arXiv subjects

Ivan Mihajlin

Publications and source records attributed to Ivan Mihajlin.

17 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

If Edge Coloring is Hard under SETH, then SETH is False

The Edge Coloring problem is notoriously hard: it is still unknown whether it can be solved in time $2^{o(n^2)}$ (let alone $2^{O(n)}$), where $n$ is the number of nodes of the input graph. Can one explain the lack of such upper bounds by deriving a lower bound $2^{\Omega(n^2)}$ from a lower bound for SAT, $3$-SUM, or APSP? In this note, we provide a negative answer for this question: if there is a reduction showing that Edge Coloring cannot be solved faster than in $\alpha^{n^2}$ (where $\alpha>1$ is an explicit constant) under a hypothesis that known algorithms for one of the problems mentioned above are optimal, then the corresponding hypothesis is false.

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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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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

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 $\omega(n)$. 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 $2^{\Omega(n / \log n)}$. This implies win-win circuit lower bounds: either $\text{E}^{\text{NP}}$ requires series-parallel circuits of size $\omega(n)$ or coNP requires monotone circuits of size $2^{\Omega(n / \log n)}$. 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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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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arXiv2023

Computations with polynomial evaluation oracle: ruling out superlinear SETH-based lower bounds

The field of fine-grained complexity aims at proving conditional lower bounds on the time complexity of computational problems. One of the most popular assumptions, Strong Exponential Time Hypothesis (SETH), implies that SAT cannot be solved in $2^{(1-ε)n}$ time. In recent years, it has been proved that known algorithms for many problems are optimal under SETH. Despite the wide applicability of SETH, for many problems, there are no known SETH-based lower bounds, so the quest for new reductions continues. Two barriers for proving SETH-based lower bounds are known. Carmosino et al. (ITCS 2016) introduced the Nondeterministic Strong Exponential Time Hypothesis (NSETH) stating that TAUT cannot be solved in time $2^{(1-ε)n}$ even if one allows nondeterminism. They used this hypothesis to show that some natural fine-grained reductions would be difficult to obtain: proving that, say, 3-SUM requires time $n^{1.5+ε}$ under SETH, breaks NSETH and this, in turn, implies strong circuit lower bounds. Recently, Belova et al. (SODA 2023) introduced the so-called polynomial formulations to show that for many NP-hard problems, proving any explicit exponential lower bound under SETH also implies strong circuit lower bounds. We prove that for a range of problems from P, including $k$-SUM and triangle detection, proving superlinear lower bounds under SETH is challenging as it implies new circuit lower bounds. To this end, we show that these problems can be solved in nearly linear time with oracle calls to evaluating a polynomial of constant degree. Then, we introduce a strengthening of SETH stating that solving SAT in time $2^{(1-\varepsilon)n}$ is difficult even if one has constant degree polynomial evaluation oracle calls. This hypothesis is stronger and less believable than SETH, but refuting it is still challenging: we show that this implies circuit lower bounds.

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arXiv2022

Polynomial formulations as a barrier for reduction-based hardness proofs

The Strong Exponential Time Hypothesis (SETH) asserts that for every $\varepsilon>0$ there exists $k$ such that $k$-SAT requires time $(2-\varepsilon)^n$. The field of fine-grained complexity has leveraged SETH to prove quite tight conditional lower bounds for dozens of problems in various domains and complexity classes, including Edit Distance, Graph Diameter, Hitting Set, Independent Set, and Orthogonal Vectors. Yet, it has been repeatedly asked in the literature whether SETH-hardness results can be proven for other fundamental problems such as Hamiltonian Path, Independent Set, Chromatic Number, MAX-$k$-SAT, and Set Cover. In this paper, we show that fine-grained reductions implying even $λ^n$-hardness of these problems from SETH for any $λ>1$, would imply new circuit lower bounds: super-linear lower bounds for Boolean series-parallel circuits or polynomial lower bounds for arithmetic circuits (each of which is a four-decade open question). We also extend this barrier result to the class of parameterized problems. Namely, for every $λ>1$ we conditionally rule out fine-grained reductions implying SETH-based lower bounds of $λ^k$ for a number of problems parameterized by the solution size $k$. Our main technical tool is a new concept called polynomial formulations. In particular, we show that many problems can be represented by relatively succinct low-degree polynomials, and that any problem with such a representation cannot be proven SETH-hard (without proving new circuit lower bounds).

