach to online matching utilizes black-box algorithms for matching on
star graphs under various models of patience. In support of this, we design
algorithms which solve the star graph problem optimally for patience with a
constant hazard rate and yield a 1/2-approximation for any patience
distribution. This 1/2-approximation also improves existing guarantees for
cascade-click models in the product ranking literature, in which a user must be
shown a sequence of items with various click-through-rates and the user's
patience could run out at any time.
We then build a framework which uses these star graph algorithms as black
boxes to solve the online matching problems under different arrival settings.
We show improved (or first-known) competitive ratios for these problems.
Finally, we present negative results that include formalizing the concept of a
stochasticity gap for LP upper bounds on these problems, bounding the
worst-case performance of some popular greedy approaches, and showing the
impossibility of having an adversarial patience in the product ranking setting.Comment: 43 page
Secretary Matching Meets Probing with Commitment
We consider the online bipartite matching problem within the context of stochastic probing with commitment. This is the one-sided online bipartite matching problem where edges adjacent to an online node must be probed to determine if they exist based on edge probabilities that become known when an online vertex arrives. If a probed edge exists, it must be used in the matching. We consider the competitiveness of online algorithms in the adversarial order model (AOM) and the secretary/random order model (ROM). More specifically, we consider an unknown bipartite stochastic graph G = (U,V,E) where U is the known set of offline vertices, V is the set of online vertices, G has edge probabilities (p_{e})_{e ? E}, and G has edge weights (w_{e})_{e ? E} or vertex weights (w_u)_{u ? U}. Additionally, G has a downward-closed set of probing constraints (?_{v})_{v ? V}, where ?_v indicates which sequences of edges adjacent to an online vertex v can be probed. This model generalizes the various settings of the classical bipartite matching problem (i.e. with and without probing). Our contributions include the introduction and analysis of probing within the random order model, and our generalization of probing constraints which includes budget (i.e. knapsack) constraints. Our algorithms run in polynomial time assuming access to a membership oracle for each ?_v.
In the vertex weighted setting, for adversarial order arrivals, we generalize the known 1/2 competitive ratio to our setting of ?_v constraints. For random order arrivals, we show that the same algorithm attains an asymptotic competitive ratio of 1-1/e, provided the edge probabilities vanish to 0 sufficiently fast. We also obtain a strict competitive ratio for non-vanishing edge probabilities when the probing constraints are sufficiently simple. For example, if each ?_v corresponds to a patience constraint ?_v (i.e., ?_v is the maximum number of probes of edges adjacent to v), and any one of following three conditions is satisfied (each studied in previous papers), then there is a conceptually simple greedy algorithm whose competitive ratio is 1-1/e.
- When the offline vertices are unweighted.
- When the online vertex probabilities are "vertex uniform"; i.e., p_{u,v} = p_v for all (u,v) ? E.
- When the patience constraint ?_v satisfies ?_v ? {[1,|U|} for every online vertex; i.e., every online vertex either has unit or full patience. Finally, in the edge weighted case, we match the known optimal 1/e asymptotic competitive ratio for the classic (i.e. without probing) secretary matching problem
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