42 research outputs found
Deterministic Primal-Dual Algorithms for Online k-way Matching with Delays
In this paper, we study the Min-cost Perfect -way Matching with Delays
(-MPMD), recently introduced by Melnyk et al. In the problem, requests
arrive one-by-one over time in a metric space. At any time, we can irrevocably
make a group of requests who arrived so far, that incurs the distance cost
among the requests in addition to the sum of the waiting cost for the
requests. The goal is to partition all the requests into groups of
requests, minimizing the total cost. The problem is a generalization of the
min-cost perfect matching with delays (corresponding to -MPMD). It is known
that no online algorithm for -MPMD can achieve a bounded competitive ratio
in general, where the competitive ratio is the worst-case ratio between its
performance and the offline optimal value. On the other hand, -MPMD is known
to admit a randomized online algorithm with competitive ratio
for a certain class of -point metrics called the -metric, where is
the size of the metric space. In this paper, we propose a deterministic online
algorithm with a competitive ratio of for the -MPMD in -metric
space. Furthermore, we show that the competitive ratio can be improved to if the metric is given as a diameter on a line
Market Pricing for Matroid Rank Valuations
In this paper, we study the problem of maximizing social welfare in
combinatorial markets through pricing schemes. We consider the existence of
prices that are capable to achieve optimal social welfare without a central
tie-breaking coordinator. In the case of two buyers with rank valuations, we
give polynomial-time algorithms that always find such prices when one of the
matroids is a simple partition matroid or both matroids are strongly base
orderable. This result partially answers a question raised by D\"uetting and
V\'egh in 2017. We further formalize a weighted variant of the conjecture of
D\"uetting and V\'egh, and show that the weighted variant can be reduced to the
unweighted one based on the weight-splitting theorem for weighted matroid
intersection by Frank. We also show that a similar reduction technique works
for M-concave functions, or equivalently, gross substitutes
functions
Complexity of the Multi-Service Center Problem
The multi-service center problem is a variant of facility location problems. In the problem, we consider locating p facilities on a graph, each of which provides distinct service required by all vertices. Each vertex incurs the cost determined by the sum of the weighted distances to the p facilities. The aim of the problem is to minimize the maximum cost among all vertices. This problem is known to be NP-hard for general graphs, while it is solvable in polynomial time when p is a fixed constant. In this paper, we give sharp analyses for the complexity of the problem from the viewpoint of graph classes and weights on vertices. We first propose a polynomial-time algorithm for trees when p is a part of input. In contrast, we prove that the problem becomes strongly NP-hard even for cycles. We also show that when vertices are allowed to have negative weights, the problem becomes NP-hard for paths of only three vertices and strongly NP-hard for stars
Parameterized Complexity of Sparse Linear Complementarity Problems
In this paper, we study the parameterized complexity of the linear complementarity problem (LCP), which is one of the most fundamental mathematical optimization problems. The parameters we focus on are the sparsities of the input and the output of the LCP: the maximum numbers of nonzero entries per row/column in the coefficient matrix and the number of nonzero entries in a solution. Our main result is to present a fixed-parameter algorithm for the LCP with all the parameters. We also show that if we drop any of the three parameters, then the LCP is fixed-parameter intractable.
In addition, we discuss the nonexistence of a polynomial kernel for the LCP
Shortest Reconfiguration of Perfect Matchings via Alternating Cycles
Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar
Streaming Algorithms for Maximizing Monotone Submodular Functions under a Knapsack Constraint
In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in the streaming setting. In particular, the elements arrive sequentially and at any point of time, the algorithm has access only to a small fraction of the data stored in primary memory. For this problem, we propose a (0.363-epsilon)-approximation algorithm, requiring only a single pass through the data; moreover, we propose a (0.4-epsilon)-approximation algorithm requiring a constant number of passes through the data. The required memory space of both algorithms depends only on the size of the knapsack capacity and epsilon