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arXiv2022

CNF Encodings of Parity

The minimum number of clauses in a CNF representation of the parity function $x_1 \oplus x_2 \oplus \dotsb \oplus x_n$ is $2^{n-1}$. One can obtain a more compact CNF encoding by using non-deterministic variables (also known as guess or auxiliary variables). In this paper, we prove the following lower bounds, that almost match known upper bounds, on the number $m$ of clauses and the maximum width $k$ of clauses: 1) if there are at most $s$ auxiliary variables, then $m \ge Ω\left(2^{n/(s+1)}/n\right)$ and $k \ge n/(s+1)$; 2) the minimum number of clauses is at least $3n$. We derive the first two bounds from the Satisfiability Coding Lemma due to Paturi, Pudlak, and Zane.

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arXiv2020

Collapsing Superstring Conjecture

In the Shortest Common Superstring (SCS) problem, one is given a collection of strings, and needs to find a shortest string containing each of them as a substring. SCS admits $2\frac{11}{23}$-approximation in polynomial time (Mucha, SODA'13). While this algorithm and its analysis are technically involved, the 30 years old Greedy Conjecture claims that the trivial and efficient Greedy Algorithm gives a 2-approximation for SCS. We develop a graph-theoretic framework for studying approximation algorithms for SCS. The framework is reminiscent of the classical 2-approximation for Traveling Salesman: take two copies of an optimal solution, apply a trivial edge-collapsing procedure, and get an approximate solution. In this framework, we observe two surprising properties of SCS solutions, and we conjecture that they hold for all input instances. The first conjecture, that we call Collapsing Superstring conjecture, claims that there is an elementary way to transform any solution repeated twice into the same graph $G$. This conjecture would give an elementary 2-approximate algorithm for SCS. The second conjecture claims that not only the resulting graph $G$ is the same for all solutions, but that $G$ can be computed by an elementary greedy procedure called Greedy Hierarchical Algorithm. While the second conjecture clearly implies the first one, perhaps surprisingly we prove their equivalence. We support these equivalent conjectures by giving a proof for the special case where all input strings have length at most 3. We prove that the standard Greedy Conjecture implies Greedy Hierarchical Conjecture, while the latter is sufficient for an efficient greedy 2-approximate approximation of SCS. Except for its (conjectured) good approximation ratio, the Greedy Hierarchical Algorithm provably finds a 3.5-approximation.

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arXiv2020

Computation of Hadwiger Number and Related Contraction Problems: Tight Lower Bounds

We prove that the Hadwiger number of an $n$-vertex graph $G$ (the maximum size of a clique minor in $G$) cannot be computed in time $n^{o(n)}$, unless the Exponential Time Hypothesis (ETH) fails. This resolves a well-known open question in the area of exact exponential algorithms. The technique developed for resolving the Hadwiger number problem has a wider applicability. We use it to rule out the existence of $n^{o(n)}$-time algorithms (up to ETH) for a large class of computational problems concerning edge contractions in graphs.

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arXiv2016

Tight Lower Bounds on Graph Embedding Problems

We prove that unless the Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $|V(H)|^{o(|V(G)|)}$. We also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of $|V(H)|^{o(|V(H)|)}$-time algorithm deciding if graph $G$ is a subgraph of $H$. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems. Moreover, as a consequence of our reductions conditional lower bounds follow for other related problems such as Locally Injective Homomorphism, Graph Minors, Topological Graph Minors, Minimum Distortion Embedding and Quadratic Assignment Problem.

