Kernelization is a strong and widely-applied technique in parameterized complexity. A kernelization algorithm, or simply a kernel, is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernel is polynomial if the size and parameter of the output are polynomially-bounded by the parameter of the input. In this paper we develop a framework which allows showing that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world (i.e. non-parametric complexity), and revolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which is of independent interest. Using the notion of distillation algorithms, we develop a generic lower-bound engine that allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These problems include k-Path, k-Cycle, k-Exact Cycle, k-Short Cheap Tour, k-Graph Minor Order Test, k-Cutwidth, k-Search Number, k-Pathwidth, k-Treewidth, k-Branchwidth, and several optimization problems parameterized by treewidth and other structural parameters.

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In this paper we study the Target Set Selection problem proposed by Kempe, Kleinberg, and Tardos; a problem which gives a nice clean combinatorial formulation for many applications arising in economy, sociology, and medicine. Its input is a graph with vertex thresholds, the social network, and the goal is to find a subset of vertices, the target set, that "activates" a prespecified number of vertices in the graph. Activation of a vertex is defined via a so-called activation process as follows: Initially, all vertices in the target set become active. Then at each step i of the process, each vertex gets activated if the number of its active neighbors at iteration i − 1 exceeds its threshold. The activation process is "monotone" in the sense that once a vertex is activated, it remains active for the entire process. Our contribution is as follows: First, we present an algorithm for Target Set Selection running in n O(w) time, for graphs with n vertices and treewidth bounded by w. This algorithm can be adopted to much more general settings, including the case of directed graphs, weighted edges, and weighted vertices. On the other hand, we also show that it is highly unlikely to find an n o(√ w) time algorithm for Target Set Selection, as this would imply a sub-exponential algorithm for all problems in SNP. Together with our upper bound result, this shows that the treewidth parameter determines the complexity of Target Set Selection to a large extent, and should be taken into consideration when tackling this problem in any scenario. In the last part of the paper we also deal with the "non-monotone" variant of Target Set Selection, and show that this problem becomes #P-hard on graphs with edge weights.

Subset Sum and k-SAT are two of the most extensively studied problems in computer science, and conjectures about their hardness are among the cornerstones of fine-grained complexity.An important open problem in this area is to base the hardness of one of these problems on the other.Our main result is a tight reduction from k-SAT to Subset Sum on dense instances, proving that Bellman's 1962 pseudo-polynomial O * (T )-time algorithm for Subset Sum on n numbers and target T cannot be improved to time T 1−ε · 2 o(n) for any ε > 0, unless the Strong Exponential Time Hypothesis (SETH) fails.As a corollary, we prove a "Direct-OR" theorem for Subset Sum under SETH, offering a new tool for proving conditional lower bounds: It is now possible to assume that deciding whether one out of N given instances of Subset Sum is a YES instance requires time (N T ) 1−o(1) . As an application of this corollary, we prove a tight SETH-based lower bound for the classical Bicriteria s, t-Path problem, which is extensively studied in Operations Research. We separate its complexity from that of Subset Sum: On graphs with m edges and edge lengths bounded by L, we show that the O(Lm) pseudopolynomial time algorithm by Joksch from 1966 cannot be improved toÕ(L + m), in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017).Subset Sum. Subset Sum is one of the most fundamental problems in computer science. Its most basic form is the following: given n integers x 1 , . . . , x n ∈ N, and a target value T ∈ N, decide whether there is a subset of the numbers that sums to T . The two most classical algorithms for the problem are the pseudo-polynomial O(T n) algorithm using dynamic programming [28], and the O(2 n/2 · poly(n, log T )) algorithm via "meet-in-the-middle" [67]. A central open question in Exact Algorithms [114] is whether faster algorithms exist, e.g., can we combine the two approaches to get a T 1/2 · n O(1) time algorithm? Such a bound was recently found in a Merlin-Arthur setting [94].The status of Subset Sum as a major problem has been established due to many applications, deep connections to other fields, and educational value. The O(T n) algorithm from 1957 is an illuminating example of dynamic programming that is taught in most undergraduate algorithms courses, and the NP-hardness proof (from Karp's original paper [78]) is a prominent example of a reduction to a problem on numbers. Interestingly, one of the earliest cryptosystems by Merkle and Hellman was based on Subset Sum [92], and was later extended to a host of Knapsack-type cryptosystems 4 (see [106,31,97,46,69] and the references therein).The version of Subset Sum where we ask for k numbers that sum to zero (the k-SUM problem) is conjectured to have n ⌈k/2⌉±o(1) time complexity. Most famously, the k = 3 case is the 3-SUM conjecture highlighted in the seminal work of Gajentaan and Overmars [57]. It has been shown that this problem lies at the core and captures the difficulty of dozens of problems in computational geometry. Searching in Google Scholar for "3su...