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

Online algorithms for multi-level aggregation

In the multilevel 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 the root of T. 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 Transmission Control Protocol acknowledgment problem, whereas for trees of depth 2, it is equivalent to the joint replenishment problem. Aggregation problems for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and 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 trees of arbitrary (fixed) depth. The competitive ratio is O(D42D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines

New results on multi-level aggregation

International audienceIn the Multi-Level Aggregation Problem (MLAP ), requests for service arrive at the nodes of an edge-weighted rooted tree . Each service is represented by a subtree X of 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 a waiting cost between its arrival and service time. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. The currently best online algorithms for the MLAP achieve competitive ratios polynomial in the tree depth, while the best lower bound is only 3.618. In this paper, we report some progress towards closing this gap, by improving this lower bound and providing several tight bounds for restricted variants of MLAP: (1) We first study a Single-Phase variant of MLAP where all requests are released at the beginning and expire at some unknown time θ, for which we provide an online algorithm with optimal competitive ratio of 4. (2) We prove a lower bound of 4 on the competitive ratio for MLAP, even when the tree is a path. We complement this with a matching upper bound for the deadline variant of MLAP on paths. Additionally, we provide two results for the offline case: (3) We prove that the Single-Phase variant can be solved optimally in polynomial time, and (4) we give a simple 2-approximation algorithm for offline MLAP with deadlines

On the Hardness of General Caching

Caching (also known as paging) is a classical problem concerning page re- placement policies in two-level memory systems. General caching is its vari- ant with pages of different sizes and fault costs. We aim at a better charac- terization of the computational complexity of general caching in the offline version. General caching in the offline version was recently shown to be strongly NP- hard, but the proof needed instances of caching with pages larger than half of the cache size. The primary result of this work addresses this problem as we prove: General caching is strongly NP-hard even when page sizes are limited to {1, 2, 3}. In the structural part of this work, a new simpler proof for the full characterization of work functions by layers for classical caching is given and then extended to caching with variable cache size. We invent two algorithms for restricted instances of general caching building on results around caching with variable cache size

On the Hardness of General Caching

Caching (also known as paging) is a classical problem concerning page re- placement policies in two-level memory systems. General caching is its vari- ant with pages of different sizes and fault costs. We aim at a better charac- terization of the computational complexity of general caching in the offline version. General caching in the offline version was recently shown to be strongly NP- hard, but the proof needed instances of caching with pages larger than half of the cache size. The primary result of this work addresses this problem as we prove: General caching is strongly NP-hard even when page sizes are limited to {1, 2, 3}. In the structural part of this work, a new simpler proof for the full characterization of work functions by layers for classical caching is given and then extended to caching with variable cache size. We invent two algorithms for restricted instances of general caching building on results around caching with variable cache size

Grafové komunikační protokoly

Graph communication protocols are a generalization of classical communi- cation protocols to the case when the underlying graph is a directed acyclic graph. Motivated by potential applications in proof complexity, we study variants of graph communication protocols and relations between them. The main result is a comparison of the strength of two types of protocols, protocols with equality and protocols with a conjunction of a constant num- ber of inequalities. We prove that protocols of the first type are at least as strong as protocols of the second type in the following sense: For a Boolean function f, if there is a protocol with a conjunction of a constant number of inequalities of polynomial size solving f, then there is a protocol with equality of polynomial size solving f. We also introduce two new types of graph communication protocols, protocols with disjointness and protocols with non-disjointness, and prove that the first type is at least as strong as the previously considered protocols and that the second type is too strong to be useful for applications.Grafové komunikační protokoly jsou zobecněním klasických komunikačních protokolů na případ, kdy je grafem protokolu orientovaný acyklický graf. Motivováni možnými aplikacemi v důkazové složitosti studujeme varianty grafových komunikačních protokolů a vztahy mezi nimi. Hlavním výsledkem je porovnání síly dvou typů protokolů, protokolů s rovností a protokolů s konjunkcí konstantního počtu nerovností. Dokazujeme, že protokoly prvního typu jsou alespoň tak silné jako protokoly druhého typu v následujícím smyslu: Necht' f je booleovská funkce. Pokud existuje protokol s konjunkcí konstantního počtu nerovností polynomiální velikosti řešící f, pak existuje protokol s rovností polynomiální velikosti řešící f. Rovněž zavádíme dva nové typy grafových komunikačních protokolů, protokoly s disjunktností a protokoly s nedisjunktností, a dokazujeme, že protokoly prvního typu jsou alespoň tak silné jako doposud uvažované protokoly a že protokoly druhého typu jsou příliš silné na to, aby mohly být aplikovány.Informatický ústav Univerzity KarlovyComputer Science Institute of Charles UniversityFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

O těžkosti obecného cachování

Cachování (také známo jako stránkování) je klasický problém modelující ob- sluhu dvouúrovňových pamět'ových systémů. Obecné cachování je varianta se stránkami různých velikostí a cen. V práci se zabýváme zpřesněním cha- rakterizace výpočetní složitosti obecného cachování v offline případě. Nedávno bylo dokázáno, že obecné cachování v offline případě je silně NP- těžké, ovšem v důkazu byly zapotřebí instance cachování se stránkami většími nežli polovina velikosti cache. Náš hlavní výsledek se vyrovnává s tímto pro- blémem: Dokazujeme, že obecné cachování je silně těžké již tehdy, když jsou velikosti stránek omezeny na {1, 2, 3}. Ve strukturální části práce pak před- stavujeme nový jednodušší důkaz úplné charakterizace work functions pomocí struktury layers v případě klasického cachování, důkaz je následně rozšířen na cachování s proměnlivou velikostí cache. Na základě těchto výsledků jsme zkonstruovali dva algoritmy pro speciální případy obecného cachování.Caching (also known as paging) is a classical problem concerning page re- placement policies in two-level memory systems. General caching is its vari- ant with pages of different sizes and fault costs. We aim at a better charac- terization of the computational complexity of general caching in the offline version. General caching in the offline version was recently shown to be strongly NP- hard, but the proof needed instances of caching with pages larger than half of the cache size. The primary result of this work addresses this problem as we prove: General caching is strongly NP-hard even when page sizes are limited to {1, 2, 3}. In the structural part of this work, a new simpler proof for the full characterization of work functions by layers for classical caching is given and then extended to caching with variable cache size. We invent two algorithms for restricted instances of general caching building on results around caching with variable cache size.Computer Science Institute of Charles UniversityInformatický ústav Univerzity KarlovyMatematicko-fyzikální fakultaFaculty of Mathematics and Physic

Graph communication protocols

Graph communication protocols are a generalization of classical communi- cation protocols to the case when the underlying graph is a directed acyclic graph. Motivated by potential applications in proof complexity, we study variants of graph communication protocols and relations between them. The main result is a comparison of the strength of two types of protocols, protocols with equality and protocols with a conjunction of a constant num- ber of inequalities. We prove that protocols of the first type are at least as strong as protocols of the second type in the following sense: For a Boolean function f, if there is a protocol with a conjunction of a constant number of inequalities of polynomial size solving f, then there is a protocol with equality of polynomial size solving f. We also introduce two new types of graph communication protocols, protocols with disjointness and protocols with non-disjointness, and prove that the first type is at least as strong as the previously considered protocols and that the second type is too strong to be useful for applications