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arXiv2015

Tight Bounds for Subgraph Isomorphism and Graph Homomorphism

We prove that unless Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $|V(H)|^{o(|V(G)|)}$. Combined with the reduction of Cygan, Pachocki, and Socała, our result rules out (subject to ETH) a possibility of $|V(G)|^{o(|V(G)|)}$-time algorithm deciding if graph $H$ is a subgraph of $G$. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems.

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arXiv2015

Lower Bounds for the Graph Homomorphism Problem

The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the $2$-CSP problem. In this paper, we prove several lower bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound $2^{Ω\left( \frac{n \log h}{\log \log h}\right)}$. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound $2^{{\mathcal O}(n\log{h})}$ is almost asymptotically tight. We also investigate what properties of graphs $G$ and $H$ make it difficult to solve HOM$(G,H)$. An easy observation is that an ${\mathcal O}(h^n)$ upper bound can be improved to ${\mathcal O}(h^{\operatorname{vc}(G)})$ where $\operatorname{vc}(G)$ is the minimum size of a vertex cover of $G$. The second lower bound $h^{Ω(\operatorname{vc}(G))}$ shows that the upper bound is asymptotically tight. As to the properties of the "right-hand side" graph $H$, it is known that HOM$(G,H)$ can be solved in time $(f(Δ(H)))^n$ and $(f(\operatorname{tw}(H)))^n$ where $Δ(H)$ is the maximum degree of $H$ and $\operatorname{tw}(H)$ is the treewidth of $H$. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number $χ(H)$ does not exceed $\operatorname{tw}(H)$ and $Δ(H)+1$, it is natural to ask whether similar upper bounds with respect to $χ(H)$ can be obtained. We provide a negative answer to this question by establishing a lower bound $(f(χ(H)))^n$ for any function $f$. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.

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arXiv2014

Families with infants: a general approach to solve hard partition problems

We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NP-hard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of so-called families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NP-hard problems as well as to simplify the proofs of several known results. For the chromatic number problem we present an algorithm with $O^*((2-\varepsilon(d))^n)$ time and exponential space for graphs of average degree $d$. This improves the algorithm by Björklund et al. [Theory Comput. Syst. 2010] that works for graphs of bounded maximum (as opposed to average) degree and closes an open problem stated by Cygan and Pilipczuk [ICALP 2013]. For the traveling salesman problem we give an algorithm working in $O^*((2-\varepsilon(d))^n)$ time and polynomial space for graphs of average degree $d$. The previously known results of this kind is a polyspace algorithm by Björklund et al. [ICALP 2008] for graphs of bounded maximum degree and an exponential space algorithm for bounded average degree by Cygan and Pilipczuk [ICALP 2013]. For counting perfect matching in graphs of average degree~$d$ we present an algorithm with running time $O^*((2-\varepsilon(d))^{n/2})$ and polynomial space. Recent algorithms of this kind due to Cygan, Pilipczuk [ICALP 2013] and Izumi, Wadayama [FOCS 2012] (for bipartite graphs only) use exponential space.

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arXiv2014

Families with infants: speeding up algorithms for NP-hard problems using FFT

Assume that a group of people is going to an excursion and our task is to seat them into buses with several constraints each saying that a pair of people does not want to see each other in the same bus. This is a well-known coloring problem and it can be solved in $O^*(2^n)$ time by the inclusion-exclusion principle as shown by Björklund, Husfeldt, and Koivisto in 2009.Another approach to solve this problem in $O^*(2^n)$ time is to use the fast Fourier transform. A graph is $k$-colorable if and only if the $k$-th power of a polynomial containing a monomial $\prod_{i=1}^n x_i^{[i \in I]}$ for each independent set $I \subseteq [n]$ of the graph, contains the monomial $x_1x_2... x_n$. Assume now that we have additional constraints: the group of people contains several infants and these infants should be accompanied by their relatives in a bus. We show that if the number of infants is linear then the problem can be solved in $O^*((2-\varepsilon)^n)$ time. We use this approach to improve known bounds for several NP-hard problems (the traveling salesman problem, the graph coloring problem, the problem of counting perfect matchings) on graphs of bounded average degree, as well as to simplify the proofs of several known results.

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