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版本更新 · v2arXiv:2609.01365

The Greedy Superstring Conjecture Is False

Nikolai Chukhin, Rishikesh Gajjala, Juha Harviainen, Alexander S. Kulikov, Ivan Mihajlin, Hiroki Shibata, Alexander V. Smal

原文摘要

In the Shortest Common Superstring problem (SCS), one is given a finite set of strings and is asked to find a shortest string containing every input string as a substring. Since the problem is NP-hard, various approximation algorithms have been proposed for it. The currently best known approximation ratio is $2.466$ due to Englert, Matsakis, and Veselý; a better approximation ratio $2$ has been announced by OpenAI just recently. In 1988, Tarhio and Ukkonen introduced a simple greedy algorithm that iteratively takes two strings with the longest overlap and merges them together. They showed that the greedy algorithm may produce a superstring that is about two times longer than the optimal one and conjectured that this bound is tight. This has become known as the Greedy Superstring Conjecture and has remained open for almost four decades. The best known upper bound on the approximation ratio of the greedy algorithm is $3.396$ due to Englert, Matsakis, and Veselý. In this paper, we present a series of results tightening the bounds of the greedy algorithm, with the main contributions being as follows: 1. We refute the Greedy Conjecture by constructing a family of instances for which a lower bound on the approximation ratio tends to $2.4$. 2. We prove that the greedy algorithm is $3$-approximate. 3. For input strings of fixed length $k$, we prove that the approximation guarantee is exactly $9/5$ for $k=3$ and at least $2$ for all $k \ge 6$.

cs.DS

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2609.03628

Random Garbage Creates Exponential Quantum Query Separations

Khaled Elbassioni, Rishikesh Gajjala, Saurabh Ray

原文摘要

We prove an exponential separation between standard (XOR) quantum queries and forward-erasing queries for a Boolean property of injective functions, answering a question of Aaronson~\cite{Aaronson2021Open}. We also obtain an exponential separation for a Boolean promise problem on permutations, where forward in-place queries are unitary but inverse queries are not supplied. Our construction follows Aaronson's suggestion of appending fixed random garbage to a Simon function. Let $X=\{0,1\}^n$ and $N=2^n$. We use the injection $f_{h,r}(x)=(h(x),x,r_x)$, where $h:X\to X$ is promised to be either a permutation or a Simon function, and $r:X\to X$ is an arbitrary tag table that is reused on every query. The resulting problem is solvable with at most $n+2$ standard queries, but requires $Θ(\sqrt N)$ forward-erasing queries, matching classical collision sampling. Embedding these instances into permutations, with the copy of $x$ replaced by a single bit that keeps the embedding injective, gives a promise problem on permutations of $L=4N^2$ elements that requires $O(\log L)$ standard queries versus $Θ(L^{1/4})$ forward in-place queries. The lower bounds use the recording method. We replace each query by a nearby contraction that records cells of a purified tag table. After $T$ queries, tracing out the purified tag table yields a sum of positive-semidefinite contributions, each depending on $h$ at no more than $T$ addresses. A query-by-query hybrid bounds the trace-norm difference between the actual and auxiliary final operators by $O(T/\sqrt N)$. Random permutations and random Simon functions have the same restriction distribution on such a set unless it contains a hidden Simon pair, an event of probability $O(T^2/N)$. Consequently, the actual final states in the two cases remain close unless $T=Ω(\sqrt N)$, giving the required lower bounds.

cs.CC · quant-ph

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04444

Binary Huffman Coding with Unequal Letter Costs Is NP-Complete

L. Sunil Chandran, Rishikesh Gajjala

原文摘要

Binary prefix coding assigns prefix-free binary words to symbols to minimize their frequency-weighted cost. Huffman solved the equal-letter-cost case in polynomial time in 1952 [Proc. IRE'52]. For unequal letter costs, Karp gave an integer-programming formulation in 1961 [IRE Trans. Inf. Theory'61]. Exact polynomial-time algorithms for fixed integer costs were given by Golin and Rote [IEEE Trans. Inf. Theory'98] and Bradford, Golin, Larmore, and Rytter [J. Algorithms'02]. For input-dependent costs, Golin, Mathieu, and Young gave a polynomial-time approximation scheme [SICOMP'12]. The exact complexity when letter costs are part of the input had remained unresolved. We resolve this $65$-year-old problem by proving that deciding whether there exists a binary prefix-free code of frequency-weighted cost at most a given threshold is NP-complete, even when the letter costs are consecutive positive integers $(M,M+1)$ supplied as part of the input. All frequencies, letter costs, and thresholds are encoded in binary. Consequently, finding an optimal code is NP-hard. Our proof uses two polynomial-time reductions, starting from Planar Independent Set. The intermediate problem prescribes the number of leaves at each depth of a full ordered binary tree and minimizes a signed linear score on their sorted right-edge counts. The first reduction represents an independent-set verification program by a tree growing level by level. A fixed-profile identity converts narrow coordinate restrictions into linear penalties that enforce the computation. The second reduction adds large dyadic terms to the symbol frequencies to force prescribed depths. These terms contribute a constant to the weighted number of ones, leaving smaller signed perturbations to reproduce the tree score.

cs.DS · cs.CC · cs.CG · cs.IT

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.03190

Asymptotically Optimal Weighted Edge Colorings via Local Bin Packing

Rishikesh Gajjala, Arindam Khan

原文摘要

We study weighted edge coloring, where each edge of a graph $G$ has a weight in $(0,1]$, and colors are assigned so that the incident weight of each color at every vertex is at most one. The objective is to minimize the number of colors, denoted by $χ'_w(G)$. This generalizes classical edge coloring (all weights are one) and bin packing ($G$ is a star). Two natural lower bounds are the maximum weighted degree $Δ_w(G)$ and the maximum local bin-packing number $\mathrm{bp}(G)$: the maximum, over vertices, of the minimum number of unit bins needed to pack the incident weights. We have $\lceilΔ_w(G)\rceil\le\mathrm{bp}(G)\leχ'_w(G)$. How closely these local lower bounds approximate the global optimum is a longstanding open problem. Our main result establishes an asymptotic weighted analogue of Vizing's theorem: every simple graph admits a coloring using at most $(1+o(1))\mathrm{bp}(G)$ colors. Thus local bin packings can be made globally consistent with asymptotically vanishing relative loss, even in nonbipartite graphs. Our local-to-global rounding framework coordinates feasible local packing distributions into a partial coloring, leaving only $o(\mathrm{bp}(G))$ uncolored weight at every vertex. For bipartite multigraphs, we prove $χ'_w(G)\le(2+o(1))Δ_w(G)$, establishing the Chung--Ross conjecture [SICOMP'91] asymptotically. For loopless multigraphs, we obtain the exact bound $χ'_w(G)\le3\lceilΔ_w(G)\rceil$. Both leading coefficients are optimal, giving weighted analogues of the classical Kőnig--Shannon bounds. Further upper and lower bounds show that the classical factor-$3/2$ separation does not persist with the local bin-packing benchmark. We also disprove the exact weighted-degree conjecture of Feige and Singh [ESA'08]. Our upper bounds have polynomial-time algorithms, and their existence statements are formalized in Lean 4.

math.CO · cs.DM · cs.DS

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arXiv ↗PDF ↗