Periodic Reporting for period 4 - TOTAL (Technology transfer between modern algorithmic paradigms)
Berichtszeitraum: 2020-10-01 bis 2021-09-30
The two most recognized algorithmic paradigms of dealing with NP-hard problems in theoretical computer science nowadays are approximation algorithms and fixed parameter tractability (FPT). Despite the fact that both fields are by now developed, they have grown mostly on their own. In our opinion the two fields have critical mass allowing for synergy effects to appear when using techniques from one area to obtain results in the other, which is the main theme of the project.
Based on our experience with parameterized complexity and approximation algorithms we proposed three research directions with potential of solving major long-standing open problems in both areas with the following objectives.
(a) Transfer from Approximation to FPT: use the PCP theorem to prove lower bounds for exact parameterized algorithms under the Exponential Time Hypothesis and take advantage of extended formulations in FPT branching algorithms.
(b) Transfer from FPT to Approximation: utilize parameterized algorithms tools in local-search- based approximation algorithms and explore the potential of proving new inapproximability results based on the Exponential Time Hypothesis.
(c) Rigorous Analysis of Practical Heuristics: unravel the practical effectiveness of randomized local search heuristics by proving their parameterized and approximation properties, when given subexponential or even moderately exponential running time.
Based on our experience with parameterized complexity and approximation algorithms we proposed three research directions with potential of solving major long-standing open problems in both areas with the following objectives.
(a) Transfer from Approximation to FPT: use the PCP theorem to prove lower bounds for exact parameterized algorithms under the Exponential Time Hypothesis and take advantage of extended formulations in FPT branching algorithms.
(b) Transfer from FPT to Approximation: utilize parameterized algorithms tools in local-search- based approximation algorithms and explore the potential of proving new inapproximability results based on the Exponential Time Hypothesis.
(c) Rigorous Analysis of Practical Heuristics: unravel the practical effectiveness of randomized local search heuristics by proving their parameterized and approximation properties, when given subexponential or even moderately exponential running time.
Below we are listing 7 concrete results achieved throughout the course of the project:
We have combined Probabilistically Checkable Proofs (PCP) and Exponential Time Hypothesis (ETH) to obtain new lower bounds for Minimum Fill-In, Interval Completion and a few its variants. To get tight results we needed to rely on an additional conjecture on subexponential approximation of min-bisection in sparse graphs. Interestingly, a follow up of our work was published at SODA'17 by Cao and Sandeep, where they show a new reduction which relies solely on ETH and PCP for the Minimum Fill-In problem.
In a paper published at ICALP'17 we have performed a systematic study of the landscape of fine-grained complexity under the assumption that the (min,+)-convolution problem does not admit trully subquadratic algorithms. Conditional lower bound were obtained for knapsack and other problems.
In an article published at ESA'17 (and awarded the Best Paper award) we have obtained faster algorithms searching through the space of all small local improvements for the travelling salesman problem (TSP). The popular k-opt heuristic for TSP finds a local optimum by starting from an arbitrary tour H and then improving it by a sequence of k-moves. Until 2016, the only known algorithm to find an improving k-move for a given tour was the naive solution in time O(n^k). At ICALP'16 de Berg, Buchin, Jansen and Woeginger showed an O(n^{floor(2/3k)+1})-time algorithm. We have shown an algorithm which runs in O(n^{(1/4 + epsilon_k)k}) time, where lim_{k -> infinity} epsilon_k = 0.
Finally, the most important result of the project so far is our work published at FOCS'17, where we have considered questions that arise from the intersection between the areas of approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms, that is whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In this paper, we have shown that both Clique and DomSet admit no non-trivial FPT-approximation algorithm. Our results hold under the Gap Exponential Time Hypothesis (GapETH) which states that no 2^o(n) -time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even (1 - ε)-satisfiable for some constant ε > 0.
A Gap-ETH-Tight Approximation Scheme for Euclidean TSP (accepted to FOCS 2022): a new approximation algorithm which runs in fpt time depending on epsilon for the classic problem of Euclidean TSP. This paper improves upon an almost 20 years old algorithm of Rao and Smith (STOC 1998) by giving a better dependence on epsilon, additionally giving a very close conditional lower bound based on the Gap-ETH problem, which directly relates to Objective 3 of the project (Objective 3. Prove approximation lower bounds under the Exponential Time Hypothesis). The result builds quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a new idea that we call sparsity-sensitive patching
Improving Schroeppel and Shamir’s algorithm for subset sum via orthogonal vectors (STOC 2021): this work presents an improvement over a celebrated result of Schroeppel and Shamir (FOCS 1979), by a new algorithm for the subset sum problem with runtime O*(2^0.5n) and space usage O*(2^(0.25-eps)n) ultimately breaking the 0.25n barrier in the exponent of the space bound. The algorithm is obtained by a space efficient reduction to the orthogonal vectors problem.
