arXiv subjects

Nikolai Chukhin

Publications and source records attributed to Nikolai Chukhin.

6 recordsLinked to original sources

arXiv2026

The Greedy Superstring Algorithm Achieves Ratio 2 for Strings of Length 6 Already

In the Shortest Common Superstring (SCS) problem, one is given a set of strings and is asked to find a string of minimum length containing each of the input strings as a substring. The greedy superstring conjecture states that the following natural greedy algorithm has approximation ratio $2$: while there is more than one string, select the pair of strings with the maximum overlap, merge them, and add the merged string back to the set. The greedy algorithm works in linear time and is probably the simplest possible approximation algorithm for SCS. If the conjecture holds, then the greedy algorithm also surpasses the approximation guarantees of the best known approximation algorithms. The conjecture is open for $40$ years already and even the approximation ratio $\rho_k$ in the special case in which input strings have length $k$ has not yet been found: for all $k \ge 3$, $2-1/k \le \rho_k \le \min\{(k+1)/2, 3.396\}$. We prove that already for strings of length $6$, the approximation ratio of the greedy algorithm is at least $2$: $\rho_k \ge 2$ for all $k \ge 6$. We also show that $\rho_3=9/5$, thus completely characterizing the worst-case behavior of the greedy algorithm for strings of length $3$.

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arXiv2026

Improved Quantum Algorithms for Subset Sum and $k$-SUM

The Subset Sum problem asks whether, given $n$ integers and a target, some subset of the integers sums to the target. Its best known worst-case running time is $O^*(2^{n/2})$ (Horowitz and Sahni, 1974), whereas the best quantum upper bound is $O^*(2^{n/3})$ (Bernstein, Jeffery, Lange, and Meurer, 2013). The $k$-SUM problem is a parameterized version of Subset Sum asking whether there are $k$ integers that sum to the target. The best classical upper bound for it is $\widetilde O(n^{\lceil k/2\rceil})$, whereas the best quantum running time is $\widetilde O(n^{k/3})$ (Tani, 2009). For random instances, a quantum algorithm with running time $\widetilde O(n^{\Phi_k})$ is known, where $$ \Phi_k=\frac{2k-\lfloor k/7\rfloor-\lfloor (k+3)/7\rfloor}{6} $$ (Schrottenloher, 2021). We present a new quantum algorithm solving worst-case $k$-SUM in time $\widetilde O(n^{\Psi_k})$, where $$ \Psi_k=\Phi_k-\frac{[k\equiv 3\bmod 7]}{9}-\frac{[k\equiv 6\bmod 7]}{18}. $$ The algorithm is not only faster for all $k$ congruent to $3$ or $6$ modulo $7$, but also gives a worst-case guarantee rather than a guarantee restricted to single-solution random instances. Combining our algorithm for $7$-SUM with the standard block reduction technique yields an $O^*(2^{2n/7})$ quantum algorithm for Subset Sum, improving the previously known $O^*(2^{n/3})$ algorithm.

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arXiv2026

Complexity of the Graph Homomorphism Problem w.r.t. Degeneracy

The graph homomorphism problem HOM is: given an $n$-vertex source graph $G$ and an $h$-vertex target graph $H$, is there a mapping from $V(G)$ to $V(H)$ that preserves edges? A straightforward brute-force algorithm for HOM has running time $O(2^{n \log h})$ and it is known that, under ETH, there are no $2^{o(n \log h)}$ algorithms. In recent years, less restrictive graph parameters $p$ have been identified that allow one to solve HOM in time $p(H)^{O(n)}$. Examples include treewidth, maximum degree, and track number. On the other hand, it is known that the chromatic number parameter is too small: under ETH, HOM cannot be solved in time $\chi(H)^{O(n)}$. We study the complexity of HOM in terms of the degeneracy of $H$. This is perhaps the most natural unresolved graph parameter between the known algorithmic and hardness regimes: on the one hand, each of bounded treewidth, bounded maximum degree, and bounded track number implies bounded degeneracy; on the other hand, bounded degeneracy implies bounded chromatic number. Our results show that, at the same time, the influence of degeneracy of $H$ on the complexity of HOM differs significantly from that of the previously studied parameters. We show that, under ETH, there is no $2^{o(degen(H) n)}$ algorithm for any value of $degen(H)$ as a function of $n$. We also show that bounded degeneracy alone does not make target size benign: even targets with $degen(H)$ at most $2$ and quasi-polynomial size force $n^{\Omega(n)}$-scale hardness. Finally, we introduce a no-compression barrier that explains why the known fine-grained lower bounds for sparse $2$-CSP are not tight under ETH. Moreover, it shows that substantially stronger lower bounds for polynomial-target degeneracy are unlikely to follow from standard reductions from sparse $3$-SAT.

