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Register a new userOn The Multiparty Communication Complexity of Testing Triangle-Freeness
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Orr Fischer
Publication date
2017
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Informatics Engineering
and research's language is
English
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In this paper we initiate the study of property testing in simultaneous and non-simultaneous multi-party communication complexity, focusing on testing triangle-freeness in graphs. We consider the $textit{coordinator}$ model, where we have $k$ players receiving private inputs, and a coordinator who receives no input; the coordinator can communicate with all the players, but the players cannot communicate with each other. In this model, we ask: if an input graph is divided between the players, with each player receiving some of the edges, how many bits do the players and the coordinator need to exchange to determine if the graph is triangle-free, or $textit{far}$ from triangle-free? For general communication protocols, we show that $tilde{O}(k(nd)^{1/4}+k^2)$ bits are sufficient to test triangle-freeness in graphs of size $n$ with average degree $d$ (the degree need not be known in advance). For $textit{simultaneous}$ protocols, where there is only one communication round, we give a protocol that uses $tilde{O}(k sqrt{n})$ bits when $d = O(sqrt{n})$ and $tilde{O}(k (nd)^{1/3})$ when $d = Omega(sqrt{n})$; here, again, the average degree $d$ does not need to be known in advance. We show that for average degree $d = O(1)$, our simultaneous protocol is asymptotically optimal up to logarithmic factors. For higher degrees, we are not able to give lower bounds on testing triangle-freeness, but we give evidence that the problem is hard by showing that finding an edge that participates in a triangle is hard, even when promised that at least a constant fraction of the edges must be removed in order to make the graph triangle-free.
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For integers $n$ and $k$, the density Hales-Jewett number $c_{n,k}$ is defined as the maximal size of a subset of $[k]^n$ that contains no combinatorial line. We show that for $k ge 3$ the density Hales-Jewett number $c_{n,k}$ is equal to the maximal size of a cylinder intersection in the problem $Part_{n,k}$ of testing whether $k$ subsets of $[n]$ form a partition. It follows that the communication complexity, in the Number On the Forehead (NOF) model, of $Part_{n,k}$, is equal to the minimal size of a partition of $[k]^n$ into subsets that do not contain a combinatorial line. Thus, the bound in cite{chattopadhyay2007languages} on $Part_{n,k}$ using the Hales-Jewett theorem is in fact tight, and the density Hales-Jewett number can be thought of as a quantity in communication complexity. This gives a new angle to this well studied quantity. As a simple application we prove a lower bound on $c_{n,k}$, similar to the lower bound in cite{polymath2010moser} which is roughly $c_{n,k}/k^n ge exp(-O(log n)^{1/lceil log_2 krceil})$. This lower bound follows from a protocol for $Part_{n,k}$. It is interesting to better understand the communication complexity of $Part_{n,k}$ as this will also lead to the better understanding of the Hales-Jewett number. The main purpose of this note is to motivate this study.
In the subgraph-freeness problem, we are given a constant-size graph $H$, and wish to determine whether the network contains $H$ as a subgraph or not. The emph{property-testing} relaxation of the problem only requires us to distinguish graphs that are $H$-free from graphs that are $epsilon$-far from $H$-free, meaning an $epsilon$-fraction of their edges must be removed to obtain an $H$-free graph. Recently, Censor-Hillel et. al. and Fraigniaud et al. showed that in the property-testing regime it is possible to test $H$-freeness for any graph $H$ of size 4 in constant time, $O(1/epsilon^2)$ rounds, regardless of the network size. However, Fraigniaud et. al. also showed that their techniques for graphs $H$ of size 4 cannot test $5$-cycle-freeness in constant time. In this paper we revisit the subgraph-freeness problem and show that $5$-cycle-freeness, and indeed $H$-freeness for many other graphs $H$ comprising more than 4 vertices, can be tested in constant time. We show that $C_k$-freeness can be tested in $O(1/epsilon)$ rounds for any cycle $C_k$, improving on the running time of $O(1/epsilon^2)$ of the previous algorithms for triangle-freeness and $C_4$-freeness. In the special case of triangles, we show that triangle-freeness can be solved in $O(1)$ rounds independently of $epsilon$, when $epsilon$ is not too small with respect to the number of nodes and edges. We also show that $T$-freeness for any constant-size tree $T$ can be tested in $O(1)$ rounds, even without the property-testing relaxation. Building on these results, we define a general class of graphs for which we can test subgraph-freeness in $O(1/epsilon)$ rounds. This class includes all graphs over 5 vertices except the 5-clique, $K_5$. For cliques $K_s$ over $s geq 3$ nodes, we show that $K_s$-freeness can be tested in $O(m^{1/2-1/(s-2)}/epsilon^{1/2+1/(s-2)})$ rounds, where $m$ is the number of edges.
The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the goal is to decide whether their two distributions are equal, or are $epsilon$-far apart in the $l_1$-distance. In the present paper we show that the quantum communication complexity of this problem is $tilde{O}(n/(tepsilon^2))$ qubits when the distributions have low $l_2$-norm, which gives a quadratic improvement over the classical communication complexity obtained by Andoni, Malkin and Nosatzki. We also obtain a matching lower bound by using the pattern matrix method. Let us stress that the samples received by each of the parties are classical, and it is only communication between them that is quantum. Our results thus give one setting where quantum protocols overcome classical protocols for a testing problem with purely classical samples.
Linear programming is a powerful method in combinatorial optimization with many applications in theory and practice. For solving a linear program quickly it is desirable to have a formulation of small size for the given problem. A useful approach for this is the construction of an extended formulation, which is a linear program in a higher dimensional space whose projection yields the original linear program. For many problems it is known that a small extended formulation cannot exist. However, most of these problems are either $mathsf{NP}$-hard (like TSP), or only quite complicated polynomial time algorithms are known for them (like for the matching problem). In this work we study the minimum makespan problem on identical machines in which we want to assign a set of $n$ given jobs to $m$ machines in order to minimize the maximum load over the machines. We prove that the canonical formulation for this problem has extension complexity $2^{Omega(n/log n)}$, even if each job has size 1 or 2 and the optimal makespan is 2. This is a case that a trivial greedy algorithm can solve optimally! For the more powerful configuration integer program, we even prove a lower bound of $2^{Omega(n)}$. On the other hand, we show that there is an abstraction of the configuration integer program admitting an extended formulation of size $f(text{opt})cdot text{poly}(n,m)$. In addition, we give an $O(log n)$-approximate integral formulation of polynomial size, even for arbitrary processing times and for the far more general setting of unrelated machines.
Hub Labeling (HL) is a data structure for distance oracles. Hierarchical HL (HHL) is a special type of HL, that received a lot of attention from a practical point of view. However, theoretical questions such as NP-hardness and approximation guarantee for HHL algorithms have been left aside. In this paper we study HL and HHL from the complexity theory point of view. We prove that both HL and HHL are NP-hard, and present upper and lower bounds for the approximation ratios of greedy HHL algorithms used in practice. We also introduce a new variant of the greedy HHL algorithm and a proof that it produces small labels for graphs with small highway dimension.
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