Degeneracy of Pt-free and C≥t-free graphs with no large complete bipartite subgraphs (accepted to JCTB): The starting point of this line of research is whether in Pt-free graphs G, the value of χ(G) can be bounded from above by a polynomial function of ω(G). We consider a relaxation of this problem, where we compare the chromatic number with the size of a largest balanced biclique contained in the graph as a (not necessarily induced) subgraph. We show that for every t there exists a constant c such that for and every C≥t-free graph which does not contain K_ℓ,ℓ as a subgraph, it holds that χ(G)≤ℓ^c.
We have combined Probabilistically Checkable Proofs (PCP) and Exponential Time Hypothesis (ETH) to obtain new lower bounds for Minimum Fill-In, Interval Completion and a few its variants. To get tight results we needed to rely on an additional conjecture on subexponential approximation of min-bisection in sparse graphs. Interestingly, a follow up of our work was published at SODA'17 by Cao and Sandeep, where they show a new reduction which relies solely on ETH and PCP for the Minimum Fill-In problem.
In a paper published at ICALP'17 we have performed a systematic study of the landscape of fine-grained complexity under the assumption that the (min,+)-convolution problem does not admit trully subquadratic algorithms. Conditional lower bound were obtained for knapsack and other problems.
In an article published at ESA'17 (and awarded the Best Paper award) we have obtained faster algorithms searching through the space of all small local improvements for the travelling salesman problem (TSP). The popular k-opt heuristic for TSP finds a local optimum by starting from an arbitrary tour H and then improving it by a sequence of k-moves. Until 2016, the only known algorithm to find an improving k-move for a given tour was the naive solution in time O(n^k). At ICALP'16 de Berg, Buchin, Jansen and Woeginger showed an O(n^{floor(2/3k)+1})-time algorithm. We have shown an algorithm which runs in O(n^{(1/4 + epsilon_k)k}) time, where lim_{k -> infinity} epsilon_k = 0.
Finally, the most important result of the project so far is our work published at FOCS'17, where we have considered questions that arise from the intersection between the areas of approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms, that is whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In this paper, we have shown that both Clique and DomSet admit no non-trivial FPT-approximation algorithm. Our results hold under the Gap Exponential Time Hypothesis (GapETH) which states that no 2^o(n) -time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even (1 - ε)-satisfiable for some constant ε > 0.
A Gap-ETH-Tight Approximation Scheme for Euclidean TSP (accepted to FOCS 2022): a new approximation algorithm which runs in fpt time depending on epsilon for the classic problem of Euclidean TSP. This paper improves upon an almost 20 years old algorithm of Rao and Smith (STOC 1998) by giving a better dependence on epsilon, additionally giving a very close conditional lower bound based on the Gap-ETH problem, which directly relates to Objective 3 of the project (Objective 3. Prove approximation lower bounds under the Exponential Time Hypothesis). The result builds quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a new idea that we call sparsity-sensitive patching
Improving Schroeppel and Shamir’s algorithm for subset sum via orthogonal vectors (STOC 2021): this work presents an improvement over a celebrated result of Schroeppel and Shamir (FOCS 1979), by a new algorithm for the subset sum problem with runtime O*(2^0.5n) and space usage O*(2^(0.25-eps)n) ultimately breaking the 0.25n barrier in the exponent of the space bound. The algorithm is obtained by a space efficient reduction to the orthogonal vectors problem.
Degeneracy of Pt-free and C≥t-free graphs with no large complete bipartite subgraphs (accepted to JCTB): The starting point of this line of research is whether in Pt-free graphs G, the value of χ(G) can be bounded from above by a polynomial function of ω(G). We consider a relaxation of this problem, where we compare the chromatic number with the size of a largest balanced biclique contained in the graph as a (not necessarily induced) subgraph. We show that for every t there exists a constant c such that for and every C≥t-free graph which does not contain K_ℓ,ℓ as a subgraph, it holds that χ(G)≤ℓ^c.
The results described above go beyond state of the art in the areas of approximation algorithms and parameterized complexity.