Reconstructing Sets of Strings from Their k-way Projections: Algorithms & Complexity

TL;DR

This work defines the String Set Reconstruction problem from $k$-way projections, formalizing $Recon_k(S)$ and three key objectives: the point of perfect reconstruction, the point of no information, and string containment. It advances both theory and practice by proving deep complexity results (including reductions to Hitting Set, DP and Pi$_2^P$–type considerations) and by introducing an overlap-graph algorithm that handles noncontiguous $k$-mers while leveraging known index sets. The paper also connects the problem to CSP counting and provides extensive experimental evidence showing the overlap-graph method outperforms simple greedy baselines across a range of parameters, with behavior that scales with $n$, $m$, $k$, and the information content in the projections. Overall, the work advances the understanding of higher-order interaction reconstruction and offers practical algorithmic tools for reconstructing sets of strings from higher-order projections under various information regimes.

Abstract

Graphs are a powerful tool for analyzing large data sets, but many real-world phenomena involve interactions that go beyond the simple pairwise relationships captured by a graph. In this paper we introduce and study a simple combinatorial model to capture higher order dependencies from an algorithms and computational complexity perspective. Specifically, we introduce the String Set Reconstruction problem, which asks when a set of strings can be reconstructed from seeing only the k-way projections of strings in the set. This problem is distinguished from genetic reconstruction problems in that we allow projections from any k indices and we maintain knowledge of those indices, but not which k-mer came from which string. We give several results on the complexity of this problem, including hardness results, inapproximability, and parametrized complexity. Our main result is the introduction of a new algorithm for this problem using a modified version of overlap graphs from genetic reconstruction algorithms. A key difference we must overcome is that in our setting the k-mers need not be contiguous, unlike the setting of genetic reconstruction. We exhibit our algorithm's efficiency in a variety of experiments, and give high-level explanations for how its complexity is observed to scale with various parameters. We back up these explanation with analytic approximations. We also consider the related problems of: whether a single string can be reconstructed from the k-way projections of a given set of strings, and finding the largest k at which we get no information about the original data set from its k-way projections (i.e., the largest $k$ for which it is "k-wise independent").

Figures (7)

• Figure 1: An example of a set $S$ of three length-$3$ strings whose point of perfection reconstruction is $< 3$. To the left is the starting data set $S = \{001, 011,100\}$, where the strings correspond to rows after the first, and the indices 1,2,3 are listed in the first row. To the right are the 2-way projections for index sets $\{1,2\}$, $\{2,3\}$, and $\{1,3\}$, respectively. From the $\{1,3\}$ projection, we see that indices 1 and 3 completely define one another: $x_1 = 1-x_3$. This, coupled with the $\{1,2\}$ projection, allows us to fill in the 2nd column and perfectly reconstruct the original data set.
• Figure 2: An example containing 7 strings of length 5 with the indices enumerated across the top of the table, along with one of its overlap graphs for $k=4$. The cycles of length 5 are highlighted yellow; the number of $5$-cycles is the same as $|S|$, so $k=4$ is the point of perfect reconstruction. The arrangement of vertices into columns is not inherent, but merely for convenience.
• Figure 3: An example data set, its overlap graph, and its adjacency matrix $A$. The diagonal entries of the matrix $A^5$ count the number of 5-cycles through each vertex. In the lower-right block we can see that each vertex in that layer has exactly 1 cycle through it, so those cycles must precisely be the original strings we started with. Thus in this case the diagonal entries of $A^n$ imply that this set of strings is perfectly reconstructed at $k=3$.
• Figure 4: Zoomed in version of Fig. \ref{['fig:big']}, focusing on $m \leq 100$. At small $k$ and for some $m \gtrsim k 2^k$, the greedy algorithm outperforms our overlap graph algorithm because $k$ is below or close to the point of no information.
• Figure 5: Comparing runtime of our overlap graph algorithm with the greedy algorithm on random data sets of length-12 strings ($n=12$), for all values of the window size $k$ (between $2$ and $11=n-1$), at (top left) $m=40$ strings, (top right) $m=90$, (bottom left) $m=500$, (bottom right) $m=2000$.

Theorems & Definitions (14)

• Definition 1.1: with Yoav Kallus
• Definition 1.3
• Theorem 2.1: Berman75
• Theorem 2.2
• proof
• Corollary 2.3
• proof
• Proposition 2.4
• proof : Proof of Prop. \ref{['thm:Pi2P']}
• proof