Naor M., Parter M., Yogev E.: (The power of distributed verifiers in interactive proofs. In: 31st ACM-SIAM symposium on discrete algorithms (SODA), pp 1096–115, 2020. https://doi.org/10.1137/1.9781611975994.67 ) have recently demonstrated the existence of a distributed interactive proof for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed IP protocols based on sequential IP protocols. The interactive proof for planarity is based on a distributed certification of the correct execution of any given sequential linear-time algorithm for planarity testing. It involves three interactions between the prover and the randomized distributed verifier (i.e., it is a dMAM protocol), and uses small certificates, on $$O(\log n)$$ bits in n-node networks. We show that a single interaction with the prover suffices, and randomization is unecessary, by providing an explicit description of a proof-labeling scheme for planarity, still using certificates on just $$O(\log n)$$ bits. We also show that there are no proof-labeling schemes—in fact, even no locally checkable proofs—for planarity using certificates on $$o(\log n)$$ bits.

Naor, Parter, and Yogev (SODA 2020) have recently demonstrated the existence of a distributed interactive proof for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed IP protocols based on sequential IP protocols. The interactive proof for planarity is based on a distributed certification of the correct execution of any given sequential linear-time algorithm for planarity testing. It involves three interactions between the prover and the randomized distributed verifier (i.e., it is a dMAM protocol), and uses small certificates, on O(log n) bits in n-node networks. We show that a single interaction from the prover suffices, and randomization is unecessary, by providing an explicit description of a proof-labeling scheme for planarity, still using certificates on just O(log n) bits. We also show that there are no proof-labeling schemes --- in fact, even no locally checkable proofs --- for planarity using certificates on o(log n) bits.

The CONGEST model is a synchronous, message-passing model of distributed computation in which each node can send (possibly different) messages of O(log n) bits along each of its incident communication links in each round, where n is the number of computing nodes in the system. In the particular case where the communication network is a complete graph, we have the unicast congested clique model. On the other end is the broadcast version of the congested clique model, in which each node can only broadcast a single message over all its links in each round. In this paper we explore the space, in terms of round complexity, that lies between these two congested clique models. Hence, we parametrize the congested clique model with the range r, the maximum number of different messages a node can send over its incident links in one round. Additionally, we study the effect of the bandwidth b, the maximum size in bits of these messages. We show that the space between the unicast and broadcast congested clique models is very rich and interesting. For instance, we show that a problem (especially designed for this work) takes Ω(n/ log n) rounds in the broadcast model (r = 1), while it can be solved in two rounds if two messages can be sent (r = 2). Other gaps are found in other parts of the spectrum of values of r. We do this by providing techniques to simulate protocols with different parameters. Therefore, we conclude that, with respect to their power to solve certain problems, there is a strict hierarchy of congested clique models.

We study the structure of the set of tilings of a polygon P with bars of fixed length. We obtain an undirected graph connecting two tilings if one can pass from one tile to the other one by a flip (i.e., a local replacement of tiles). Using algebraic tools (such as tiling groups and their quotients and subgroups), we give a formula to compute the distance in this graph (i.e., the minimal number of necessary flips) between two tilings. Moreover, we prove that, for each pair (T, T') of tilings, the set $\Upsilon_{T, T'}$ consisting of tilings which are in a path of minimal length from T to T' canonically has a structure of distributive lattice.

We present an approximation scheme for strip-packing, or packing rectangles into a rectangle of fixed width and minimum height, a classical NP-hard cutting-stock problem. The algorithm finds a packing of n rectangles whose total height is within a factor of (1+/spl epsiv/) of optimal, and has running time polynomial both in n and in 1//spl epsiv/. It is based on a reduction to fractional bin-packing, and can be performed by 5 stages of guillotine cuts.