arXiv subjects

Pavel Veselý

Publications and source records attributed to Pavel Veselý.

24 records · Page 2Linked to original sources

arXiv2017

Online Chromatic Number is PSPACE-Complete

In the online graph coloring problem, vertices from a graph G, known in advance, arrive in an online fashion and an algorithm must immediately assign a color to each incoming vertex v so that the revealed graph is properly colored. The exact location of v in the graph G is not known to the algorithm. The online chromatic number of G is the smallest number of colors such that some online algorithm is able to properly color G for any incoming order. We prove that computing the online chromatic number of a graph is PSPACE-complete.

cs.CC↗

arXiv2016

Online Algorithms for Multi-Level Aggregation

In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edge-weighted tree T, and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while for trees of depth 2, it is equivalent to the Joint Replenishment Problem. Aggregation problem for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests. Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for networks of arbitrary (fixed) number of levels. The competitive ratio is O(D^4 2^D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines. We also show several additional lower and upper bound results for some special cases of MLAP, including the Single-Phase variant and the case when the tree is a path.

cs.DS↗

arXiv2016

A Two-Phase Algorithm for Bin Stretching with Stretching Factor 1.5

Online Bin Stretching is a semi-online variant of bin packing in which the algorithm has to use the same number of bins as an optimal packing, but is allowed to slightly overpack the bins. The goal is to minimize the amount of overpacking, i.e., the maximum size packed into any bin. We give an algorithm for Online Bin Stretching with a stretching factor of 1.5 for any number of bins. We build on previous algorithms and use a two-phase approach. However, our analysis is technically more complicated and uses amortization over the bins with the help of two weight functions.

cs.DS↗

arXiv2016

Online Packet Scheduling with Bounded Delay and Lookahead

We study the online bounded-delay packet scheduling problem (BDPS), where packets of unit size arrive at a router over time and need to be transmitted over a network link. Each packet has two attributes: a non-negative weight and a deadline for its transmission. The objective is to maximize the total weight of the transmitted packets. This problem has been well studied in the literature, yet its optimal competitive ratio remains unknown: the best upper bound is $1.828$, still quite far from the best lower bound of $ϕ\approx 1.618$. In the variant of BDPS with $s$-bounded instances, each packet can be scheduled in at most $s$ consecutive slots, starting at its release time. The lower bound of $ϕ$ applies even to the special case of $2$-bounded instances, and a $ϕ$-competitive algorithm for $3$-bounded instances was given in Chin et al. Improving that result, and addressing a question posed by Goldwasser, we present a $ϕ$-competitive algorithm for $4$-bounded instances. We also study a variant of BDPS where an online algorithm has the additional power of $1$-lookahead, knowing at time $t$ which packets will arrive at time $t+1$. For BDPS with $1$-lookahead restricted to $2$-bounded instances, we present an online algorithm with competitive ratio $(\sqrt{13} - 1)/2 \approx 1.303$ and we prove a nearly tight lower bound of $(1 + \sqrt{17})/4 \approx 1.281$.

cs.DS↗

arXiv2016

Online Bin Stretching with Three Bins

Online Bin Stretching is a semi-online variant of bin packing in which the algorithm has to use the same number of bins as an optimal packing, but is allowed to slightly overpack the bins. The goal is to minimize the amount of overpacking, i.e., the maximum size packed into any bin. We give an algorithm for Online Bin Stretching with a stretching factor of $11/8 = 1.375$ for three bins. Additionally, we present a lower bound of $45/33 = 1.\overline{36}$ for Online Bin Stretching on three bins and a lower bound of $19/14$ for four and five bins that were discovered using a computer search.

cs.DS↗

arXiv2014

Online Colored Bin Packing

In the Colored Bin Packing problem a sequence of items of sizes up to $1$ arrives to be packed into bins of unit capacity. Each item has one of $c\geq 2$ colors and an additional constraint is that we cannot pack two items of the same color next to each other in the same bin. The objective is to minimize the number of bins. In the important special case when all items have size zero, we characterize the optimal value to be equal to color discrepancy. As our main result, we give an (asymptotically) 1.5-competitive algorithm which is optimal. In fact, the algorithm always uses at most $\lceil1.5\cdot OPT\rceil$ bins and we show a matching lower bound of $\lceil1.5\cdot OPT\rceil$ for any value of $OPT\geq 2$. In particular, the absolute ratio of our algorithm is $5/3$ and this is optimal. For items of unrestricted sizes we give an asymptotically $3.5$-competitive algorithm. When the items have sizes at most $1/d$ for a real $d \geq 2$ the asymptotic competitive ratio is $1.5+d/(d-1)$. We also show that classical algorithms First Fit, Best Fit and Worst Fit are not constant competitive, which holds already for three colors and small items. In the case of two colors---the Black and White Bin Packing problem---we prove that all Any Fit algorithms have absolute competitive ratio $3$. When the items have sizes at most $1/d$ for a real $d \geq 2$ we show that the Worst Fit algorithm is absolutely $(1+d/(d-1))$-competitive.

cs.DS↗