arXiv subjects

Adam Y. Shavit

Publications and source records attributed to Adam Y. Shavit.

4 recordsLinked to original sources

arXiv2026

Minimum-makespan completion and vertex selection leave the Wang-Sitters constant at 11/6

The 11/6 worst-case constant of the Wang-Sitters rounding scheme, which a companion note establishes, can naturally be attributed to the freedom in Step 3, where an arbitrary valid slot matching is permitted. We show that eliminating that freedom does not improve the constant. A minimum-makespan completion oracle still has worst-case constant exactly 11/6 against the optimum; both natural 7/4 statements about it are false; and restricting Step 1 to vertices of the relaxation does not help. The loss therefore cannot be attributed solely to the freedom in Step 3. We also record what structure survives: a reduction confining every overload to two shapes, a seven-machine instance defeating the natural two-phase repair, and a strict 7/4 bound on the generalized three-path family.

cs.DS↗

arXiv2026

The 11/6 supremum of the Wang-Sitters rounding scheme for graph balancing

Wang and Sitters' 11/6-approximation for graph balancing is not one algorithm but a set of permitted executions: Step 1 may return any feasible solution of the relaxation and Step 3 any of the many ways to match the remaining jobs into the slots the rounding opens. We determine exactly what that latitude permits: ratios arbitrarily close to 11/6, and none reaching it, so 11/6 is the least constant that bounds every permitted run, and no run attains it. We then determine the worst-case guarantee as a function of the big-job threshold beta, measured against the optimum itself. On Wang and Sitters' own range 1/2 < beta < 1 the guarantee is exactly max{3/2 + beta/2, 5/2 - beta}. We then extend the same eligibility rule to 0 < beta <= 1/2 -- outside the range they state, and where a big job's two shares can both reach the threshold, so Step 2 acquires a third choice -- and determine the guarantee there as well: exactly 3/2 + (1-beta)floor(1/beta), hence unbounded as beta falls. The worst-case ratio is therefore known at every threshold in (0,1), and attained at none. Consequently 2/3 is the unique optimal threshold, and the guarantee jumps at one half rather than degrading smoothly. A companion note asks what does not fix the constant.

cs.DS↗

arXiv2026

A minimum witness for the 3/2 configuration-linear-program gap in two-weight graph balancing, unique at its size

In restricted assignment - makespan minimization where each job has one size and a set of allowed machines - the configuration LP is the tightest studied relaxation, and its integrality gap is open in general. On two-weight graph balancing - each job allowed on at most two machines, sizes from two values - the value is known, both bounds due to Jansen, Land, and Maack (2016): their Table 1 instance attains 3/2, and their Corollary 11 bound of 2 - s/b for sizes s < b meets it at {1,2}. We ask how small such an instance - a witness - can be. We give I*, a six-job witness: the complete graph on four machines, unit jobs on a Hamiltonian cycle, weight-2 jobs on the complementary perfect matching, with integral optimum 3 against relaxation value 2. That is one job fewer than the smallest previously in print, and we prove it minimum and unique at its size. No instance of the class with at most five jobs reaches gap 3/2, on any number of machines; at six jobs, again on any number of machines, I* is the only witness, up to relabeling machines and adding machines no job can use. At seven jobs uniqueness fails: exactly thirteen witnesses, classified - the Jansen-Land-Maack instance among them - and at eight jobs exactly 154. Three machines never suffice, at any size: four are necessary for the gap. Results of this shape are in print for the same relaxation in one-dimensional cutting stock, where the extremal non-round-up instances have been enumerated and classified for small demand; the Discussion sets out the relation. Recognizing witnesses at relaxation value 2 - where all of ours live - is coNP-complete, so no min-max characterization exists unless NP = coNP. Every feasibility decision behind the exhaustive claims was made twice, in floating point and in exact rational arithmetic, with full agreement, and the pipeline must rediscover I* before its negatives are believed.

cs.DS↗

arXiv2026

Two base rates, two weights: base-rate neglect has a second axis

Base-rate neglect is usually treated as one mistake: giving the prior too little weight. Turning the co-occurrences you see into a useful judgment, though, means correcting for two base rates, not one. The first is the familiar prior, how common the outcome is. The second is how common the cue itself is. Those are two separate mistakes, and a learner can make either one alone. Under-correcting the prior is classical base-rate neglect; under-correcting the cue is the cue-density effect of contingency learning, long studied but not previously recognised as a kind of base-rate neglect. We write both corrections as two weights in one Bayesian equation. The task decides which weight it can measure: the cue-frequency weight appears only in graded ratings, because a two-choice test cancels it. At their extremes the two weights recover familiar quantities: base-rate neglect, the signal-detection criterion, the contiguity/sensitivity/validity triple, and the "lift" measure of causal strength. The same cue-frequency weight also sits inside six standard learning-and-memory models; they seem to agree, but only because the usual experiments squeeze the data into a form where they cannot disagree. Above all, the two neglects should be separately manipulable: an experimenter can move one without moving the other. That is a double dissociation, and a one-parameter account cannot produce it. That prediction is the framework's centre, and it has not yet been tested. This paper lays out the framework and the rating experiment that would settle it; a companion paper fits the two weights to an existing colour-flavour dataset.

stat.ME↗