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

中文翻译：

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r -Simple k -Path 和由 k / r 参数化的相关问题

阿巴西等人。（2014）引入了以下两个问题。在里面r-S实现 ķ-Path问题，给定一个有向图G在n顶点和正整数r,ķ， 决定是否G有一个r-简单的ķ-path，即每个顶点最多出现的路径r次，顶点出现的总数为ķ. 在里面 （r,ķ)-M名词性的D检测问题，给定一个简洁地编码一些多项式的算术电路磷在n变量和正整数ķ,r， 决定是否磷有一个总度数的单项式ķ其中每个变量的度数最多为r. 阿巴西等人。获得运行时间的随机算法 4(ķ/r）日志r ⋅n ○(1)对于这两个问题。加比松等人。(2015) 设计确定性 2 ○((ķ/r）日志r)⋅n ○(1)两个问题的时间算法（但是，对于（r,ķ)-M名词性的D检测问题输入电路被限制为不可取消）。加比松等人。还研究了以下问题。在里面磷-S等(r,q)-P确认磷问题，给定一个宇宙五, 正整数 (p,q,r)，以及一组大小集合 H磷其元素属于五，判断是否存在子集合H'H的大小q其中每个元素最多出现在rH 组'. 加比松等人。获得了确定性 2 ○((pq/r）日志r)⋅n ○(1)-时间算法磷-S等(r,q)-P确认. 以上结果证明了这三个问题是单指数由乘积参数化的固定参数可处理 (FPT)二参数，即ķ/r并记录r， 在哪里ķ=pq为了磷-S等(r,q)-P确认. 阿巴西等人。和加比松等人。询问是否日志r可以避免指数中的因素。博纳米等人。（2017）回答了（r,ķ)-M名词性的D检测通过证明除非指数时间假设 (ETH) 失败，否则没有 2 ○((ķ/r） 日志r)⋅ (n+ 日志ķ) ○(1)-时间算法（r,ķ)-M名词性的D检测， IE， （r,ķ)-M名词性的D检测参数化时不太可能是单指数 FPTķ/r独自的。这个问题仍然悬而未决r-S实现 ķ-Path和磷-S等(r,q)-P确认. 我们从更广泛的角度考虑这个问题：上述问题在参数化时是 FPTķ/ronly，即是否存在可计算函数F这样的问题承认F(ķ/r)(n+日志ķ) ○(1)时间算法？自从r可以大大大于输入大小，Abasi 等人的算法。和加比松等人。甚至没有表明这三个问题中的任何一个都存在于 XP 参数化中ķ/r独自的。我们通过 (a) 获得 2 ○((ķ/r) 2 日志（ķ/r))⋅ (n+ 日志ķ) ○(1)-时间算法r-S实现 ķ-Path关于有向图和 2 ○(ķ/r)&sdot (n+ 日志ķ) ○(1)-时间算法r-S实现 ķ-Path在无向图上（即，对于无向图，我们肯定地回答原始问题），（b）表明磷-S等(r,q)-P确认是 FPT（相反，我们证明磷-M超集(r,q)-P确认是 W[1]-hard)，并且 (c) 证明 (r,ķ)-M名词性的D检测即使只有两个不同的变量在多项式中，也是对 NP 难的磷并且电路是不可取消的。对于 (r,ķ)-M名词性的D检测在哪里ķ是由输入大小（在 XP 中）以多项式为界的，我们展示了 W[1]-硬度。一路解决磷-S等(r,q)-P确认，我们得到一个多项式核对于任何固定磷，这解决了 Gabizon 等人提出的问题。关于具有松弛不相交约束的问题的多项式核的存在。我们所有的算法都是确定性的。