#lower bounds

Define what is a PHP tree. Here we need to fix some n. This n will be sufficiently large and fixed for all of the steps. PHP tells us that there is no bijection from n+1 pigeons to n holes. PHP tree is a tree that asks about the location of i-th pigeon or which pigeon is in j-th hole at each node.

Define what is k-evaluation of a set of formulas. This is a mapping from formulas to tuples (H,S). Morally, H is some set of partial maps from n+1 to n which makes the formula true and S is a PHP tree that asks about holes and pigeons in the formula (atoms tell us i-th pigeon is in the hole j). Here the depth of the tree (the size of the domain) is at most k.

Show that all clauses of the propositional CNF expressing negation of PHP are true with respect to any k-evaluation where k<d-1.

  Biare / Desired. Biaty / As desired. Bibav / As desired and find       1   boxes will stay open or shut, as desired. / and find it does well.       2 as desired, and find the whole       3 much or little set, as desired,   and find such set lasts longer than the spring       4 can run into any one of the four pockets as desired, and find platforms on either side       5       of the horn as desired,   and find corresponding values of L, or distance from       6 Bibet / Destination. Biboy / Destination of.   Bibux / Destination unknown       7       value of n to make the error as small as desired and find λ with       8 BIBAV / Idaho White Pine   BIBEV / Place BIBEW / A favorite place BIBEX / A good place (for)       9  

sources

1 ex A Telegraphic Code for the Use of the United States Public Health Service (1915) : 93 (NLM/NIH copy, same) 2 OCR cross-column misread, involving “Another flour-box” and “Keeping Hams” (the latter submitted by “E. R. A. S.”, Hardinsburgh, Indiana), at American Agriculturist (for the Farm, Garden, and Household; published also in German) 31 (Orange Judd & Co., publishers and proprietors, New York; June 1872) : 226 for Orange Judd (1822-92), see wikipedia 3 ex observation by a microscopist, referring to “Vines in a narrow border” mentioned on page 512 of December 13 issue, in The Journal of Horticulture, Cottage Gardener, and Home Farmer (“A chronicle of country pursuits and country life, including poultry, pigeon, and bee-keeping;” Conducted by Robert Hogg) 7 (December 20, 1883) : 529 4 ex “Swaged teeth for circular saws” (communication from Judd), in The Wood-worker (a journal for machine wood-workers) 21:9 (Indianapolis, Ind., November 1902) : 28 5 ex communication from Wm. H. Burr, Wm. Barclay Parsons, and Ira A. McCormack, commissioners, to J. W. Stevenson, Commissioner of Bridges, City of New York, in Minutes of the Board of Estimate and Apportionment of the City of New York, Financial and Franchise Matters, from January 1 to March 31, 1907 (January 11, 1907) : 100-109 (104) 6 ex RCA Review : A Technical Journal (1939) : 101 (snippet only) 7 ex A Telegraphic Code for the Use of the United States Public Health Service (1915) : 93 8 Hans F. Weinberger, A Theory of Lower Bounds for Eigenvalues (United States Air Force, Office of Scientific Research, 1959) : 53 9 ex Charles J. Hoffman, comp., Universal Lumber Code (2nd—3rd Edition; 1909, 1923) : 291

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encountering the eponymous phrase in the Public Health Service code, wondered in what contexts it might be wanted.

In the rendezvous problem, two parties with different labelings of the vertices of a complete graph are trying to meet at some vertex at the same time. It is well-known that if the parties have predetermined roles, then the strategy where one of them waits at one vertex, while the other visits all $n$ vertices in random order is optimal, taking at most $n$ steps and averaging about $n/2$. Anderson and Weber considered the symmetric rendezvous problem, where both parties must use the same randomized strategy. They analyzed strategies where the parties repeatedly play the optimal asymmetric strategy, determining their role independently each time by a biased coin-flip. By tuning the bias, Anderson and Weber achieved an expected meeting time of about $0.829 n$, which they conjectured to be asymptotically optimal.

We change perspective slightly: instead of minimizing the expected meeting time, we seek to maximize the probability of meeting within a specified time $T$. The Anderson-Weber strategy, which fails with constant probability when $T= \Theta(n)$, is not asymptotically optimal for large $T$ in this setting. Specifically, we exhibit a symmetric strategy that succeeds with probability $1-o(1)$ in $T=4n$ steps. This is tight: for any $\alpha < 4$, any symmetric strategy with $T = \alpha n$ fails with constant probability. Our strategy uses a new combinatorial object that we dub a "rendezvous code," which may be of independent interest.

When $T \le n$, we show that the probability of meeting within $T$ steps is indeed asymptotically maximized by the Anderson-Weber strategy. Our results imply new lower bounds, showing that the best symmetric strategy takes at least $0.638 n$ steps in expectation. We also present some partial results for the symmetric rendezvous problem on other vertex-transitive graphs.