<i>r</i> -Simple <i>k</i> -Path and Related Problems Parameterized by <i>k</i> / <i>r</i>

Abasi et al. (2014) introduced the following two problems. In r -S imple k -P ath problem, given a digraph G on n vertices and positive integers , decide whether has an -simple -path, which is walk where every vertex occurs at most times total number of occurrences . ( )-M onomial D etection arithmetic circuit that succinctly encodes some polynomial P variables monomial degree each variable obtained randomized algorithms running time 4 / )log ? O (1) for both Gabizon (2015) designed deterministic 2 (( ) -time problems (however, problem input restricted to be non-canceling). also studied problem. q )-P acking roblem universe V p ), collection H sets size whose elements belong there exists subcollection ? element in pq algorithm The above results prove three are single-exponentially fixed-parameter tractable (FPT) parameterized by product parameters, is, log = asked factor exponent can avoided. Bonamy (2017) answered question proving unless Exponential Time Hypothesis (ETH) fails no o + i.e., unlikely FPT when alone. remains open We consider from wider perspective: only, computable function f such admit )( +log algorithm? Since substantially larger than size, do not even show any these XP resolve (a) obtaining log( )) digraphs &amp;sdot undirected graphs (i.e., graphs, we answer original affirmative), (b) showing (in contrast, -M ultiset W[1]-hard), (c) para-NP-hard if only distinct non-canceling. For special case polynomially bounded (which XP), W[1]-hardness. Along way solve obtain kernel fixed resolves posed regarding existence kernels with relaxed disjointness constraints. All our deterministic.