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arXiv2026

Conditional Complexity Hardness: Monotone Circuit Size, Matrix Rigidity, and Tensor Rank

Proving complexity lower bounds remains a challenging task: we only know how to prove conditional uniform lower bounds and nonuniform lower bounds in restricted circuit models. Williams (STOC 2010) showed how to derive nonuniform lower bounds from uniform upper bounds: by designing a fast algorithm for checking satisfiability of circuits, one gets a lower bound for this circuit class. Since then, a number of results of this kind have been proved. For example, Jahanjou et al. (ICALP 2015) and Carmosino et al. (ITCS 2016) proved that if NSETH fails, then $\text{E}^{\text{NP}}$ has series-parallel circuit size $ω(n)$. One can also derive nonuniform lower bounds from nondeterministic uniform lower bounds. Recent examples include lower bounds on tensor rank, arithmetic circuit size, $\text{ETHR} \circ \text{ETHR}$ circuit size under assumptions that various problems (like TSP, MAX-3-SAT, SAT, Set Cover) cannot be solved faster than in $2^n$ time. In this paper, we continue developing this line of research and show how uniform nondeterministic lower bounds can be used to construct generators of various types of combinatorial objects: Boolean functions of high circuit size, matrices of high rigidity, and tensors of high rank. Specifically, we prove the following. If $k$-SAT cannot be solved in input-oblivious co-nondeterministic time $O(2^{(1/2+\varepsilon)n})$, then there exists a monotone Boolean function family in coNP of monotone circuit size $2^{Ω(n / \log n)}$. This implies win-win circuit lower bounds: either $\text{E}^{\text{NP}}$ requires series-parallel circuits of size $ω(n)$ or coNP requires monotone circuits of size $2^{Ω(n / \log n)}$. If MAX-3-SAT cannot be solved in co-nondeterministic time $O(2^{(1 - \varepsilon)n})$, then there exist small families of matrices with high rigidity as well as small families of three-dimensional tensors of high rank.

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arXiv2025

Toward Better Depth Lower Bounds: Strong Composition of XOR and a Random Function

Proving formula depth lower bounds is a fundamental challenge in complexity theory, with the strongest known bound of $(3 - o(1))\log n$ established by Hastad over 25 years ago. The Karchmer-Raz-Wigderson (KRW) conjecture offers a promising approach to advance these bounds and separate P from NC$^{1}$. It suggests that the depth complexity of a function composition $f \diamond g$ approximates the sum of the depth complexities of $f$ and $g$. The Karchmer-Wigderson (KW) relation framework translates formula depth into communication complexity, restating the KRW conjecture as $\mathsf{CC}(\mathsf{KW}_f \diamond \mathsf{KW}_g) \approx \mathsf{CC}(\mathsf{KW}_f) + \mathsf{CC}(\mathsf{KW}_g)$. Prior work has confirmed the conjecture under various relaxations, often replacing one or both KW relations with the universal relation or constraining the communication game through strong composition. In this paper, we examine the strong composition $\mathsf{KW}_{\mathsf{XOR}} \circledast \mathsf{KW}_f$ of the parity function and a random Boolean function $f$. We prove that with probability $1-o(1)$, any protocol solving this composition requires at least $n^{3 - o(1)}$ leaves. This result establishes a depth lower bound of $(3 - o(1))\log n$, matching Hastad's bound, but is applicable to a broader class of inner functions, even when the outer function is simple. Though bounds for the strong composition do not translate directly to formula depth bounds, they usually help to analyze the standard composition (of the corresponding two functions) which is directly related to formula depth.

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arXiv2024

Improved Space Bounds for Subset Sum

More than 40 years ago, Schroeppel and Shamir presented an algorithm that solves the Subset Sum problem for $n$ integers in time $O^*(2^{0.5n})$ and space $O^*(2^{0.25n})$. The time upper bound remains unbeaten, but the space upper bound has been improved to $O^*(2^{0.249999n})$ in a recent breakthrough paper by Nederlof and Węgrzycki (STOC 2021). Their algorithm is a clever combination of a number of previously known techniques with a new reduction and a new algorithm for the Orthogonal Vectors problem. In this paper, we improve the space bound by Nederlof and Węgrzycki to $O^*(2^{0.246n})$ and also simplify their algorithm and its analysis. We achieve this by using an idea, due to Howgrave-Graham and Joux, of using a random prime number to filter the family of subsets. We incorporate it into the algorithm by Schroeppel and Shamir and then use this amalgam inside the representation technique. This allows us to reduce an instance of Subset Sum to a larger number of instances of weighted orthogonal vector.

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